TPTP Problem File: SLH0216^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Eval_FO/0005_Ailamazyan/prob_01959_072999__15768006_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1656 ( 710 unt; 370 typ; 0 def)
% Number of atoms : 2937 (1657 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 10768 ( 182 ~; 21 |; 132 &;9387 @)
% ( 0 <=>;1046 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 47 ( 46 usr)
% Number of type conns : 2619 (2619 >; 0 *; 0 +; 0 <<)
% Number of symbols : 327 ( 324 usr; 24 con; 0-5 aty)
% Number of variables : 4044 ( 374 ^;3595 !; 75 ?;4044 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:06:45.899
%------------------------------------------------------------------------------
% 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_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
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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,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
list_P3773045032060140154_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
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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,
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thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Option__Ooption_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_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,
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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__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $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__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (324)
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,
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thf(sy_c_Ailamazyan_Oext__tuple_001tf__a,type,
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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,
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thf(sy_c_Ailamazyan_Onall__tuples_001tf__a,type,
nall_tuples_a: set_a > nat > set_li6526943997496501093_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__vals_001t__Nat__Onat,type,
proj_vals_nat: set_nat_nat > list_nat > set_list_nat ).
thf(sy_c_Ailamazyan_Oproj__vals_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
proj_v7785048252284376108_a_nat: set_na5864254054690418060_a_nat > list_nat > set_li7803226448621193899_a_nat ).
thf(sy_c_Ailamazyan_Oproj__vals_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
proj_v2875961069841262333_a_nat: set_na964695563613990089_a_nat > list_nat > set_li5408732054060274282_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_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 ).
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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_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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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__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_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comple1247738100258233164_a_nat: set_se4904748513628223167_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
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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,
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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,
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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 ).
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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 ).
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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 ).
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fun_up4481430433955130898_a_nat: ( list_P5056861408695629236_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_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
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fun_up8866717756282968183_a_nat: ( option_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 ).
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vimage8470460917905013169at_nat: ( produc5454855657571582244_a_nat > nat ) > set_nat > set_Pr1021281732701644634_a_nat ).
thf(sy_c_Set_Ovimage_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,
vimage2715053500650648030_a_nat: ( produc5454855657571582244_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Pr1021281732701644634_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_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_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_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__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member486768635318144366_a_nat: option_Sum_sum_a_nat > set_op253850770698000421_a_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_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____,type,
sigma: nat > sum_sum_a_nat ).
thf(sy_v__092_060sigma_062____,type,
sigma2: 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 (1276)
thf(fact_0_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_1_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_2_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_3_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_4_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_5_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_6_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_7_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_8_List_Omap_Ocomp,axiom,
! [F: sum_sum_a_nat > nat,G: nat > sum_sum_a_nat] :
( ( comp_l5314616558136524580st_nat @ ( map_Su5227373005390213643at_nat @ F ) @ ( map_na823391071729141993_a_nat @ G ) )
= ( map_nat_nat @ ( comp_S7873419708350676862at_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_9_List_Omap_Ocomp,axiom,
! [F: option_Sum_sum_a_nat > nat,G: nat > option_Sum_sum_a_nat] :
( ( comp_l3532711467655825258st_nat @ ( map_op589859327486859985at_nat @ F ) @ ( map_na4778969939102885679_a_nat @ G ) )
= ( map_nat_nat @ ( comp_o8557203714689770820at_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_10_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_11_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_12_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_13_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_14_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_15_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_16_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_17_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_18_list_Omap__comp,axiom,
! [G: sum_sum_a_nat > nat,F: nat > sum_sum_a_nat,V: list_nat] :
( ( map_Su5227373005390213643at_nat @ G @ ( map_na823391071729141993_a_nat @ F @ V ) )
= ( map_nat_nat @ ( comp_S7873419708350676862at_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_19_list_Omap__comp,axiom,
! [G: option_Sum_sum_a_nat > nat,F: nat > option_Sum_sum_a_nat,V: list_nat] :
( ( map_op589859327486859985at_nat @ G @ ( map_na4778969939102885679_a_nat @ F @ V ) )
= ( map_nat_nat @ ( comp_o8557203714689770820at_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_List_Omap_Ocompositionality,axiom,
! [F: sum_sum_a_nat > nat,G: nat > sum_sum_a_nat,List: list_nat] :
( ( map_Su5227373005390213643at_nat @ F @ ( map_na823391071729141993_a_nat @ G @ List ) )
= ( map_nat_nat @ ( comp_S7873419708350676862at_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_29_List_Omap_Ocompositionality,axiom,
! [F: option_Sum_sum_a_nat > nat,G: nat > option_Sum_sum_a_nat,List: list_nat] :
( ( map_op589859327486859985at_nat @ F @ ( map_na4778969939102885679_a_nat @ G @ List ) )
= ( map_nat_nat @ ( comp_o8557203714689770820at_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_map__map,axiom,
! [F: sum_sum_a_nat > nat,G: nat > sum_sum_a_nat,Xs: list_nat] :
( ( map_Su5227373005390213643at_nat @ F @ ( map_na823391071729141993_a_nat @ G @ Xs ) )
= ( map_nat_nat @ ( comp_S7873419708350676862at_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_39_map__map,axiom,
! [F: option_Sum_sum_a_nat > nat,G: nat > option_Sum_sum_a_nat,Xs: list_nat] :
( ( map_op589859327486859985at_nat @ F @ ( map_na4778969939102885679_a_nat @ G @ Xs ) )
= ( map_nat_nat @ ( comp_o8557203714689770820at_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_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_48_map__comp__map,axiom,
! [F: sum_sum_a_nat > nat,G: nat > sum_sum_a_nat] :
( ( comp_l5314616558136524580st_nat @ ( map_Su5227373005390213643at_nat @ F ) @ ( map_na823391071729141993_a_nat @ G ) )
= ( map_nat_nat @ ( comp_S7873419708350676862at_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_49_map__comp__map,axiom,
! [F: option_Sum_sum_a_nat > nat,G: nat > option_Sum_sum_a_nat] :
( ( comp_l3532711467655825258st_nat @ ( map_op589859327486859985at_nat @ F ) @ ( map_na4778969939102885679_a_nat @ G ) )
= ( map_nat_nat @ ( comp_o8557203714689770820at_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_50__092_060sigma_062_H__def,axiom,
( sigma
= ( ^ [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_51_comp__apply,axiom,
( comp_l7562833077401626166_a_nat
= ( ^ [F2: list_nat > list_Sum_sum_a_nat,G2: list_P5056861408695629236_a_nat > list_nat,X: list_P5056861408695629236_a_nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_52_comp__apply,axiom,
( comp_l4895937012117054562st_nat
= ( ^ [F2: list_nat > list_Sum_sum_a_nat,G2: list_nat > list_nat,X: list_nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_53_comp__apply,axiom,
( comp_n848383228264675746at_nat
= ( ^ [F2: nat > option_Sum_sum_a_nat,G2: nat > nat,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_54_comp__apply,axiom,
( comp_n6276394432912108976_a_nat
= ( ^ [F2: nat > sum_sum_a_nat,G2: produc5454855657571582244_a_nat > nat,X: produc5454855657571582244_a_nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_55_comp__apply,axiom,
( comp_n1522318729830440540at_nat
= ( ^ [F2: nat > sum_sum_a_nat,G2: nat > nat,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_56_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_57_fun_Omap__comp,axiom,
! [G: option_Sum_sum_a_nat > sum_sum_a_nat,F: nat > option_Sum_sum_a_nat,V: nat > nat] :
( ( comp_o2721059924169077421at_nat @ G @ ( comp_n848383228264675746at_nat @ F @ V ) )
= ( comp_n1522318729830440540at_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_58_fun_Omap__comp,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat,V: nat > option_Sum_sum_a_nat] :
( ( comp_S2395038772458240499at_nat @ G @ ( comp_o2721059924169077421at_nat @ F @ V ) )
= ( comp_o2721059924169077421at_nat @ ( comp_S2748962947160188642_a_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_59_fun_Omap__comp,axiom,
! [G: option_Sum_sum_a_nat > sum_sum_a_nat,F: option_Sum_sum_a_nat > option_Sum_sum_a_nat,V: nat > option_Sum_sum_a_nat] :
( ( comp_o2721059924169077421at_nat @ G @ ( comp_o5487264423576322163at_nat @ F @ V ) )
= ( comp_o2721059924169077421at_nat @ ( comp_o8686311273808465576_a_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_60_fun_Omap__comp,axiom,
! [G: nat > sum_sum_a_nat,F: nat > nat,V: nat > nat] :
( ( comp_n1522318729830440540at_nat @ G @ ( comp_nat_nat_nat @ F @ V ) )
= ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_61_fun_Omap__comp,axiom,
! [G: nat > option_Sum_sum_a_nat,F: nat > nat,V: nat > nat] :
( ( comp_n848383228264675746at_nat @ G @ ( comp_nat_nat_nat @ F @ V ) )
= ( comp_n848383228264675746at_nat @ ( comp_n848383228264675746at_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_62_fun_Omap__comp,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: nat > sum_sum_a_nat,V: nat > nat] :
( ( comp_S2395038772458240499at_nat @ G @ ( comp_n1522318729830440540at_nat @ F @ V ) )
= ( comp_n1522318729830440540at_nat @ ( comp_S2395038772458240499at_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_63_fun_Omap__comp,axiom,
! [G: sum_sum_a_nat > option_Sum_sum_a_nat,F: nat > sum_sum_a_nat,V: nat > nat] :
( ( comp_S4056802848252454585at_nat @ G @ ( comp_n1522318729830440540at_nat @ F @ V ) )
= ( comp_n848383228264675746at_nat @ ( comp_S4056802848252454585at_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_64_fun_Omap__comp,axiom,
! [G: nat > sum_sum_a_nat,F: option_Sum_sum_a_nat > nat,V: nat > option_Sum_sum_a_nat] :
( ( comp_n1522318729830440540at_nat @ G @ ( comp_o8557203714689770820at_nat @ F @ V ) )
= ( comp_o2721059924169077421at_nat @ ( comp_n3164697776794079673_a_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_65_fun_Omap__comp,axiom,
! [G: option_Sum_sum_a_nat > option_Sum_sum_a_nat,F: nat > option_Sum_sum_a_nat,V: nat > nat] :
( ( comp_o5487264423576322163at_nat @ G @ ( comp_n848383228264675746at_nat @ F @ V ) )
= ( comp_n848383228264675746at_nat @ ( comp_o5487264423576322163at_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_66_fun_Omap__comp,axiom,
! [G: list_nat > list_Sum_sum_a_nat,F: list_nat > list_nat,V: list_nat > list_nat] :
( ( comp_l4895937012117054562st_nat @ G @ ( comp_l7223822213492037765st_nat @ F @ V ) )
= ( comp_l4895937012117054562st_nat @ ( comp_l4895937012117054562st_nat @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_67_comp__def,axiom,
( comp_l7562833077401626166_a_nat
= ( ^ [F2: list_nat > list_Sum_sum_a_nat,G2: list_P5056861408695629236_a_nat > list_nat,X: list_P5056861408695629236_a_nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_68_comp__def,axiom,
( comp_l4895937012117054562st_nat
= ( ^ [F2: list_nat > list_Sum_sum_a_nat,G2: list_nat > list_nat,X: list_nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_69_comp__def,axiom,
( comp_n848383228264675746at_nat
= ( ^ [F2: nat > option_Sum_sum_a_nat,G2: nat > nat,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_70_comp__def,axiom,
( comp_n6276394432912108976_a_nat
= ( ^ [F2: nat > sum_sum_a_nat,G2: produc5454855657571582244_a_nat > nat,X: produc5454855657571582244_a_nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_71_comp__def,axiom,
( comp_n1522318729830440540at_nat
= ( ^ [F2: nat > sum_sum_a_nat,G2: nat > nat,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_72_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_73_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_74_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_75_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_76_comp__assoc,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,H: nat > nat] :
( ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_77_comp__assoc,axiom,
! [F: nat > option_Sum_sum_a_nat,G: nat > nat,H: nat > nat] :
( ( comp_n848383228264675746at_nat @ ( comp_n848383228264675746at_nat @ F @ G ) @ H )
= ( comp_n848383228264675746at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_78_comp__assoc,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,H: nat > nat] :
( ( comp_n1522318729830440540at_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) @ H )
= ( comp_S2395038772458240499at_nat @ F @ ( comp_n1522318729830440540at_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_79_comp__assoc,axiom,
! [F: nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > nat,H: nat > option_Sum_sum_a_nat] :
( ( comp_o2721059924169077421at_nat @ ( comp_n3164697776794079673_a_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ F @ ( comp_o8557203714689770820at_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_80_comp__assoc,axiom,
! [F: sum_sum_a_nat > option_Sum_sum_a_nat,G: nat > sum_sum_a_nat,H: nat > nat] :
( ( comp_n848383228264675746at_nat @ ( comp_S4056802848252454585at_nat @ F @ G ) @ H )
= ( comp_S4056802848252454585at_nat @ F @ ( comp_n1522318729830440540at_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_81_comp__assoc,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,H: nat > produc5454855657571582244_a_nat] :
( ( comp_P2433158451856970632at_nat @ ( comp_n6276394432912108976_a_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ F @ ( comp_P2340930289473004329at_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_82_comp__assoc,axiom,
! [F: list_nat > list_Sum_sum_a_nat,G: list_nat > list_nat,H: list_nat > list_nat] :
( ( comp_l4895937012117054562st_nat @ ( comp_l4895937012117054562st_nat @ F @ G ) @ H )
= ( comp_l4895937012117054562st_nat @ F @ ( comp_l7223822213492037765st_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_83_comp__eq__dest,axiom,
! [A: option_Sum_sum_a_nat > sum_sum_a_nat,B: nat > option_Sum_sum_a_nat,C: option_Sum_sum_a_nat > sum_sum_a_nat,D: nat > option_Sum_sum_a_nat,V: nat] :
( ( ( comp_o2721059924169077421at_nat @ A @ B )
= ( comp_o2721059924169077421at_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_84_comp__eq__dest,axiom,
! [A: option_Sum_sum_a_nat > sum_sum_a_nat,B: nat > option_Sum_sum_a_nat,C: nat > sum_sum_a_nat,D: nat > nat,V: nat] :
( ( ( comp_o2721059924169077421at_nat @ A @ B )
= ( comp_n1522318729830440540at_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_85_comp__eq__dest,axiom,
! [A: list_nat > list_Sum_sum_a_nat,B: list_P5056861408695629236_a_nat > list_nat,C: list_nat > list_Sum_sum_a_nat,D: list_P5056861408695629236_a_nat > list_nat,V: list_P5056861408695629236_a_nat] :
( ( ( comp_l7562833077401626166_a_nat @ A @ B )
= ( comp_l7562833077401626166_a_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_86_comp__eq__dest,axiom,
! [A: list_nat > list_Sum_sum_a_nat,B: list_nat > list_nat,C: list_nat > list_Sum_sum_a_nat,D: list_nat > list_nat,V: list_nat] :
( ( ( comp_l4895937012117054562st_nat @ A @ B )
= ( comp_l4895937012117054562st_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_87_comp__eq__dest,axiom,
! [A: nat > option_Sum_sum_a_nat,B: nat > nat,C: nat > option_Sum_sum_a_nat,D: nat > nat,V: nat] :
( ( ( comp_n848383228264675746at_nat @ A @ B )
= ( comp_n848383228264675746at_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_88_comp__eq__dest,axiom,
! [A: nat > sum_sum_a_nat,B: produc5454855657571582244_a_nat > nat,C: nat > sum_sum_a_nat,D: produc5454855657571582244_a_nat > nat,V: produc5454855657571582244_a_nat] :
( ( ( comp_n6276394432912108976_a_nat @ A @ B )
= ( comp_n6276394432912108976_a_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_89_comp__eq__dest,axiom,
! [A: nat > sum_sum_a_nat,B: nat > nat,C: option_Sum_sum_a_nat > sum_sum_a_nat,D: nat > option_Sum_sum_a_nat,V: nat] :
( ( ( comp_n1522318729830440540at_nat @ A @ B )
= ( comp_o2721059924169077421at_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_90_comp__eq__dest,axiom,
! [A: nat > sum_sum_a_nat,B: nat > nat,C: nat > sum_sum_a_nat,D: nat > nat,V: nat] :
( ( ( comp_n1522318729830440540at_nat @ A @ B )
= ( comp_n1522318729830440540at_nat @ C @ D ) )
=> ( ( A @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_91_comp__eq__elim,axiom,
! [A: option_Sum_sum_a_nat > sum_sum_a_nat,B: nat > option_Sum_sum_a_nat,C: option_Sum_sum_a_nat > sum_sum_a_nat,D: nat > option_Sum_sum_a_nat] :
( ( ( comp_o2721059924169077421at_nat @ A @ B )
= ( comp_o2721059924169077421at_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_92_comp__eq__elim,axiom,
! [A: option_Sum_sum_a_nat > sum_sum_a_nat,B: nat > option_Sum_sum_a_nat,C: nat > sum_sum_a_nat,D: nat > nat] :
( ( ( comp_o2721059924169077421at_nat @ A @ B )
= ( comp_n1522318729830440540at_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_93_comp__eq__elim,axiom,
! [A: list_nat > list_Sum_sum_a_nat,B: list_P5056861408695629236_a_nat > list_nat,C: list_nat > list_Sum_sum_a_nat,D: list_P5056861408695629236_a_nat > list_nat] :
( ( ( comp_l7562833077401626166_a_nat @ A @ B )
= ( comp_l7562833077401626166_a_nat @ C @ D ) )
=> ! [V2: list_P5056861408695629236_a_nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_94_comp__eq__elim,axiom,
! [A: list_nat > list_Sum_sum_a_nat,B: list_nat > list_nat,C: list_nat > list_Sum_sum_a_nat,D: list_nat > list_nat] :
( ( ( comp_l4895937012117054562st_nat @ A @ B )
= ( comp_l4895937012117054562st_nat @ C @ D ) )
=> ! [V2: list_nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_95_comp__eq__elim,axiom,
! [A: nat > option_Sum_sum_a_nat,B: nat > nat,C: nat > option_Sum_sum_a_nat,D: nat > nat] :
( ( ( comp_n848383228264675746at_nat @ A @ B )
= ( comp_n848383228264675746at_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_96_comp__eq__elim,axiom,
! [A: nat > sum_sum_a_nat,B: produc5454855657571582244_a_nat > nat,C: nat > sum_sum_a_nat,D: produc5454855657571582244_a_nat > nat] :
( ( ( comp_n6276394432912108976_a_nat @ A @ B )
= ( comp_n6276394432912108976_a_nat @ C @ D ) )
=> ! [V2: produc5454855657571582244_a_nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_97_comp__eq__elim,axiom,
! [A: nat > sum_sum_a_nat,B: nat > nat,C: option_Sum_sum_a_nat > sum_sum_a_nat,D: nat > option_Sum_sum_a_nat] :
( ( ( comp_n1522318729830440540at_nat @ A @ B )
= ( comp_o2721059924169077421at_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_98_comp__eq__elim,axiom,
! [A: nat > sum_sum_a_nat,B: nat > nat,C: nat > sum_sum_a_nat,D: nat > nat] :
( ( ( comp_n1522318729830440540at_nat @ A @ B )
= ( comp_n1522318729830440540at_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_99_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_100_map__eq__conv,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_o6767316136206324043_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( ( map_op8672217432457595070_a_nat @ F @ Xs )
= ( map_op8672217432457595070_a_nat @ G @ Xs ) )
= ( ! [X: option_Sum_sum_a_nat] :
( ( member486768635318144366_a_nat @ X @ ( set_op1665301557670659034_a_nat @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_101_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_102_map__eq__conv,axiom,
! [F: nat > produc5454855657571582244_a_nat,Xs: list_nat,G: nat > produc5454855657571582244_a_nat] :
( ( ( map_na4983825868277236666_a_nat @ F @ Xs )
= ( map_na4983825868277236666_a_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_103_map__eq__conv,axiom,
! [F: nat > option_Sum_sum_a_nat,Xs: list_nat,G: nat > option_Sum_sum_a_nat] :
( ( ( map_na4778969939102885679_a_nat @ F @ Xs )
= ( map_na4778969939102885679_a_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_104_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_105_map__eq__conv,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Xs ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_106_ex__map__conv,axiom,
! [Ys: list_P5056861408695629236_a_nat,F: nat > produc5454855657571582244_a_nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_na4983825868277236666_a_nat @ F @ Xs2 ) ) )
= ( ! [X: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X @ ( set_Pr2196415342531872975_a_nat @ Ys ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_107_ex__map__conv,axiom,
! [Ys: list_o6767316136206324043_a_nat,F: nat > option_Sum_sum_a_nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_na4778969939102885679_a_nat @ F @ Xs2 ) ) )
= ( ! [X: option_Sum_sum_a_nat] :
( ( member486768635318144366_a_nat @ X @ ( set_op1665301557670659034_a_nat @ Ys ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_108_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_109_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 ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_110_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 ) )
=> ? [Y: produc5454855657571582244_a_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_111_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( ? [Xs2: list_o6767316136206324043_a_nat] :
( Ys
= ( map_op8672217432457595070_a_nat @ F @ Xs2 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y: option_Sum_sum_a_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_112_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ? [Xs2: list_Sum_sum_a_nat] :
( Ys
= ( map_Su2790769393171190532_a_nat @ F @ Xs2 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y: sum_sum_a_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_113_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 )
=> ( ! [X2: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X2 @ ( set_Pr2196415342531872975_a_nat @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_114_map__cong,axiom,
! [Xs: list_o6767316136206324043_a_nat,Ys: list_o6767316136206324043_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: option_Sum_sum_a_nat] :
( ( member486768635318144366_a_nat @ X2 @ ( set_op1665301557670659034_a_nat @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_op8672217432457595070_a_nat @ F @ Xs )
= ( map_op8672217432457595070_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_115_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_116_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > produc5454855657571582244_a_nat,G: nat > produc5454855657571582244_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na4983825868277236666_a_nat @ F @ Xs )
= ( map_na4983825868277236666_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_117_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > option_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na4778969939102885679_a_nat @ F @ Xs )
= ( map_na4778969939102885679_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_118_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_119_map__cong,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_120_map__idI,axiom,
! [Xs: list_n989787106983797996_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_na722425985163074756_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_121_map__idI,axiom,
! [Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_122_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_123_map__idI,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_124_map__ext,axiom,
! [Xs: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [X2: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X2 @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_125_map__ext,axiom,
! [Xs: list_o6767316136206324043_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ! [X2: option_Sum_sum_a_nat] :
( ( member486768635318144366_a_nat @ X2 @ ( set_op1665301557670659034_a_nat @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_op8672217432457595070_a_nat @ F @ Xs )
= ( map_op8672217432457595070_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_126_map__ext,axiom,
! [Xs: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_127_map__ext,axiom,
! [Xs: list_nat,F: nat > produc5454855657571582244_a_nat,G: nat > produc5454855657571582244_a_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na4983825868277236666_a_nat @ F @ Xs )
= ( map_na4983825868277236666_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_128_map__ext,axiom,
! [Xs: list_nat,F: nat > option_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na4778969939102885679_a_nat @ F @ Xs )
= ( map_na4778969939102885679_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_129_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_130_map__ext,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_131_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_132_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_133_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_134_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_135_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_136_list_Oinj__map__strong,axiom,
! [X3: list_o6767316136206324043_a_nat,Xa: list_o6767316136206324043_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat,Fa: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z: option_Sum_sum_a_nat,Za: option_Sum_sum_a_nat] :
( ( member486768635318144366_a_nat @ Z @ ( set_op1665301557670659034_a_nat @ X3 ) )
=> ( ( member486768635318144366_a_nat @ Za @ ( set_op1665301557670659034_a_nat @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_op8672217432457595070_a_nat @ F @ X3 )
= ( map_op8672217432457595070_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_137_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_138_list_Oinj__map__strong,axiom,
! [X3: list_nat,Xa: list_nat,F: nat > produc5454855657571582244_a_nat,Fa: nat > produc5454855657571582244_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_na4983825868277236666_a_nat @ F @ X3 )
= ( map_na4983825868277236666_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_139_list_Oinj__map__strong,axiom,
! [X3: list_nat,Xa: list_nat,F: nat > option_Sum_sum_a_nat,Fa: nat > option_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_na4778969939102885679_a_nat @ F @ X3 )
= ( map_na4778969939102885679_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_140_list_Oinj__map__strong,axiom,
! [X3: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X3 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X3 )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_141_list_Oinj__map__strong,axiom,
! [X3: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,Fa: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z: sum_sum_a_nat,Za: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( set_Sum_sum_a_nat2 @ X3 ) )
=> ( ( member_Sum_sum_a_nat @ Za @ ( set_Sum_sum_a_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Su2790769393171190532_a_nat @ F @ X3 )
= ( map_Su2790769393171190532_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_142_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_143_list_Omap__cong0,axiom,
! [X3: list_o6767316136206324043_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z: option_Sum_sum_a_nat] :
( ( member486768635318144366_a_nat @ Z @ ( set_op1665301557670659034_a_nat @ X3 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_op8672217432457595070_a_nat @ F @ X3 )
= ( map_op8672217432457595070_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_144_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_145_list_Omap__cong0,axiom,
! [X3: list_nat,F: nat > produc5454855657571582244_a_nat,G: nat > produc5454855657571582244_a_nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X3 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na4983825868277236666_a_nat @ F @ X3 )
= ( map_na4983825868277236666_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_146_list_Omap__cong0,axiom,
! [X3: list_nat,F: nat > option_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X3 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na4778969939102885679_a_nat @ F @ X3 )
= ( map_na4778969939102885679_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_147_list_Omap__cong0,axiom,
! [X3: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X3 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X3 )
= ( map_nat_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_148_list_Omap__cong0,axiom,
! [X3: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( set_Sum_sum_a_nat2 @ X3 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ X3 )
= ( map_Su2790769393171190532_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_149_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_150_list_Omap__cong,axiom,
! [X3: list_o6767316136206324043_a_nat,Ya: list_o6767316136206324043_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z: option_Sum_sum_a_nat] :
( ( member486768635318144366_a_nat @ Z @ ( set_op1665301557670659034_a_nat @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_op8672217432457595070_a_nat @ F @ X3 )
= ( map_op8672217432457595070_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_151_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_152_list_Omap__cong,axiom,
! [X3: list_nat,Ya: list_nat,F: nat > produc5454855657571582244_a_nat,G: nat > produc5454855657571582244_a_nat] :
( ( X3 = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na4983825868277236666_a_nat @ F @ X3 )
= ( map_na4983825868277236666_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_153_list_Omap__cong,axiom,
! [X3: list_nat,Ya: list_nat,F: nat > option_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na4778969939102885679_a_nat @ F @ X3 )
= ( map_na4778969939102885679_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_154_list_Omap__cong,axiom,
! [X3: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X3 = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X3 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_155_list_Omap__cong,axiom,
! [X3: list_Sum_sum_a_nat,Ya: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( set_Sum_sum_a_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ X3 )
= ( map_Su2790769393171190532_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_156_comp__eq__dest__lhs,axiom,
! [A: option_Sum_sum_a_nat > sum_sum_a_nat,B: nat > option_Sum_sum_a_nat,C: nat > sum_sum_a_nat,V: nat] :
( ( ( comp_o2721059924169077421at_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_157_comp__eq__dest__lhs,axiom,
! [A: list_nat > list_Sum_sum_a_nat,B: list_P5056861408695629236_a_nat > list_nat,C: list_P5056861408695629236_a_nat > list_Sum_sum_a_nat,V: list_P5056861408695629236_a_nat] :
( ( ( comp_l7562833077401626166_a_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_158_comp__eq__dest__lhs,axiom,
! [A: list_nat > list_Sum_sum_a_nat,B: list_nat > list_nat,C: list_nat > list_Sum_sum_a_nat,V: list_nat] :
( ( ( comp_l4895937012117054562st_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_159_comp__eq__dest__lhs,axiom,
! [A: nat > option_Sum_sum_a_nat,B: nat > nat,C: nat > option_Sum_sum_a_nat,V: nat] :
( ( ( comp_n848383228264675746at_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_160_comp__eq__dest__lhs,axiom,
! [A: nat > sum_sum_a_nat,B: produc5454855657571582244_a_nat > nat,C: produc5454855657571582244_a_nat > sum_sum_a_nat,V: produc5454855657571582244_a_nat] :
( ( ( comp_n6276394432912108976_a_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_161_comp__eq__dest__lhs,axiom,
! [A: nat > sum_sum_a_nat,B: nat > nat,C: nat > sum_sum_a_nat,V: nat] :
( ( ( comp_n1522318729830440540at_nat @ A @ B )
= C )
=> ( ( A @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_162__092_060sigma_062_H__S,axiom,
member8690443509505302927_a_nat @ sigma @ r ).
% \<sigma>'_S
thf(fact_163__092_060tau_062__R,axiom,
member8690443509505302927_a_nat @ tau @ r ).
% \<tau>_R
thf(fact_164_map__fv__sub,axiom,
( ( map_na823391071729141993_a_nat @ sigma @ fv_sub )
= ( map_na823391071729141993_a_nat @ tau @ fv_sub ) ) ).
% map_fv_sub
thf(fact_165__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,
! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ fv_sub ) )
=> ( ( tau @ X4 )
= ( sigma @ X4 ) ) ) ).
% \<open>\<forall>n\<in>set fv_sub. \<tau> n = \<sigma>' n\<close>
thf(fact_166__092_060tau_062__def_I1_J,axiom,
( as
= ( map_na823391071729141993_a_nat @ tau @ fv_sub ) ) ).
% \<tau>_def(1)
thf(fact_167_map__eq__map__tailrec,axiom,
map_na823391071729141993_a_nat = map_ta1136998156224711455_a_nat ).
% map_eq_map_tailrec
thf(fact_168_map__eq__map__tailrec,axiom,
map_Pr2515564165894599445_a_nat = map_ta8991945714069663307_a_nat ).
% map_eq_map_tailrec
thf(fact_169_map__eq__map__tailrec,axiom,
map_op8672217432457595070_a_nat = map_ta6533011271814402568_a_nat ).
% map_eq_map_tailrec
thf(fact_170_map__eq__map__tailrec,axiom,
map_Su2790769393171190532_a_nat = map_ta197872344851190606_a_nat ).
% map_eq_map_tailrec
thf(fact_171_map__eq__map__tailrec,axiom,
map_na4983825868277236666_a_nat = map_ta6871343446014289412_a_nat ).
% map_eq_map_tailrec
thf(fact_172_map__eq__map__tailrec,axiom,
map_na4778969939102885679_a_nat = map_ta1041444099103809637_a_nat ).
% map_eq_map_tailrec
thf(fact_173_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_174_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: nat > nat,R1: nat > nat,R2: nat > nat,F: nat > sum_sum_a_nat,L: nat > sum_sum_a_nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= ( comp_nat_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_n1522318729830440540at_nat @ F @ R1 )
= L )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_175_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: nat > nat,R1: nat > nat,R2: nat > nat,F: nat > option_Sum_sum_a_nat,L: nat > option_Sum_sum_a_nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= ( comp_nat_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_n848383228264675746at_nat @ F @ R1 )
= L )
=> ( ( comp_n848383228264675746at_nat @ ( comp_n848383228264675746at_nat @ F @ G ) @ H )
= ( comp_n848383228264675746at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_176_rewriteR__comp__comp2,axiom,
! [G: nat > sum_sum_a_nat,H: nat > nat,R1: nat > sum_sum_a_nat,R2: nat > nat,F: sum_sum_a_nat > sum_sum_a_nat,L: nat > sum_sum_a_nat] :
( ( ( comp_n1522318729830440540at_nat @ G @ H )
= ( comp_n1522318729830440540at_nat @ R1 @ R2 ) )
=> ( ( ( comp_S2395038772458240499at_nat @ F @ R1 )
= L )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_177_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: nat > nat,R1: option_Sum_sum_a_nat > nat,R2: nat > option_Sum_sum_a_nat,F: nat > sum_sum_a_nat,L: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= ( comp_o8557203714689770820at_nat @ R1 @ R2 ) )
=> ( ( ( comp_n3164697776794079673_a_nat @ F @ R1 )
= L )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ H )
= ( comp_o2721059924169077421at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_178_rewriteR__comp__comp2,axiom,
! [G: option_Sum_sum_a_nat > nat,H: nat > option_Sum_sum_a_nat,R1: nat > nat,R2: nat > nat,F: nat > sum_sum_a_nat,L: nat > sum_sum_a_nat] :
( ( ( comp_o8557203714689770820at_nat @ G @ H )
= ( comp_nat_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_n1522318729830440540at_nat @ F @ R1 )
= L )
=> ( ( comp_o2721059924169077421at_nat @ ( comp_n3164697776794079673_a_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_179_rewriteR__comp__comp2,axiom,
! [G: nat > option_Sum_sum_a_nat,H: nat > nat,R1: nat > option_Sum_sum_a_nat,R2: nat > nat,F: option_Sum_sum_a_nat > sum_sum_a_nat,L: nat > sum_sum_a_nat] :
( ( ( comp_n848383228264675746at_nat @ G @ H )
= ( comp_n848383228264675746at_nat @ R1 @ R2 ) )
=> ( ( ( comp_o2721059924169077421at_nat @ F @ R1 )
= L )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_o2721059924169077421at_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_180_rewriteR__comp__comp2,axiom,
! [G: nat > sum_sum_a_nat,H: nat > nat,R1: nat > sum_sum_a_nat,R2: nat > nat,F: sum_sum_a_nat > option_Sum_sum_a_nat,L: nat > option_Sum_sum_a_nat] :
( ( ( comp_n1522318729830440540at_nat @ G @ H )
= ( comp_n1522318729830440540at_nat @ R1 @ R2 ) )
=> ( ( ( comp_S4056802848252454585at_nat @ F @ R1 )
= L )
=> ( ( comp_n848383228264675746at_nat @ ( comp_S4056802848252454585at_nat @ F @ G ) @ H )
= ( comp_n848383228264675746at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_181_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: nat > nat,R1: produc5454855657571582244_a_nat > nat,R2: nat > produc5454855657571582244_a_nat,F: nat > sum_sum_a_nat,L: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= ( comp_P2340930289473004329at_nat @ R1 @ R2 ) )
=> ( ( ( comp_n6276394432912108976_a_nat @ F @ R1 )
= L )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ H )
= ( comp_P2433158451856970632at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_182_rewriteR__comp__comp2,axiom,
! [G: produc5454855657571582244_a_nat > nat,H: nat > produc5454855657571582244_a_nat,R1: nat > nat,R2: nat > nat,F: nat > sum_sum_a_nat,L: nat > sum_sum_a_nat] :
( ( ( comp_P2340930289473004329at_nat @ G @ H )
= ( comp_nat_nat_nat @ R1 @ R2 ) )
=> ( ( ( comp_n1522318729830440540at_nat @ F @ R1 )
= L )
=> ( ( comp_P2433158451856970632at_nat @ ( comp_n6276394432912108976_a_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_183_rewriteR__comp__comp2,axiom,
! [G: nat > nat,H: produc5454855657571582244_a_nat > nat,R1: nat > nat,R2: produc5454855657571582244_a_nat > nat,F: nat > sum_sum_a_nat,L: nat > sum_sum_a_nat] :
( ( ( comp_n652616413913648681_a_nat @ G @ H )
= ( comp_n652616413913648681_a_nat @ R1 @ R2 ) )
=> ( ( ( comp_n1522318729830440540at_nat @ F @ R1 )
= L )
=> ( ( comp_n6276394432912108976_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ H )
= ( comp_n6276394432912108976_a_nat @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_184_rewriteL__comp__comp2,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,L1: nat > sum_sum_a_nat,L2: nat > nat,H: nat > nat,R: nat > nat] :
( ( ( comp_n1522318729830440540at_nat @ F @ G )
= ( comp_n1522318729830440540at_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_nat @ L2 @ H )
= R )
=> ( ( comp_n1522318729830440540at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_n1522318729830440540at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_185_rewriteL__comp__comp2,axiom,
! [F: nat > option_Sum_sum_a_nat,G: nat > nat,L1: nat > option_Sum_sum_a_nat,L2: nat > nat,H: nat > nat,R: nat > nat] :
( ( ( comp_n848383228264675746at_nat @ F @ G )
= ( comp_n848383228264675746at_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_nat @ L2 @ H )
= R )
=> ( ( comp_n848383228264675746at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_n848383228264675746at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_186_rewriteL__comp__comp2,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,L1: nat > sum_sum_a_nat,L2: nat > nat,H: nat > nat,R: nat > nat] :
( ( ( comp_S2395038772458240499at_nat @ F @ G )
= ( comp_n1522318729830440540at_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_nat @ L2 @ H )
= R )
=> ( ( comp_S2395038772458240499at_nat @ F @ ( comp_n1522318729830440540at_nat @ G @ H ) )
= ( comp_n1522318729830440540at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_187_rewriteL__comp__comp2,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,L1: sum_sum_a_nat > sum_sum_a_nat,L2: nat > sum_sum_a_nat,H: nat > nat,R: nat > sum_sum_a_nat] :
( ( ( comp_n1522318729830440540at_nat @ F @ G )
= ( comp_S2395038772458240499at_nat @ L1 @ L2 ) )
=> ( ( ( comp_n1522318729830440540at_nat @ L2 @ H )
= R )
=> ( ( comp_n1522318729830440540at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_S2395038772458240499at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_188_rewriteL__comp__comp2,axiom,
! [F: sum_sum_a_nat > option_Sum_sum_a_nat,G: nat > sum_sum_a_nat,L1: nat > option_Sum_sum_a_nat,L2: nat > nat,H: nat > nat,R: nat > nat] :
( ( ( comp_S4056802848252454585at_nat @ F @ G )
= ( comp_n848383228264675746at_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_nat @ L2 @ H )
= R )
=> ( ( comp_S4056802848252454585at_nat @ F @ ( comp_n1522318729830440540at_nat @ G @ H ) )
= ( comp_n848383228264675746at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_189_rewriteL__comp__comp2,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,L1: nat > sum_sum_a_nat,L2: nat > nat,H: nat > nat,R: nat > nat] :
( ( ( comp_o2721059924169077421at_nat @ F @ G )
= ( comp_n1522318729830440540at_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_nat @ L2 @ H )
= R )
=> ( ( comp_o2721059924169077421at_nat @ F @ ( comp_n848383228264675746at_nat @ G @ H ) )
= ( comp_n1522318729830440540at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_190_rewriteL__comp__comp2,axiom,
! [F: nat > option_Sum_sum_a_nat,G: nat > nat,L1: sum_sum_a_nat > option_Sum_sum_a_nat,L2: nat > sum_sum_a_nat,H: nat > nat,R: nat > sum_sum_a_nat] :
( ( ( comp_n848383228264675746at_nat @ F @ G )
= ( comp_S4056802848252454585at_nat @ L1 @ L2 ) )
=> ( ( ( comp_n1522318729830440540at_nat @ L2 @ H )
= R )
=> ( ( comp_n848383228264675746at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_S4056802848252454585at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_191_rewriteL__comp__comp2,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,L1: option_Sum_sum_a_nat > sum_sum_a_nat,L2: nat > option_Sum_sum_a_nat,H: nat > nat,R: nat > option_Sum_sum_a_nat] :
( ( ( comp_n1522318729830440540at_nat @ F @ G )
= ( comp_o2721059924169077421at_nat @ L1 @ L2 ) )
=> ( ( ( comp_n848383228264675746at_nat @ L2 @ H )
= R )
=> ( ( comp_n1522318729830440540at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_o2721059924169077421at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_192_rewriteL__comp__comp2,axiom,
! [F: option_Sum_sum_a_nat > option_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,L1: nat > option_Sum_sum_a_nat,L2: nat > nat,H: nat > nat,R: nat > nat] :
( ( ( comp_o5487264423576322163at_nat @ F @ G )
= ( comp_n848383228264675746at_nat @ L1 @ L2 ) )
=> ( ( ( comp_nat_nat_nat @ L2 @ H )
= R )
=> ( ( comp_o5487264423576322163at_nat @ F @ ( comp_n848383228264675746at_nat @ G @ H ) )
= ( comp_n848383228264675746at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_193_rewriteL__comp__comp2,axiom,
! [F: nat > option_Sum_sum_a_nat,G: nat > nat,L1: option_Sum_sum_a_nat > option_Sum_sum_a_nat,L2: nat > option_Sum_sum_a_nat,H: nat > nat,R: nat > option_Sum_sum_a_nat] :
( ( ( comp_n848383228264675746at_nat @ F @ G )
= ( comp_o5487264423576322163at_nat @ L1 @ L2 ) )
=> ( ( ( comp_n848383228264675746at_nat @ L2 @ H )
= R )
=> ( ( comp_n848383228264675746at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_o5487264423576322163at_nat @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_194_rewriteR__comp__comp,axiom,
! [G: nat > nat,H: nat > nat,R: nat > nat,F: nat > sum_sum_a_nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= R )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_195_rewriteR__comp__comp,axiom,
! [G: nat > nat,H: nat > nat,R: nat > nat,F: nat > option_Sum_sum_a_nat] :
( ( ( comp_nat_nat_nat @ G @ H )
= R )
=> ( ( comp_n848383228264675746at_nat @ ( comp_n848383228264675746at_nat @ F @ G ) @ H )
= ( comp_n848383228264675746at_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_196_rewriteR__comp__comp,axiom,
! [G: nat > sum_sum_a_nat,H: nat > nat,R: nat > sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ( comp_n1522318729830440540at_nat @ G @ H )
= R )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) @ H )
= ( comp_S2395038772458240499at_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_197_rewriteR__comp__comp,axiom,
! [G: option_Sum_sum_a_nat > nat,H: nat > option_Sum_sum_a_nat,R: nat > nat,F: nat > sum_sum_a_nat] :
( ( ( comp_o8557203714689770820at_nat @ G @ H )
= R )
=> ( ( comp_o2721059924169077421at_nat @ ( comp_n3164697776794079673_a_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_198_rewriteR__comp__comp,axiom,
! [G: nat > option_Sum_sum_a_nat,H: nat > nat,R: nat > option_Sum_sum_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( ( comp_n848383228264675746at_nat @ G @ H )
= R )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_o2721059924169077421at_nat @ F @ G ) @ H )
= ( comp_o2721059924169077421at_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_199_rewriteR__comp__comp,axiom,
! [G: nat > sum_sum_a_nat,H: nat > nat,R: nat > sum_sum_a_nat,F: sum_sum_a_nat > option_Sum_sum_a_nat] :
( ( ( comp_n1522318729830440540at_nat @ G @ H )
= R )
=> ( ( comp_n848383228264675746at_nat @ ( comp_S4056802848252454585at_nat @ F @ G ) @ H )
= ( comp_S4056802848252454585at_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_200_rewriteR__comp__comp,axiom,
! [G: produc5454855657571582244_a_nat > nat,H: nat > produc5454855657571582244_a_nat,R: nat > nat,F: nat > sum_sum_a_nat] :
( ( ( comp_P2340930289473004329at_nat @ G @ H )
= R )
=> ( ( comp_P2433158451856970632at_nat @ ( comp_n6276394432912108976_a_nat @ F @ G ) @ H )
= ( comp_n1522318729830440540at_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_201_rewriteR__comp__comp,axiom,
! [G: list_nat > list_nat,H: list_nat > list_nat,R: list_nat > list_nat,F: list_nat > list_Sum_sum_a_nat] :
( ( ( comp_l7223822213492037765st_nat @ G @ H )
= R )
=> ( ( comp_l4895937012117054562st_nat @ ( comp_l4895937012117054562st_nat @ F @ G ) @ H )
= ( comp_l4895937012117054562st_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_202_rewriteR__comp__comp,axiom,
! [G: nat > nat,H: produc5454855657571582244_a_nat > nat,R: produc5454855657571582244_a_nat > nat,F: nat > sum_sum_a_nat] :
( ( ( comp_n652616413913648681_a_nat @ G @ H )
= R )
=> ( ( comp_n6276394432912108976_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ H )
= ( comp_n6276394432912108976_a_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_203_rewriteR__comp__comp,axiom,
! [G: nat > option_Sum_sum_a_nat,H: nat > nat,R: nat > option_Sum_sum_a_nat,F: option_Sum_sum_a_nat > option_Sum_sum_a_nat] :
( ( ( comp_n848383228264675746at_nat @ G @ H )
= R )
=> ( ( comp_n848383228264675746at_nat @ ( comp_o5487264423576322163at_nat @ F @ G ) @ H )
= ( comp_o5487264423576322163at_nat @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_204_rewriteL__comp__comp,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,L: nat > sum_sum_a_nat,H: nat > nat] :
( ( ( comp_n1522318729830440540at_nat @ F @ G )
= L )
=> ( ( comp_n1522318729830440540at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_n1522318729830440540at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_205_rewriteL__comp__comp,axiom,
! [F: nat > option_Sum_sum_a_nat,G: nat > nat,L: nat > option_Sum_sum_a_nat,H: nat > nat] :
( ( ( comp_n848383228264675746at_nat @ F @ G )
= L )
=> ( ( comp_n848383228264675746at_nat @ F @ ( comp_nat_nat_nat @ G @ H ) )
= ( comp_n848383228264675746at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_206_rewriteL__comp__comp,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,L: nat > sum_sum_a_nat,H: nat > nat] :
( ( ( comp_S2395038772458240499at_nat @ F @ G )
= L )
=> ( ( comp_S2395038772458240499at_nat @ F @ ( comp_n1522318729830440540at_nat @ G @ H ) )
= ( comp_n1522318729830440540at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_207_rewriteL__comp__comp,axiom,
! [F: sum_sum_a_nat > option_Sum_sum_a_nat,G: nat > sum_sum_a_nat,L: nat > option_Sum_sum_a_nat,H: nat > nat] :
( ( ( comp_S4056802848252454585at_nat @ F @ G )
= L )
=> ( ( comp_S4056802848252454585at_nat @ F @ ( comp_n1522318729830440540at_nat @ G @ H ) )
= ( comp_n848383228264675746at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_208_rewriteL__comp__comp,axiom,
! [F: nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > nat,L: option_Sum_sum_a_nat > sum_sum_a_nat,H: nat > option_Sum_sum_a_nat] :
( ( ( comp_n3164697776794079673_a_nat @ F @ G )
= L )
=> ( ( comp_n1522318729830440540at_nat @ F @ ( comp_o8557203714689770820at_nat @ G @ H ) )
= ( comp_o2721059924169077421at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_209_rewriteL__comp__comp,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,L: nat > sum_sum_a_nat,H: nat > nat] :
( ( ( comp_o2721059924169077421at_nat @ F @ G )
= L )
=> ( ( comp_o2721059924169077421at_nat @ F @ ( comp_n848383228264675746at_nat @ G @ H ) )
= ( comp_n1522318729830440540at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_210_rewriteL__comp__comp,axiom,
! [F: option_Sum_sum_a_nat > option_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,L: nat > option_Sum_sum_a_nat,H: nat > nat] :
( ( ( comp_o5487264423576322163at_nat @ F @ G )
= L )
=> ( ( comp_o5487264423576322163at_nat @ F @ ( comp_n848383228264675746at_nat @ G @ H ) )
= ( comp_n848383228264675746at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_211_rewriteL__comp__comp,axiom,
! [F: list_nat > list_Sum_sum_a_nat,G: list_nat > list_nat,L: list_nat > list_Sum_sum_a_nat,H: list_nat > list_nat] :
( ( ( comp_l4895937012117054562st_nat @ F @ G )
= L )
=> ( ( comp_l4895937012117054562st_nat @ F @ ( comp_l7223822213492037765st_nat @ G @ H ) )
= ( comp_l4895937012117054562st_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_212_rewriteL__comp__comp,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,L: produc5454855657571582244_a_nat > sum_sum_a_nat,H: nat > produc5454855657571582244_a_nat] :
( ( ( comp_n6276394432912108976_a_nat @ F @ G )
= L )
=> ( ( comp_n1522318729830440540at_nat @ F @ ( comp_P2340930289473004329at_nat @ G @ H ) )
= ( comp_P2433158451856970632at_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_213_rewriteL__comp__comp,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,L: nat > sum_sum_a_nat,H: produc5454855657571582244_a_nat > nat] :
( ( ( comp_n1522318729830440540at_nat @ F @ G )
= L )
=> ( ( comp_n6276394432912108976_a_nat @ F @ ( comp_n652616413913648681_a_nat @ G @ H ) )
= ( comp_n6276394432912108976_a_nat @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_214_type__copy__map__cong0,axiom,
! [M: nat > nat,G: nat > nat,X3: nat,N2: nat > nat,H: nat > nat,F: nat > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ M ) @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_215_type__copy__map__cong0,axiom,
! [M: nat > nat,G: nat > nat,X3: nat,N2: nat > nat,H: nat > nat,F: nat > option_Sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_n848383228264675746at_nat @ ( comp_n848383228264675746at_nat @ F @ M ) @ G @ X3 )
= ( comp_n848383228264675746at_nat @ ( comp_n848383228264675746at_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_216_type__copy__map__cong0,axiom,
! [M: option_Sum_sum_a_nat > nat,G: nat > option_Sum_sum_a_nat,X3: nat,N2: nat > nat,H: nat > nat,F: nat > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_o2721059924169077421at_nat @ ( comp_n3164697776794079673_a_nat @ F @ M ) @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_217_type__copy__map__cong0,axiom,
! [M: nat > option_Sum_sum_a_nat,G: nat > 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_n1522318729830440540at_nat @ ( comp_o2721059924169077421at_nat @ F @ M ) @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ ( comp_o2721059924169077421at_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_218_type__copy__map__cong0,axiom,
! [M: nat > nat,G: nat > nat,X3: nat,N2: option_Sum_sum_a_nat > nat,H: nat > option_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ M ) @ G @ X3 )
= ( comp_o2721059924169077421at_nat @ ( comp_n3164697776794079673_a_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_219_type__copy__map__cong0,axiom,
! [M: produc5454855657571582244_a_nat > nat,G: nat > produc5454855657571582244_a_nat,X3: nat,N2: nat > nat,H: nat > nat,F: nat > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_P2433158451856970632at_nat @ ( comp_n6276394432912108976_a_nat @ F @ M ) @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_220_type__copy__map__cong0,axiom,
! [M: nat > nat,G: produc5454855657571582244_a_nat > nat,X3: produc5454855657571582244_a_nat,N2: nat > nat,H: produc5454855657571582244_a_nat > nat,F: nat > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_n6276394432912108976_a_nat @ ( comp_n1522318729830440540at_nat @ F @ M ) @ G @ X3 )
= ( comp_n6276394432912108976_a_nat @ ( comp_n1522318729830440540at_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_221_type__copy__map__cong0,axiom,
! [M: nat > nat,G: nat > nat,X3: nat,N2: produc5454855657571582244_a_nat > nat,H: nat > produc5454855657571582244_a_nat,F: nat > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_n1522318729830440540at_nat @ F @ M ) @ G @ X3 )
= ( comp_P2433158451856970632at_nat @ ( comp_n6276394432912108976_a_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_222_type__copy__map__cong0,axiom,
! [M: list_nat > list_nat,G: list_nat > list_nat,X3: list_nat,N2: list_nat > list_nat,H: list_nat > list_nat,F: list_nat > list_Sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_l4895937012117054562st_nat @ ( comp_l4895937012117054562st_nat @ F @ M ) @ G @ X3 )
= ( comp_l4895937012117054562st_nat @ ( comp_l4895937012117054562st_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_223_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_224_comp__apply__eq,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,X3: nat,H: option_Sum_sum_a_nat > sum_sum_a_nat,K: nat > option_Sum_sum_a_nat] :
( ( ( F @ ( G @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp_o2721059924169077421at_nat @ F @ G @ X3 )
= ( comp_o2721059924169077421at_nat @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_225_comp__apply__eq,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,X3: nat,H: nat > sum_sum_a_nat,K: nat > nat] :
( ( ( F @ ( G @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp_o2721059924169077421at_nat @ F @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_226_comp__apply__eq,axiom,
! [F: list_nat > list_Sum_sum_a_nat,G: list_P5056861408695629236_a_nat > list_nat,X3: list_P5056861408695629236_a_nat,H: list_nat > list_Sum_sum_a_nat,K: list_P5056861408695629236_a_nat > list_nat] :
( ( ( F @ ( G @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp_l7562833077401626166_a_nat @ F @ G @ X3 )
= ( comp_l7562833077401626166_a_nat @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_227_comp__apply__eq,axiom,
! [F: list_nat > list_Sum_sum_a_nat,G: list_nat > list_nat,X3: list_nat,H: list_nat > list_Sum_sum_a_nat,K: list_nat > list_nat] :
( ( ( F @ ( G @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp_l4895937012117054562st_nat @ F @ G @ X3 )
= ( comp_l4895937012117054562st_nat @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_228_comp__apply__eq,axiom,
! [F: nat > option_Sum_sum_a_nat,G: nat > nat,X3: nat,H: nat > option_Sum_sum_a_nat,K: nat > nat] :
( ( ( F @ ( G @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp_n848383228264675746at_nat @ F @ G @ X3 )
= ( comp_n848383228264675746at_nat @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_229_comp__apply__eq,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,X3: produc5454855657571582244_a_nat,H: nat > sum_sum_a_nat,K: produc5454855657571582244_a_nat > nat] :
( ( ( F @ ( G @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp_n6276394432912108976_a_nat @ F @ G @ X3 )
= ( comp_n6276394432912108976_a_nat @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_230_comp__apply__eq,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,X3: nat,H: option_Sum_sum_a_nat > sum_sum_a_nat,K: nat > option_Sum_sum_a_nat] :
( ( ( F @ ( G @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ F @ G @ X3 )
= ( comp_o2721059924169077421at_nat @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_231_comp__apply__eq,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,X3: nat,H: nat > sum_sum_a_nat,K: nat > nat] :
( ( ( F @ ( G @ X3 ) )
= ( H @ ( K @ X3 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ F @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ H @ K @ X3 ) ) ) ).
% comp_apply_eq
thf(fact_232__092_060sigma_062__def_I1_J,axiom,
member8690443509505302927_a_nat @ sigma2 @ r ).
% \<sigma>_def(1)
thf(fact_233_as__fs__def_I1_J,axiom,
member408289922725080238_a_nat @ as @ ass ).
% as_fs_def(1)
thf(fact_234_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_235__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,
~ ! [Sigma: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma @ r )
=> ( as
!= ( fo_nmlz_a @ ad @ ( map_na823391071729141993_a_nat @ Sigma @ 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_236_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_237_assms_I7_J,axiom,
! [Sigma2: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ fv_sub ) @ ( set_nat2 @ fv_sub ) @ ad @ Sigma2 @ Tau )
=> ( ( member8690443509505302927_a_nat @ Sigma2 @ r )
= ( member8690443509505302927_a_nat @ Tau @ r ) ) ) ).
% assms(7)
thf(fact_238__092_060tau_062__def_I2_J,axiom,
ad_agr_list_a_nat @ ad @ ( map_na823391071729141993_a_nat @ sigma2 @ fv_sub ) @ ( map_na823391071729141993_a_nat @ tau @ fv_sub ) ).
% \<tau>_def(2)
thf(fact_239__092_060sigma_062__def_I2_J,axiom,
( as
= ( fo_nmlz_a @ ad @ ( map_na823391071729141993_a_nat @ sigma2 @ fv_sub ) ) ) ).
% \<sigma>_def(2)
thf(fact_240_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_241_comp__cong,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,X3: nat,F3: nat > sum_sum_a_nat,G3: nat > nat,X5: nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ F @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_242_comp__cong,axiom,
! [F: nat > option_Sum_sum_a_nat,G: nat > nat,X3: nat,F3: nat > option_Sum_sum_a_nat,G3: nat > nat,X5: nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_n848383228264675746at_nat @ F @ G @ X3 )
= ( comp_n848383228264675746at_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_243_comp__cong,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,X3: nat,F3: nat > sum_sum_a_nat,G3: nat > nat,X5: nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_o2721059924169077421at_nat @ F @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_244_comp__cong,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,X3: nat,F3: option_Sum_sum_a_nat > sum_sum_a_nat,G3: nat > option_Sum_sum_a_nat,X5: nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ F @ G @ X3 )
= ( comp_o2721059924169077421at_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_245_comp__cong,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,X3: produc5454855657571582244_a_nat,F3: nat > sum_sum_a_nat,G3: nat > nat,X5: nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_n6276394432912108976_a_nat @ F @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_246_comp__cong,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,X3: nat,F3: nat > sum_sum_a_nat,G3: produc5454855657571582244_a_nat > nat,X5: produc5454855657571582244_a_nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ F @ G @ X3 )
= ( comp_n6276394432912108976_a_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_247_comp__cong,axiom,
! [F: list_nat > list_Sum_sum_a_nat,G: list_nat > list_nat,X3: list_nat,F3: list_nat > list_Sum_sum_a_nat,G3: list_nat > list_nat,X5: list_nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_l4895937012117054562st_nat @ F @ G @ X3 )
= ( comp_l4895937012117054562st_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_248_comp__cong,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,X3: nat,F3: option_Sum_sum_a_nat > sum_sum_a_nat,G3: nat > option_Sum_sum_a_nat,X5: nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_o2721059924169077421at_nat @ F @ G @ X3 )
= ( comp_o2721059924169077421at_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_249_comp__cong,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,X3: nat,F3: nat > sum_sum_a_nat,G3: produc5454855657571582244_a_nat > nat,X5: produc5454855657571582244_a_nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_o2721059924169077421at_nat @ F @ G @ X3 )
= ( comp_n6276394432912108976_a_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_250_comp__cong,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,X3: produc5454855657571582244_a_nat,F3: option_Sum_sum_a_nat > sum_sum_a_nat,G3: nat > option_Sum_sum_a_nat,X5: nat] :
( ( ( F @ ( G @ X3 ) )
= ( F3 @ ( G3 @ X5 ) ) )
=> ( ( comp_n6276394432912108976_a_nat @ F @ G @ X3 )
= ( comp_o2721059924169077421at_nat @ F3 @ G3 @ X5 ) ) ) ).
% comp_cong
thf(fact_251_length__fs,axiom,
( ( size_s5283204784079214577_a_nat @ fs )
= ( size_size_list_nat @ fv_sub_comp ) ) ).
% length_fs
thf(fact_252_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_253_mem__Collect__eq,axiom,
! [A: nat > sum_sum_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ A @ ( collec5629555741568564177_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_254_mem__Collect__eq,axiom,
! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_255_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_256_Collect__mem__eq,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_257_Collect__mem__eq,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_258_map__fun__upd,axiom,
! [Y2: produc5454855657571582244_a_nat,Xs: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,V: sum_sum_a_nat] :
( ~ ( member3650934779329710139_a_nat @ Y2 @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( ( map_Pr2515564165894599445_a_nat @ ( fun_up19502095292272284_a_nat @ F @ Y2 @ V ) @ Xs )
= ( map_Pr2515564165894599445_a_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_259_map__fun__upd,axiom,
! [Y2: option_Sum_sum_a_nat,Xs: list_o6767316136206324043_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat,V: sum_sum_a_nat] :
( ~ ( member486768635318144366_a_nat @ Y2 @ ( set_op1665301557670659034_a_nat @ Xs ) )
=> ( ( map_op8672217432457595070_a_nat @ ( fun_up8866717756282968183_a_nat @ F @ Y2 @ V ) @ Xs )
= ( map_op8672217432457595070_a_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_260_map__fun__upd,axiom,
! [Y2: nat,Xs: list_nat,F: nat > sum_sum_a_nat,V: sum_sum_a_nat] :
( ~ ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( map_na823391071729141993_a_nat @ ( fun_up180537416982607344_a_nat @ F @ Y2 @ V ) @ Xs )
= ( map_na823391071729141993_a_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_261_map__fun__upd,axiom,
! [Y2: nat,Xs: list_nat,F: nat > produc5454855657571582244_a_nat,V: produc5454855657571582244_a_nat] :
( ~ ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( map_na4983825868277236666_a_nat @ ( fun_up1054661830135108083_a_nat @ F @ Y2 @ V ) @ Xs )
= ( map_na4983825868277236666_a_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_262_map__fun__upd,axiom,
! [Y2: nat,Xs: list_nat,F: nat > option_Sum_sum_a_nat,V: option_Sum_sum_a_nat] :
( ~ ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( map_na4778969939102885679_a_nat @ ( fun_up6974907568698703286_a_nat @ F @ Y2 @ V ) @ Xs )
= ( map_na4778969939102885679_a_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_263_map__fun__upd,axiom,
! [Y2: nat,Xs: list_nat,F: nat > nat,V: nat] :
( ~ ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( map_nat_nat @ ( fun_upd_nat_nat @ F @ Y2 @ V ) @ Xs )
= ( map_nat_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_264_map__fun__upd,axiom,
! [Y2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,V: sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ Y2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( map_Su2790769393171190532_a_nat @ ( fun_up6086130847573437501_a_nat @ F @ Y2 @ V ) @ Xs )
= ( map_Su2790769393171190532_a_nat @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_265__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,
~ ! [Tau2: nat > sum_sum_a_nat] :
( ( as
= ( map_na823391071729141993_a_nat @ Tau2 @ fv_sub ) )
=> ~ ( ad_agr_list_a_nat @ ad @ ( map_na823391071729141993_a_nat @ sigma2 @ fv_sub ) @ ( map_na823391071729141993_a_nat @ Tau2 @ 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_266_length__as,axiom,
( ( size_s5283204784079214577_a_nat @ as )
= ( size_size_list_nat @ fv_sub ) ) ).
% length_as
thf(fact_267_length__map,axiom,
! [F: nat > produc5454855657571582244_a_nat,Xs: list_nat] :
( ( size_s4076174644546656840_a_nat @ ( map_na4983825868277236666_a_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_268_length__map,axiom,
! [F: nat > option_Sum_sum_a_nat,Xs: list_nat] :
( ( size_s7896677892106675127_a_nat @ ( map_na4778969939102885679_a_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_269_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_270_length__map,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_o6767316136206324043_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_op8672217432457595070_a_nat @ F @ Xs ) )
= ( size_s7896677892106675127_a_nat @ Xs ) ) ).
% length_map
thf(fact_271_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_272_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_273_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_274_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_275_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_276_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_277_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_278_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_279_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_280_ad__agr__sets__comm,axiom,
! [FV: set_nat,S: set_nat,X6: set_a,Sigma2: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ FV @ S @ X6 @ Sigma2 @ Tau )
=> ( ad_agr_sets_a_nat @ FV @ S @ X6 @ Tau @ Sigma2 ) ) ).
% ad_agr_sets_comm
thf(fact_281_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_282_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_283_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_284_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_285_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_286_ad__agr__list__comm,axiom,
! [X6: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X6 @ Xs @ Ys )
=> ( ad_agr_list_a_nat @ X6 @ Ys @ Xs ) ) ).
% ad_agr_list_comm
thf(fact_287_ad__agr__list__refl,axiom,
! [X6: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ X6 @ Xs @ Xs ) ).
% ad_agr_list_refl
thf(fact_288_ad__agr__list__trans,axiom,
! [X6: 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 @ X6 @ Xs @ Ys )
=> ( ( ad_agr_list_a_nat @ X6 @ Ys @ Zs )
=> ( ad_agr_list_a_nat @ X6 @ Xs @ Zs ) ) ) ).
% ad_agr_list_trans
thf(fact_289_ad__agr__list__length,axiom,
! [X6: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X6 @ Xs @ Ys )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% ad_agr_list_length
thf(fact_290_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_291_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_292_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_293_Ex__list__of__length,axiom,
! [N3: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N3 ) ).
% Ex_list_of_length
thf(fact_294_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_295_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_Su5227373005390213643at_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_296_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_297_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_298_map__eq__imp__length__eq,axiom,
! [F: nat > option_Sum_sum_a_nat,Xs: list_nat,G: nat > option_Sum_sum_a_nat,Ys: list_nat] :
( ( ( map_na4778969939102885679_a_nat @ F @ Xs )
= ( map_na4778969939102885679_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_299_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_300_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_301_map__eq__imp__length__eq,axiom,
! [F: nat > produc5454855657571582244_a_nat,Xs: list_nat,G: nat > produc5454855657571582244_a_nat,Ys: list_nat] :
( ( ( map_na4983825868277236666_a_nat @ F @ Xs )
= ( map_na4983825868277236666_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_302_map__eq__imp__length__eq,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_o6767316136206324043_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_op8672217432457595070_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s7896677892106675127_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_303_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > option_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Su5109359166326516938_a_nat @ F @ Xs )
= ( map_na4778969939102885679_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_304_fun__upd__comp,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,X3: nat,Y2: option_Sum_sum_a_nat] :
( ( comp_o2721059924169077421at_nat @ F @ ( fun_up6974907568698703286_a_nat @ G @ X3 @ Y2 ) )
= ( fun_up180537416982607344_a_nat @ ( comp_o2721059924169077421at_nat @ F @ G ) @ X3 @ ( F @ Y2 ) ) ) ).
% fun_upd_comp
thf(fact_305_fun__upd__comp,axiom,
! [F: list_nat > list_Sum_sum_a_nat,G: list_P5056861408695629236_a_nat > list_nat,X3: list_P5056861408695629236_a_nat,Y2: list_nat] :
( ( comp_l7562833077401626166_a_nat @ F @ ( fun_up2507851329316450195st_nat @ G @ X3 @ Y2 ) )
= ( fun_up4481430433955130898_a_nat @ ( comp_l7562833077401626166_a_nat @ F @ G ) @ X3 @ ( F @ Y2 ) ) ) ).
% fun_upd_comp
thf(fact_306_fun__upd__comp,axiom,
! [F: list_nat > list_Sum_sum_a_nat,G: list_nat > list_nat,X3: list_nat,Y2: list_nat] :
( ( comp_l4895937012117054562st_nat @ F @ ( fun_up3599288762871520703st_nat @ G @ X3 @ Y2 ) )
= ( fun_up556463684815361254_a_nat @ ( comp_l4895937012117054562st_nat @ F @ G ) @ X3 @ ( F @ Y2 ) ) ) ).
% fun_upd_comp
thf(fact_307_fun__upd__comp,axiom,
! [F: nat > option_Sum_sum_a_nat,G: nat > nat,X3: nat,Y2: nat] :
( ( comp_n848383228264675746at_nat @ F @ ( fun_upd_nat_nat @ G @ X3 @ Y2 ) )
= ( fun_up6974907568698703286_a_nat @ ( comp_n848383228264675746at_nat @ F @ G ) @ X3 @ ( F @ Y2 ) ) ) ).
% fun_upd_comp
thf(fact_308_fun__upd__comp,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,X3: produc5454855657571582244_a_nat,Y2: nat] :
( ( comp_n6276394432912108976_a_nat @ F @ ( fun_up7352287552899701875at_nat @ G @ X3 @ Y2 ) )
= ( fun_up19502095292272284_a_nat @ ( comp_n6276394432912108976_a_nat @ F @ G ) @ X3 @ ( F @ Y2 ) ) ) ).
% fun_upd_comp
thf(fact_309_fun__upd__comp,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,X3: nat,Y2: nat] :
( ( comp_n1522318729830440540at_nat @ F @ ( fun_upd_nat_nat @ G @ X3 @ Y2 ) )
= ( fun_up180537416982607344_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ X3 @ ( F @ Y2 ) ) ) ).
% fun_upd_comp
thf(fact_310_ad__agr__list__link,axiom,
! [Ns: list_nat,AD: set_a,Sigma2: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ns ) @ AD @ Sigma2 @ Tau )
= ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau @ Ns ) ) ) ).
% ad_agr_list_link
thf(fact_311_fo__nmlz__map,axiom,
! [AD: set_a,Sigma2: nat > sum_sum_a_nat,Ns: list_nat] :
? [Tau2: nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) )
= ( map_na823391071729141993_a_nat @ Tau2 @ Ns ) ) ).
% fo_nmlz_map
thf(fact_312_fo__nmlz__map,axiom,
! [AD: set_a,Sigma2: produc5454855657571582244_a_nat > sum_sum_a_nat,Ns: list_P5056861408695629236_a_nat] :
? [Tau2: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ns ) )
= ( map_Pr2515564165894599445_a_nat @ Tau2 @ Ns ) ) ).
% fo_nmlz_map
thf(fact_313_fo__nmlz__map,axiom,
! [AD: set_a,Sigma2: option_Sum_sum_a_nat > sum_sum_a_nat,Ns: list_o6767316136206324043_a_nat] :
? [Tau2: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_op8672217432457595070_a_nat @ Sigma2 @ Ns ) )
= ( map_op8672217432457595070_a_nat @ Tau2 @ Ns ) ) ).
% fo_nmlz_map
thf(fact_314_fo__nmlz__map,axiom,
! [AD: set_a,Sigma2: sum_sum_a_nat > sum_sum_a_nat,Ns: list_Sum_sum_a_nat] :
? [Tau2: sum_sum_a_nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ns ) )
= ( map_Su2790769393171190532_a_nat @ Tau2 @ Ns ) ) ).
% fo_nmlz_map
thf(fact_315_assms_I6_J,axiom,
( ass
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( proj_v3643391342904276326_a_nat @ r @ fv_sub ) ) ) ).
% assms(6)
thf(fact_316_vs__norm,axiom,
fo_nmlzd_a @ ad @ xs ).
% vs_norm
thf(fact_317_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_318_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_319_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_320_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_321_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_322_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_323_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_324_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_325_Un__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_326_image__eqI,axiom,
! [B: nat,F: nat > nat,X3: nat,A2: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_327_image__eqI,axiom,
! [B: nat > sum_sum_a_nat,F: nat > nat > sum_sum_a_nat,X3: nat,A2: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ B @ ( image_1051037728736664655_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_328_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,X3: nat,A2: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ B @ ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_329_image__eqI,axiom,
! [B: nat,F: ( nat > sum_sum_a_nat ) > nat,X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( image_5786201776793816049at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_330_image__eqI,axiom,
! [B: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ B @ ( image_6222892899998961285_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_331_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ B @ ( image_6721470456781115300_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_332_image__eqI,axiom,
! [B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member5553968465346197646_a_nat @ B @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_333_image__eqI,axiom,
! [B: nat,F: list_Sum_sum_a_nat > nat,X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_334_image__eqI,axiom,
! [B: nat > sum_sum_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ B @ ( image_701559317304863014_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_335_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_336_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_337_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_338_all__not__in__conv,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( ! [X: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ X @ A2 ) )
= ( A2 = bot_bo6441361344521902642_a_nat ) ) ).
% all_not_in_conv
thf(fact_339_all__not__in__conv,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ! [X: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X @ A2 ) )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% all_not_in_conv
thf(fact_340_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_341_empty__iff,axiom,
! [C: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ C @ bot_bo6441361344521902642_a_nat ) ).
% empty_iff
thf(fact_342_empty__iff,axiom,
! [C: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_343_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_344_Int__iff,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) )
= ( ( member8690443509505302927_a_nat @ C @ A2 )
& ( member8690443509505302927_a_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_345_Int__iff,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
= ( ( member408289922725080238_a_nat @ C @ A2 )
& ( member408289922725080238_a_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_346_Int__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_347_IntI,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ A2 )
=> ( ( member8690443509505302927_a_nat @ C @ B2 )
=> ( member8690443509505302927_a_nat @ C @ ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_348_IntI,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A2 )
=> ( ( member408289922725080238_a_nat @ C @ B2 )
=> ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_349_IntI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_350_Un__iff,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( sup_su3329769938372955546_a_nat @ A2 @ B2 ) )
= ( ( member8690443509505302927_a_nat @ C @ A2 )
| ( member8690443509505302927_a_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_351_Un__iff,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( ( member408289922725080238_a_nat @ C @ A2 )
| ( member408289922725080238_a_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_352_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_353_UnCI,axiom,
! [C: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( ~ ( member8690443509505302927_a_nat @ C @ B2 )
=> ( member8690443509505302927_a_nat @ C @ A2 ) )
=> ( member8690443509505302927_a_nat @ C @ ( sup_su3329769938372955546_a_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_354_UnCI,axiom,
! [C: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ C @ B2 )
=> ( member408289922725080238_a_nat @ C @ A2 ) )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_355_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_356_image__is__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A2 )
= bot_bo1033123847703346641_a_nat )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% image_is_empty
thf(fact_357_image__is__empty,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( image_7676043921908783909_a_nat @ F @ A2 )
= bot_bo3237059034911209905_a_nat )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% image_is_empty
thf(fact_358_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_359_empty__is__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% empty_is_image
thf(fact_360_empty__is__image,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( bot_bo3237059034911209905_a_nat
= ( image_7676043921908783909_a_nat @ F @ A2 ) )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% empty_is_image
thf(fact_361_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_362_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_363_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_364_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_365_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_366_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_367_list_Oset__map,axiom,
! [F: nat > produc5454855657571582244_a_nat,V: list_nat] :
( ( set_Pr2196415342531872975_a_nat @ ( map_na4983825868277236666_a_nat @ F @ V ) )
= ( image_4603156464795991419_a_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_368_list_Oset__map,axiom,
! [F: nat > option_Sum_sum_a_nat,V: list_nat] :
( ( set_op1665301557670659034_a_nat @ ( map_na4778969939102885679_a_nat @ F @ V ) )
= ( image_2070734891404281902_a_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_369_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_370_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_371_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_372_list_Oset__map,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,V: list_o6767316136206324043_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_op8672217432457595070_a_nat @ F @ V ) )
= ( image_6550528888111247359_a_nat @ F @ ( set_op1665301557670659034_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_373_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_374_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_375_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_376_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: nat > sum_sum_a_nat,F: nat > nat > sum_sum_a_nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member8690443509505302927_a_nat @ B @ ( image_1051037728736664655_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_377_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_378_rev__image__eqI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: nat,F: ( nat > sum_sum_a_nat ) > nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_5786201776793816049at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_379_rev__image__eqI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member8690443509505302927_a_nat @ B @ ( image_6222892899998961285_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_380_rev__image__eqI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: list_Sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_6721470456781115300_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_381_rev__image__eqI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member5553968465346197646_a_nat @ B @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_382_rev__image__eqI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_383_rev__image__eqI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: nat > sum_sum_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member8690443509505302927_a_nat @ B @ ( image_701559317304863014_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_384_rev__image__eqI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_385_ball__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_386_ball__imageD,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ! [X2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X2 @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_387_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_388_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_389_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 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_5081948215111134021_a_nat @ F @ M )
= ( image_5081948215111134021_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_390_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 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_7676043921908783909_a_nat @ F @ M )
= ( image_7676043921908783909_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_391_bex__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_392_bex__imageD,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ? [X4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_393_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_394_image__iff,axiom,
! [Z2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Z2 @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_395_image__iff,axiom,
! [Z2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_396_image__iff,axiom,
! [Z2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ Z2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_397_imageI,axiom,
! [X3: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_398_imageI,axiom,
! [X3: nat,A2: set_nat,F: nat > nat > sum_sum_a_nat] :
( ( member_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F @ X3 ) @ ( image_1051037728736664655_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_399_imageI,axiom,
! [X3: nat,A2: set_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_400_imageI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ ( image_5786201776793816049at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_401_imageI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F @ X3 ) @ ( image_6222892899998961285_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_402_imageI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ ( image_6721470456781115300_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_403_imageI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member5553968465346197646_a_nat @ ( F @ X3 ) @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_404_imageI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_405_imageI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F @ X3 ) @ ( image_701559317304863014_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_406_imageI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_407_image__Un,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_408_image__Un,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( sup_su499249268922660121_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) @ ( image_7676043921908783909_a_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_409_image__Un,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_410_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_411_image__comp,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,R: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( image_nat_nat @ G @ R ) )
= ( image_7293268710728258664_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_412_image__comp,axiom,
! [F: nat > option_Sum_sum_a_nat,G: nat > nat,R: set_nat] :
( ( image_2070734891404281902_a_nat @ F @ ( image_nat_nat @ G @ R ) )
= ( image_2070734891404281902_a_nat @ ( comp_n848383228264675746at_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_413_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_414_image__comp,axiom,
! [F: list_nat > list_Sum_sum_a_nat,G: list_nat > list_nat,R: set_list_nat] :
( ( image_1891931749380356958_a_nat @ F @ ( image_7976474329151083847st_nat @ G @ R ) )
= ( image_1891931749380356958_a_nat @ ( comp_l4895937012117054562st_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_415_image__comp,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,R: set_Pr1021281732701644634_a_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( image_1677410150705809403at_nat @ G @ R ) )
= ( image_5908522764169838868_a_nat @ ( comp_n6276394432912108976_a_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_416_image__comp,axiom,
! [F: list_nat > list_Sum_sum_a_nat,G: list_P5056861408695629236_a_nat > list_nat,R: set_li5408732054060274282_a_nat] :
( ( image_1891931749380356958_a_nat @ F @ ( image_8202059914250588955st_nat @ G @ R ) )
= ( image_5519482102954158218_a_nat @ ( comp_l7562833077401626166_a_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_417_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_418_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_419_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_420_image__eq__imp__comp,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat,H: nat > nat] :
( ( ( image_nat_nat @ F @ A2 )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F ) @ A2 )
= ( image_nat_nat @ ( comp_nat_nat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_421_image__eq__imp__comp,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat,H: nat > sum_sum_a_nat] :
( ( ( image_nat_nat @ F @ A2 )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_7293268710728258664_a_nat @ ( comp_n1522318729830440540at_nat @ H @ F ) @ A2 )
= ( image_7293268710728258664_a_nat @ ( comp_n1522318729830440540at_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_422_image__eq__imp__comp,axiom,
! [F: nat > nat,A2: set_nat,G: nat > nat,B2: set_nat,H: nat > option_Sum_sum_a_nat] :
( ( ( image_nat_nat @ F @ A2 )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_2070734891404281902_a_nat @ ( comp_n848383228264675746at_nat @ H @ F ) @ A2 )
= ( image_2070734891404281902_a_nat @ ( comp_n848383228264675746at_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_423_image__eq__imp__comp,axiom,
! [F: nat > option_Sum_sum_a_nat,A2: set_nat,G: nat > option_Sum_sum_a_nat,B2: set_nat,H: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( ( image_2070734891404281902_a_nat @ F @ A2 )
= ( image_2070734891404281902_a_nat @ G @ B2 ) )
=> ( ( image_7293268710728258664_a_nat @ ( comp_o2721059924169077421at_nat @ H @ F ) @ A2 )
= ( image_7293268710728258664_a_nat @ ( comp_o2721059924169077421at_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_424_image__eq__imp__comp,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,G: list_nat > list_nat,B2: set_list_nat,H: list_nat > list_Sum_sum_a_nat] :
( ( ( image_7976474329151083847st_nat @ F @ A2 )
= ( image_7976474329151083847st_nat @ G @ B2 ) )
=> ( ( image_1891931749380356958_a_nat @ ( comp_l4895937012117054562st_nat @ H @ F ) @ A2 )
= ( image_1891931749380356958_a_nat @ ( comp_l4895937012117054562st_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_425_image__eq__imp__comp,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: set_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,H: list_Sum_sum_a_nat > nat] :
( ( ( image_6262589752765146990_a_nat @ F @ A2 )
= ( image_5081948215111134021_a_nat @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_l6255619850357176452at_nat @ H @ F ) @ A2 )
= ( image_2535339886381165584at_nat @ ( comp_l6309227251058912209_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_426_image__eq__imp__comp,axiom,
! [F: produc5454855657571582244_a_nat > nat,A2: set_Pr1021281732701644634_a_nat,G: nat > nat,B2: set_nat,H: nat > sum_sum_a_nat] :
( ( ( image_1677410150705809403at_nat @ F @ A2 )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_5908522764169838868_a_nat @ ( comp_n6276394432912108976_a_nat @ H @ F ) @ A2 )
= ( image_7293268710728258664_a_nat @ ( comp_n1522318729830440540at_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_427_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat,G: nat > nat,B2: set_nat,H: nat > list_Sum_sum_a_nat] :
( ( ( image_2535339886381165584at_nat @ F @ A2 )
= ( image_nat_nat @ G @ B2 ) )
=> ( ( image_5081948215111134021_a_nat @ ( comp_n8501611615020050419_a_nat @ H @ F ) @ A2 )
= ( image_6262589752765146990_a_nat @ ( comp_n3932894967291182050at_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_428_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,G: nat > list_Sum_sum_a_nat,B2: set_nat,H: list_Sum_sum_a_nat > nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A2 )
= ( image_6262589752765146990_a_nat @ G @ B2 ) )
=> ( ( image_2535339886381165584at_nat @ ( comp_l6309227251058912209_a_nat @ H @ F ) @ A2 )
= ( image_nat_nat @ ( comp_l6255619850357176452at_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_429_image__eq__imp__comp,axiom,
! [F: nat > nat,A2: set_nat,G: produc5454855657571582244_a_nat > nat,B2: set_Pr1021281732701644634_a_nat,H: nat > sum_sum_a_nat] :
( ( ( image_nat_nat @ F @ A2 )
= ( image_1677410150705809403at_nat @ G @ B2 ) )
=> ( ( image_7293268710728258664_a_nat @ ( comp_n1522318729830440540at_nat @ H @ F ) @ A2 )
= ( image_5908522764169838868_a_nat @ ( comp_n6276394432912108976_a_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_430_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_431_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_432_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_433_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_434_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_435_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_436_image__set,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_o6767316136206324043_a_nat] :
( ( image_6550528888111247359_a_nat @ F @ ( set_op1665301557670659034_a_nat @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_op8672217432457595070_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_437_image__set,axiom,
! [F: nat > produc5454855657571582244_a_nat,Xs: list_nat] :
( ( image_4603156464795991419_a_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_Pr2196415342531872975_a_nat @ ( map_na4983825868277236666_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_438_image__set,axiom,
! [F: nat > option_Sum_sum_a_nat,Xs: list_nat] :
( ( image_2070734891404281902_a_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_op1665301557670659034_a_nat @ ( map_na4778969939102885679_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_439_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_440_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_441_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_442_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_443_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_444_ex__in__conv,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( ? [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A2 ) )
= ( A2 != bot_bo6441361344521902642_a_nat ) ) ).
% ex_in_conv
thf(fact_445_ex__in__conv,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ? [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A2 ) )
= ( A2 != bot_bo1033123847703346641_a_nat ) ) ).
% ex_in_conv
thf(fact_446_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_447_equals0I,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ! [Y3: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ Y3 @ A2 )
=> ( A2 = bot_bo6441361344521902642_a_nat ) ) ).
% equals0I
thf(fact_448_equals0I,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ! [Y3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y3 @ A2 )
=> ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_449_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_450_equals0D,axiom,
! [A2: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat] :
( ( A2 = bot_bo6441361344521902642_a_nat )
=> ~ ( member8690443509505302927_a_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_451_equals0D,axiom,
! [A2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat] :
( ( A2 = bot_bo1033123847703346641_a_nat )
=> ~ ( member408289922725080238_a_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_452_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_453_emptyE,axiom,
! [A: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ).
% emptyE
thf(fact_454_emptyE,axiom,
! [A: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_455_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_456_Int__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_457_Int__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_458_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_459_Int__absorb,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_460_Int__assoc,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 ) ) ) ).
% Int_assoc
thf(fact_461_IntD2,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) )
=> ( member8690443509505302927_a_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_462_IntD2,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
=> ( member408289922725080238_a_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_463_IntD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_464_IntD1,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) )
=> ( member8690443509505302927_a_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_465_IntD1,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
=> ( member408289922725080238_a_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_466_IntD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_467_IntE,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) )
=> ~ ( ( member8690443509505302927_a_nat @ C @ A2 )
=> ~ ( member8690443509505302927_a_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_468_IntE,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
=> ~ ( ( member408289922725080238_a_nat @ C @ A2 )
=> ~ ( member408289922725080238_a_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_469_IntE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_470_Un__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_471_Un__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_472_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_473_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_474_Un__assoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_475_ball__Un,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( P @ X ) )
& ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_476_bex__Un,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) )
| ? [X: nat] :
( ( member_nat @ X @ B2 )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_477_UnI2,axiom,
! [C: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ B2 )
=> ( member8690443509505302927_a_nat @ C @ ( sup_su3329769938372955546_a_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_478_UnI2,axiom,
! [C: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ B2 )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_479_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_480_UnI1,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ A2 )
=> ( member8690443509505302927_a_nat @ C @ ( sup_su3329769938372955546_a_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_481_UnI1,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A2 )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_482_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_483_UnE,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( sup_su3329769938372955546_a_nat @ A2 @ B2 ) )
=> ( ~ ( member8690443509505302927_a_nat @ C @ A2 )
=> ( member8690443509505302927_a_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_484_UnE,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
=> ( ~ ( member408289922725080238_a_nat @ C @ A2 )
=> ( member408289922725080238_a_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_485_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_486_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y: nat] :
( ( member_nat @ Y @ B2 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_487_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_488_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_489_disjoint__iff,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ( inf_in8399021836546144180_a_nat @ A2 @ B2 )
= bot_bo6441361344521902642_a_nat )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A2 )
=> ~ ( member8690443509505302927_a_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_490_disjoint__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A2 @ B2 )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ~ ( member408289922725080238_a_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_491_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_492_Int__emptyI,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ~ ( member8690443509505302927_a_nat @ X2 @ B2 ) )
=> ( ( inf_in8399021836546144180_a_nat @ A2 @ B2 )
= bot_bo6441361344521902642_a_nat ) ) ).
% Int_emptyI
thf(fact_493_Int__emptyI,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ~ ( member408289922725080238_a_nat @ X2 @ B2 ) )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ B2 )
= bot_bo1033123847703346641_a_nat ) ) ).
% Int_emptyI
thf(fact_494_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_495_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_496_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_497_Un__Int__distrib2,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ C2 ) @ A2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ A2 ) @ ( sup_sup_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_498_Int__Un__distrib2,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ A2 ) @ ( inf_inf_set_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_499_Un__Int__distrib,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_500_Int__Un__distrib,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_501_Un__Int__crazy,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A2 ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ B2 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_502_sup__inf__absorb,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y2 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_503_inf__sup__absorb,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_504_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_505_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_506_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_507_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_508_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_509_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_510_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_511_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_512_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_513_sup__idem,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_514_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_515_sup__left__idem,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_516_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_517_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_518_inf__idem,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_519_inf_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_520_inf__left__idem,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y2 ) )
= ( inf_inf_set_nat @ X3 @ Y2 ) ) ).
% inf_left_idem
thf(fact_521_inf_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_522_inf__right__idem,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Y2 )
= ( inf_inf_set_nat @ X3 @ Y2 ) ) ).
% inf_right_idem
thf(fact_523_sup__bot__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_524_sup__bot__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_525_bot__eq__sup__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_526_sup__eq__bot__iff,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y2 )
= bot_bot_set_nat )
= ( ( X3 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_527_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_528_inf__sup__aci_I8_J,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) )
= ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_529_inf__sup__aci_I7_J,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_530_inf__sup__aci_I6_J,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_531_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_532_sup_Oassoc,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 ) ) ) ).
% sup.assoc
thf(fact_533_sup__assoc,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_534_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_535_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X ) ) ) ).
% sup_commute
thf(fact_536_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_537_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( sup_sup_set_nat @ K @ B ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_538_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_539_sup__left__commute,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_540_inf__sup__aci_I4_J,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y2 ) )
= ( inf_inf_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_541_inf__sup__aci_I3_J,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ Y2 @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_542_inf__sup__aci_I2_J,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_543_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y: set_nat] : ( inf_inf_set_nat @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_544_inf_Oassoc,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 ) ) ) ).
% inf.assoc
thf(fact_545_inf__assoc,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_546_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_547_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y: set_nat] : ( inf_inf_set_nat @ Y @ X ) ) ) ).
% inf_commute
thf(fact_548_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( inf_inf_set_nat @ K @ A ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_549_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( inf_inf_set_nat @ K @ B ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_550_inf_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_551_inf__left__commute,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ Y2 @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_552_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_553_distrib__imp1,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) )
=> ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_554_distrib__imp2,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) )
=> ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_555_inf__sup__distrib1,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_556_inf__sup__distrib2,axiom,
! [Y2: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_557_sup__inf__distrib1,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_558_sup__inf__distrib2,axiom,
! [Y2: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_559_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_560_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_561_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_562_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_563_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_564_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_565_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_566_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A2: set_nat] :
( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_567_Sup_OSUP__image,axiom,
! [Sup: set_Sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,F: nat > nat,A2: set_nat] :
( ( Sup @ ( image_7293268710728258664_a_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
= ( Sup @ ( image_7293268710728258664_a_nat @ ( comp_n1522318729830440540at_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_568_Sup_OSUP__image,axiom,
! [Sup: set_op253850770698000421_a_nat > option_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,F: nat > nat,A2: set_nat] :
( ( Sup @ ( image_2070734891404281902_a_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
= ( Sup @ ( image_2070734891404281902_a_nat @ ( comp_n848383228264675746at_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_569_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,A2: set_nat] :
( ( Sup @ ( image_6550528888111247359_a_nat @ G @ ( image_2070734891404281902_a_nat @ F @ A2 ) ) )
= ( Sup @ ( image_7293268710728258664_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_570_Sup_OSUP__image,axiom,
! [Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_nat > list_Sum_sum_a_nat,F: list_nat > list_nat,A2: set_list_nat] :
( ( Sup @ ( image_1891931749380356958_a_nat @ G @ ( image_7976474329151083847st_nat @ F @ A2 ) ) )
= ( Sup @ ( image_1891931749380356958_a_nat @ ( comp_l4895937012117054562st_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_571_Sup_OSUP__image,axiom,
! [Sup: set_Sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,F: produc5454855657571582244_a_nat > nat,A2: set_Pr1021281732701644634_a_nat] :
( ( Sup @ ( image_7293268710728258664_a_nat @ G @ ( image_1677410150705809403at_nat @ F @ A2 ) ) )
= ( Sup @ ( image_5908522764169838868_a_nat @ ( comp_n6276394432912108976_a_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_572_Sup_OSUP__image,axiom,
! [Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_nat > list_Sum_sum_a_nat,F: list_P5056861408695629236_a_nat > list_nat,A2: set_li5408732054060274282_a_nat] :
( ( Sup @ ( image_1891931749380356958_a_nat @ G @ ( image_8202059914250588955st_nat @ F @ A2 ) ) )
= ( Sup @ ( image_5519482102954158218_a_nat @ ( comp_l7562833077401626166_a_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_573_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,A2: set_li6526943997496501093_a_nat] :
( ( Sup @ ( image_5081948215111134021_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_574_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,A2: set_li6526943997496501093_a_nat] :
( ( Sup @ ( image_1711104591301794661_a_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ ( comp_s2318311659520683906_a_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_575_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,A2: set_li6526943997496501093_a_nat] :
( ( Sup @ ( image_7676043921908783909_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) )
= ( Sup @ ( image_7676043921908783909_a_nat @ ( comp_l6800670225517141954_a_nat @ G @ F ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_576_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A2: set_nat] :
( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_577_Inf_OINF__image,axiom,
! [Inf: set_Sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,F: nat > nat,A2: set_nat] :
( ( Inf @ ( image_7293268710728258664_a_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
= ( Inf @ ( image_7293268710728258664_a_nat @ ( comp_n1522318729830440540at_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_578_Inf_OINF__image,axiom,
! [Inf: set_op253850770698000421_a_nat > option_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,F: nat > nat,A2: set_nat] :
( ( Inf @ ( image_2070734891404281902_a_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
= ( Inf @ ( image_2070734891404281902_a_nat @ ( comp_n848383228264675746at_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_579_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,A2: set_nat] :
( ( Inf @ ( image_6550528888111247359_a_nat @ G @ ( image_2070734891404281902_a_nat @ F @ A2 ) ) )
= ( Inf @ ( image_7293268710728258664_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_580_Inf_OINF__image,axiom,
! [Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_nat > list_Sum_sum_a_nat,F: list_nat > list_nat,A2: set_list_nat] :
( ( Inf @ ( image_1891931749380356958_a_nat @ G @ ( image_7976474329151083847st_nat @ F @ A2 ) ) )
= ( Inf @ ( image_1891931749380356958_a_nat @ ( comp_l4895937012117054562st_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_581_Inf_OINF__image,axiom,
! [Inf: set_Sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,F: produc5454855657571582244_a_nat > nat,A2: set_Pr1021281732701644634_a_nat] :
( ( Inf @ ( image_7293268710728258664_a_nat @ G @ ( image_1677410150705809403at_nat @ F @ A2 ) ) )
= ( Inf @ ( image_5908522764169838868_a_nat @ ( comp_n6276394432912108976_a_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_582_Inf_OINF__image,axiom,
! [Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_nat > list_Sum_sum_a_nat,F: list_P5056861408695629236_a_nat > list_nat,A2: set_li5408732054060274282_a_nat] :
( ( Inf @ ( image_1891931749380356958_a_nat @ G @ ( image_8202059914250588955st_nat @ F @ A2 ) ) )
= ( Inf @ ( image_5519482102954158218_a_nat @ ( comp_l7562833077401626166_a_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_583_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,A2: set_li6526943997496501093_a_nat] :
( ( Inf @ ( image_5081948215111134021_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_584_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,A2: set_li6526943997496501093_a_nat] :
( ( Inf @ ( image_1711104591301794661_a_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ ( comp_s2318311659520683906_a_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_585_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,A2: set_li6526943997496501093_a_nat] :
( ( Inf @ ( image_7676043921908783909_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) )
= ( Inf @ ( image_7676043921908783909_a_nat @ ( comp_l6800670225517141954_a_nat @ G @ F ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_586__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_587_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_588_map__map__of,axiom,
! [Xs: list_o6767316136206324043_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s7896677892106675127_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin4230969608812606142_a_nat @ Xs )
=> ( Ys
= ( map_op8672217432457595070_a_nat @ ( comp_o8686311273808465576_a_nat @ the_Sum_sum_a_nat @ ( map_of421381886531501277_a_nat @ ( zip_op1551033228026260913_a_nat @ Xs @ Ys ) ) ) @ Xs ) ) ) ) ).
% map_map_of
thf(fact_589_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_590_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_591_map__map__of,axiom,
! [Xs: list_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s4076174644546656840_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ( Ys
= ( map_na4983825868277236666_a_nat @ ( comp_o7556894165571034829at_nat @ the_Pr1944682135967112592_a_nat @ ( map_of3667836346982782297_a_nat @ ( zip_na7231373358936250413_a_nat @ Xs @ Ys ) ) ) @ Xs ) ) ) ) ).
% map_map_of
thf(fact_592_map__map__of,axiom,
! [Xs: list_nat,Ys: list_o6767316136206324043_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s7896677892106675127_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ( Ys
= ( map_na4778969939102885679_a_nat @ ( comp_o3275820428381651373at_nat @ the_op4356739967077272025_a_nat @ ( map_of5350075150544978640_a_nat @ ( zip_na2714845588454090492_a_nat @ Xs @ Ys ) ) ) @ Xs ) ) ) ) ).
% map_map_of
thf(fact_593_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_594_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_595_nall__tuplesI,axiom,
! [Vs: list_Sum_sum_a_nat,N3: nat,AD: set_a] :
( ( ( size_s5283204784079214577_a_nat @ Vs )
= N3 )
=> ( ( fo_nmlzd_a @ AD @ Vs )
=> ( member408289922725080238_a_nat @ Vs @ ( nall_tuples_a @ AD @ N3 ) ) ) ) ).
% nall_tuplesI
thf(fact_596_vimageI,axiom,
! [F: nat > nat,A: nat,B: nat,B2: set_nat] :
( ( ( F @ A )
= B )
=> ( ( member_nat @ B @ B2 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_597_vimageI,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat,A: nat > sum_sum_a_nat,B: nat,B2: set_nat] :
( ( ( F @ A )
= B )
=> ( ( member_nat @ B @ B2 )
=> ( member8690443509505302927_a_nat @ A @ ( vimage1507083042557951675at_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_598_vimageI,axiom,
! [F: list_Sum_sum_a_nat > nat,A: list_Sum_sum_a_nat,B: nat,B2: set_nat] :
( ( ( F @ A )
= B )
=> ( ( member_nat @ B @ B2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_599_vimageI,axiom,
! [F: nat > nat > sum_sum_a_nat,A: nat,B: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member8690443509505302927_a_nat @ B @ B2 )
=> ( member_nat @ A @ ( vimage5995291031355576089_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_600_vimageI,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,A: nat > sum_sum_a_nat,B: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member8690443509505302927_a_nat @ B @ B2 )
=> ( member8690443509505302927_a_nat @ A @ ( vimage8281748387122106427_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_601_vimageI,axiom,
! [F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,A: list_Sum_sum_a_nat,B: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member8690443509505302927_a_nat @ B @ B2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage2518184964943814876_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_602_vimageI,axiom,
! [F: nat > list_Sum_sum_a_nat,A: nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member408289922725080238_a_nat @ B @ B2 )
=> ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_603_vimageI,axiom,
! [F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,A: nat > sum_sum_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member408289922725080238_a_nat @ B @ B2 )
=> ( member8690443509505302927_a_nat @ A @ ( vimage8538096104420067162_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_604_vimageI,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member408289922725080238_a_nat @ B @ B2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage7538740559230589179_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_605_vimageI,axiom,
! [F: nat > sum_sum_a_nat,A: nat,B: sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( F @ A )
= B )
=> ( ( member_Sum_sum_a_nat @ B @ B2 )
=> ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B2 ) ) ) ) ).
% vimageI
thf(fact_606_vimage__eq,axiom,
! [A: nat,F: nat > nat,B2: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ B2 ) )
= ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_607_vimage__eq,axiom,
! [A: nat,F: nat > nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member_nat @ A @ ( vimage5995291031355576089_a_nat @ F @ B2 ) )
= ( member8690443509505302927_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_608_vimage__eq,axiom,
! [A: nat,F: nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ B2 ) )
= ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_609_vimage__eq,axiom,
! [A: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat,B2: set_nat] :
( ( member8690443509505302927_a_nat @ A @ ( vimage1507083042557951675at_nat @ F @ B2 ) )
= ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_610_vimage__eq,axiom,
! [A: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( vimage8281748387122106427_a_nat @ F @ B2 ) )
= ( member8690443509505302927_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_611_vimage__eq,axiom,
! [A: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( vimage8538096104420067162_a_nat @ F @ B2 ) )
= ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_612_vimage__eq,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,B2: set_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ B2 ) )
= ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_613_vimage__eq,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage2518184964943814876_a_nat @ F @ B2 ) )
= ( member8690443509505302927_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_614_vimage__eq,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage7538740559230589179_a_nat @ F @ B2 ) )
= ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_615_vimage__eq,axiom,
! [A: nat,F: nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B2 ) )
= ( member_Sum_sum_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimage_eq
thf(fact_616_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_617_Sup__bot__conv_I2_J,axiom,
! [A2: set_se4330304633200676677_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( comple1686717674086456018_a_nat @ A2 ) )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A2 )
=> ( X = bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_618_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_619_Sup__bot__conv_I1_J,axiom,
! [A2: set_se4330304633200676677_a_nat] :
( ( ( comple1686717674086456018_a_nat @ A2 )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A2 )
=> ( X = bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_620_Union__Un__distrib,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_621_Union__Un__distrib,axiom,
! [A2: set_se4330304633200676677_a_nat,B2: set_se4330304633200676677_a_nat] :
( ( comple1686717674086456018_a_nat @ ( sup_su499249268922660121_a_nat @ A2 @ B2 ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ A2 ) @ ( comple1686717674086456018_a_nat @ B2 ) ) ) ).
% Union_Un_distrib
thf(fact_622_ball__UN,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: 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 @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ! [Y: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% ball_UN
thf(fact_623_bex__UN,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: 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 @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ? [Y: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% bex_UN
thf(fact_624_UN__ball__bex__simps_I2_J,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: 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 @ B2 @ A2 ) ) )
=> ( P @ X ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ! [Y: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y @ ( B2 @ X ) )
=> ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_625_UN__ball__bex__simps_I4_J,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: 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 @ B2 @ A2 ) ) )
& ( P @ X ) ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ? [Y: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y @ ( B2 @ X ) )
& ( P @ Y ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_626_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_627_vimage__empty,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_628_vimage__Int,axiom,
! [F: nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A2 ) @ ( vimage3040984495076556338_a_nat @ F @ B2 ) ) ) ).
% vimage_Int
thf(fact_629_vimage__Int,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( vimage_nat_nat @ F @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A2 ) @ ( vimage_nat_nat @ F @ B2 ) ) ) ).
% vimage_Int
thf(fact_630_vimage__Un,axiom,
! [F: nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A2 ) @ ( vimage3040984495076556338_a_nat @ F @ B2 ) ) ) ).
% vimage_Un
thf(fact_631_vimage__Un,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( vimage_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ A2 ) @ ( vimage_nat_nat @ F @ B2 ) ) ) ).
% vimage_Un
thf(fact_632_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_633_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_634_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_635_Sup__empty,axiom,
( ( comple1686717674086456018_a_nat @ bot_bo3237059034911209905_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% Sup_empty
thf(fact_636_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > set_li6526943997496501093_a_nat,D2: nat > set_li6526943997496501093_a_nat] :
( ( A2 = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ C2 @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_637_SUP__cong,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,C2: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,D2: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ( A2 = B2 )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ C2 @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_638_SUP__cong,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,D2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( A2 = B2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ C2 @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_639_SUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat] :
( ( A2 = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C2 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_640_SUP__cong,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,C2: ( nat > sum_sum_a_nat ) > nat,D2: ( nat > sum_sum_a_nat ) > nat] :
( ( A2 = B2 )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ C2 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_641_SUP__cong,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C2: list_Sum_sum_a_nat > nat,D2: list_Sum_sum_a_nat > nat] :
( ( A2 = B2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ C2 @ A2 ) )
= ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ D2 @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_642_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_643_Union__empty,axiom,
( ( comple1686717674086456018_a_nat @ bot_bo3237059034911209905_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% Union_empty
thf(fact_644_Union__empty__conv,axiom,
! [A2: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A2 )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_645_Union__empty__conv,axiom,
! [A2: set_se4330304633200676677_a_nat] :
( ( ( comple1686717674086456018_a_nat @ A2 )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A2 )
=> ( X = bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_646_empty__Union__conv,axiom,
! [A2: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_647_empty__Union__conv,axiom,
! [A2: set_se4330304633200676677_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( comple1686717674086456018_a_nat @ A2 ) )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A2 )
=> ( X = bot_bo1033123847703346641_a_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_648_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_649_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_650_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_651_vimageD,axiom,
! [A: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ A2 ) )
=> ( member_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_652_vimageD,axiom,
! [A: nat,F: nat > nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member_nat @ A @ ( vimage5995291031355576089_a_nat @ F @ A2 ) )
=> ( member8690443509505302927_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_653_vimageD,axiom,
! [A: nat,F: nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ A2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_654_vimageD,axiom,
! [A: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat,A2: set_nat] :
( ( member8690443509505302927_a_nat @ A @ ( vimage1507083042557951675at_nat @ F @ A2 ) )
=> ( member_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_655_vimageD,axiom,
! [A: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( vimage8281748387122106427_a_nat @ F @ A2 ) )
=> ( member8690443509505302927_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_656_vimageD,axiom,
! [A: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( vimage8538096104420067162_a_nat @ F @ A2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_657_vimageD,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,A2: set_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ A2 ) )
=> ( member_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_658_vimageD,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage2518184964943814876_a_nat @ F @ A2 ) )
=> ( member8690443509505302927_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_659_vimageD,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage7538740559230589179_a_nat @ F @ A2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_660_vimageD,axiom,
! [A: nat,F: nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ A2 ) )
=> ( member_Sum_sum_a_nat @ ( F @ A ) @ A2 ) ) ).
% vimageD
thf(fact_661_vimageE,axiom,
! [A: nat,F: nat > nat,B2: set_nat] :
( ( member_nat @ A @ ( vimage_nat_nat @ F @ B2 ) )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_662_vimageE,axiom,
! [A: nat,F: nat > nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member_nat @ A @ ( vimage5995291031355576089_a_nat @ F @ B2 ) )
=> ( member8690443509505302927_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_663_vimageE,axiom,
! [A: nat,F: nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ B2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_664_vimageE,axiom,
! [A: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat,B2: set_nat] :
( ( member8690443509505302927_a_nat @ A @ ( vimage1507083042557951675at_nat @ F @ B2 ) )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_665_vimageE,axiom,
! [A: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( vimage8281748387122106427_a_nat @ F @ B2 ) )
=> ( member8690443509505302927_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_666_vimageE,axiom,
! [A: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( vimage8538096104420067162_a_nat @ F @ B2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_667_vimageE,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,B2: set_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ B2 ) )
=> ( member_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_668_vimageE,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage2518184964943814876_a_nat @ F @ B2 ) )
=> ( member8690443509505302927_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_669_vimageE,axiom,
! [A: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( vimage7538740559230589179_a_nat @ F @ B2 ) )
=> ( member408289922725080238_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_670_vimageE,axiom,
! [A: nat,F: nat > sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B2 ) )
=> ( member_Sum_sum_a_nat @ ( F @ A ) @ B2 ) ) ).
% vimageE
thf(fact_671_vimageI2,axiom,
! [F: nat > nat,A: nat,A2: set_nat] :
( ( member_nat @ ( F @ A ) @ A2 )
=> ( member_nat @ A @ ( vimage_nat_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_672_vimageI2,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat,A: nat > sum_sum_a_nat,A2: set_nat] :
( ( member_nat @ ( F @ A ) @ A2 )
=> ( member8690443509505302927_a_nat @ A @ ( vimage1507083042557951675at_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_673_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > nat,A: list_Sum_sum_a_nat,A2: set_nat] :
( ( member_nat @ ( F @ A ) @ A2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage4660099214859102426at_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_674_vimageI2,axiom,
! [F: nat > nat > sum_sum_a_nat,A: nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ ( F @ A ) @ A2 )
=> ( member_nat @ A @ ( vimage5995291031355576089_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_675_vimageI2,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ ( F @ A ) @ A2 )
=> ( member8690443509505302927_a_nat @ A @ ( vimage8281748387122106427_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_676_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,A: list_Sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ ( F @ A ) @ A2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage2518184964943814876_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_677_vimageI2,axiom,
! [F: nat > list_Sum_sum_a_nat,A: nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A ) @ A2 )
=> ( member_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_678_vimageI2,axiom,
! [F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,A: nat > sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A ) @ A2 )
=> ( member8690443509505302927_a_nat @ A @ ( vimage8538096104420067162_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_679_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A ) @ A2 )
=> ( member408289922725080238_a_nat @ A @ ( vimage7538740559230589179_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_680_vimageI2,axiom,
! [F: nat > sum_sum_a_nat,A: nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A ) @ A2 )
=> ( member_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ A2 ) ) ) ).
% vimageI2
thf(fact_681_vimage__Collect,axiom,
! [P: sum_sum_a_nat > $o,F: nat > sum_sum_a_nat,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage3040984495076556338_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_682_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_683_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_684_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_685_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_686_set_Ocomp,axiom,
! [F: list_P5056861408695629236_a_nat > list_nat,G: list_nat > list_Sum_sum_a_nat] :
( ( comp_s1894432362582861504_a_nat @ ( vimage4515968288297180881st_nat @ F ) @ ( vimage6850595178808314664_a_nat @ G ) )
= ( vimage7285588754567757652_a_nat @ ( comp_l7562833077401626166_a_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_687_set_Ocomp,axiom,
! [F: list_nat > list_nat,G: list_nat > list_Sum_sum_a_nat] :
( ( comp_s3180485948757189140_a_nat @ ( vimage8207059779087378429st_nat @ F ) @ ( vimage6850595178808314664_a_nat @ G ) )
= ( vimage6850595178808314664_a_nat @ ( comp_l4895937012117054562st_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_688_set_Ocomp,axiom,
! [F: nat > nat,G: nat > option_Sum_sum_a_nat] :
( ( comp_s6617926381671661684_a_nat @ ( vimage_nat_nat @ F ) @ ( vimage7014988194023193336_a_nat @ G ) )
= ( vimage7014988194023193336_a_nat @ ( comp_n848383228264675746at_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_689_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_690_set_Ocomp,axiom,
! [F: produc5454855657571582244_a_nat > nat,G: nat > sum_sum_a_nat] :
( ( comp_s5330836805869514842_a_nat @ ( vimage8470460917905013169at_nat @ F ) @ ( vimage3040984495076556338_a_nat @ G ) )
= ( vimage2715053500650648030_a_nat @ ( comp_n6276394432912108976_a_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_691_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_692_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_693_set_Ocompositionality,axiom,
! [F: list_P5056861408695629236_a_nat > list_nat,G: list_nat > list_Sum_sum_a_nat,Set: set_li6526943997496501093_a_nat] :
( ( vimage4515968288297180881st_nat @ F @ ( vimage6850595178808314664_a_nat @ G @ Set ) )
= ( vimage7285588754567757652_a_nat @ ( comp_l7562833077401626166_a_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_694_set_Ocompositionality,axiom,
! [F: list_nat > list_nat,G: list_nat > list_Sum_sum_a_nat,Set: set_li6526943997496501093_a_nat] :
( ( vimage8207059779087378429st_nat @ F @ ( vimage6850595178808314664_a_nat @ G @ Set ) )
= ( vimage6850595178808314664_a_nat @ ( comp_l4895937012117054562st_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_695_set_Ocompositionality,axiom,
! [F: nat > nat,G: nat > option_Sum_sum_a_nat,Set: set_op253850770698000421_a_nat] :
( ( vimage_nat_nat @ F @ ( vimage7014988194023193336_a_nat @ G @ Set ) )
= ( vimage7014988194023193336_a_nat @ ( comp_n848383228264675746at_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_696_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_697_set_Ocompositionality,axiom,
! [F: produc5454855657571582244_a_nat > nat,G: nat > sum_sum_a_nat,Set: set_Sum_sum_a_nat] :
( ( vimage8470460917905013169at_nat @ F @ ( vimage3040984495076556338_a_nat @ G @ Set ) )
= ( vimage2715053500650648030_a_nat @ ( comp_n6276394432912108976_a_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_698_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_699_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_700_vimage__comp,axiom,
! [F: list_P5056861408695629236_a_nat > list_nat,G: list_nat > list_Sum_sum_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( vimage4515968288297180881st_nat @ F @ ( vimage6850595178808314664_a_nat @ G @ X3 ) )
= ( vimage7285588754567757652_a_nat @ ( comp_l7562833077401626166_a_nat @ G @ F ) @ X3 ) ) ).
% vimage_comp
thf(fact_701_vimage__comp,axiom,
! [F: list_nat > list_nat,G: list_nat > list_Sum_sum_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( vimage8207059779087378429st_nat @ F @ ( vimage6850595178808314664_a_nat @ G @ X3 ) )
= ( vimage6850595178808314664_a_nat @ ( comp_l4895937012117054562st_nat @ G @ F ) @ X3 ) ) ).
% vimage_comp
thf(fact_702_vimage__comp,axiom,
! [F: nat > nat,G: nat > option_Sum_sum_a_nat,X3: set_op253850770698000421_a_nat] :
( ( vimage_nat_nat @ F @ ( vimage7014988194023193336_a_nat @ G @ X3 ) )
= ( vimage7014988194023193336_a_nat @ ( comp_n848383228264675746at_nat @ G @ F ) @ X3 ) ) ).
% vimage_comp
thf(fact_703_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_704_vimage__comp,axiom,
! [F: produc5454855657571582244_a_nat > nat,G: nat > sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( vimage8470460917905013169at_nat @ F @ ( vimage3040984495076556338_a_nat @ G @ X3 ) )
= ( vimage2715053500650648030_a_nat @ ( comp_n6276394432912108976_a_nat @ G @ F ) @ X3 ) ) ).
% vimage_comp
thf(fact_705_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_706_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_707_vimage__inter__cong,axiom,
! [S: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,Y2: set_Sum_sum_a_nat] :
( ! [W: nat] :
( ( member_nat @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ F @ Y2 ) @ S )
= ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ G @ Y2 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_708_SUP__eq__const,axiom,
! [I: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( I != bot_bo6441361344521902642_a_nat )
=> ( ! [I2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I2 @ I )
=> ( ( F @ I2 )
= X3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ I ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_709_SUP__eq__const,axiom,
! [I: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( I != bot_bo1033123847703346641_a_nat )
=> ( ! [I2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I2 @ I )
=> ( ( F @ I2 )
= X3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ I ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_710_SUP__eq__const,axiom,
! [I: set_nat,F: nat > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( F @ I2 )
= X3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ I ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_711_SUP__image,axiom,
! [G: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ ( comp_s3377174507952090018_a_nat @ G @ F ) @ A2 ) ) ) ).
% SUP_image
thf(fact_712_SUP__image,axiom,
! [G: nat > set_li6526943997496501093_a_nat,F: nat > nat,A2: set_nat] :
( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
= ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ ( comp_n2527386428755934146at_nat @ G @ F ) @ A2 ) ) ) ).
% SUP_image
thf(fact_713_SUP__image,axiom,
! [G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ ( comp_l6800670225517141954_a_nat @ G @ F ) @ A2 ) ) ) ).
% SUP_image
thf(fact_714_SUP__image,axiom,
! [G: list_Sum_sum_a_nat > nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) )
= ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ ( comp_l6309227251058912209_a_nat @ G @ F ) @ A2 ) ) ) ).
% SUP_image
thf(fact_715_SUP__image,axiom,
! [G: set_li6526943997496501093_a_nat > nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( complete_Sup_Sup_nat @ ( image_4755997627985507312at_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) )
= ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ ( comp_s8210616430925637617_a_nat @ G @ F ) @ A2 ) ) ) ).
% SUP_image
thf(fact_716_SUP__image,axiom,
! [G: nat > nat,F: nat > nat,A2: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A2 ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A2 ) ) ) ).
% SUP_image
thf(fact_717_Sup__union__distrib,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_718_Sup__union__distrib,axiom,
! [A2: set_se4330304633200676677_a_nat,B2: set_se4330304633200676677_a_nat] :
( ( comple1686717674086456018_a_nat @ ( sup_su499249268922660121_a_nat @ A2 @ B2 ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ A2 ) @ ( comple1686717674086456018_a_nat @ B2 ) ) ) ).
% Sup_union_distrib
thf(fact_719_Union__disjoint,axiom,
! [C2: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C2 ) @ A2 )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ C2 )
=> ( ( inf_inf_set_nat @ X @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_720_Union__disjoint,axiom,
! [C2: set_se4330304633200676677_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ C2 ) @ A2 )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ C2 )
=> ( ( inf_in3249246906714053971_a_nat @ X @ A2 )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Union_disjoint
thf(fact_721_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_722_exists__map,axiom,
! [Xs: list_o6767316136206324043_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s7896677892106675127_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin4230969608812606142_a_nat @ Xs )
=> ? [F4: option_Sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( map_op8672217432457595070_a_nat @ F4 @ Xs ) ) ) ) ).
% exists_map
thf(fact_723_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_724_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_725_exists__map,axiom,
! [Xs: list_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s4076174644546656840_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ? [F4: nat > produc5454855657571582244_a_nat] :
( Ys
= ( map_na4983825868277236666_a_nat @ F4 @ Xs ) ) ) ) ).
% exists_map
thf(fact_726_exists__map,axiom,
! [Xs: list_nat,Ys: list_o6767316136206324043_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s7896677892106675127_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ? [F4: nat > option_Sum_sum_a_nat] :
( Ys
= ( map_na4778969939102885679_a_nat @ F4 @ Xs ) ) ) ) ).
% exists_map
thf(fact_727_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_728_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_729_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_730_Union__image__empty,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_731_Union__image__empty,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ bot_bo1033123847703346641_a_nat ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_732_Union__image__empty,axiom,
! [A2: set_li6526943997496501093_a_nat,F: nat > set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ bot_bot_set_nat ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_733_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A2 ) )
= ( Sup @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_734_Sup_OSUP__cong,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C2: 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] :
( ( A2 = B2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_5081948215111134021_a_nat @ C2 @ A2 ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_735_Sup_OSUP__cong,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C2: 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] :
( ( A2 = B2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_7676043921908783909_a_nat @ C2 @ A2 ) )
= ( Sup @ ( image_7676043921908783909_a_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_736_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A2 ) )
= ( Inf @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_737_Inf_OINF__cong,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C2: 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] :
( ( A2 = B2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_5081948215111134021_a_nat @ C2 @ A2 ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_738_Inf_OINF__cong,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C2: 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] :
( ( A2 = B2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ B2 )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_7676043921908783909_a_nat @ C2 @ A2 ) )
= ( Inf @ ( image_7676043921908783909_a_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_739_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_740_exists__fo__nmlzd,axiom,
! [Xs: list_o6767316136206324043_a_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_s7896677892106675127_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin4230969608812606142_a_nat @ Xs )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F4: option_Sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_op8672217432457595070_a_nat @ F4 @ Xs ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_741_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_742_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_743_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_744_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_745_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_746_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_747_Sup__inf__eq__bot__iff,axiom,
! [B2: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B2 ) @ A )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ B2 )
=> ( ( inf_inf_set_nat @ X @ A )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_748_Sup__inf__eq__bot__iff,axiom,
! [B2: set_se4330304633200676677_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ B2 ) @ A )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ B2 )
=> ( ( inf_in3249246906714053971_a_nat @ X @ A )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_749_inf__Sup,axiom,
! [A: set_nat,B2: set_set_nat] :
( ( inf_inf_set_nat @ A @ ( comple7399068483239264473et_nat @ B2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A ) @ B2 ) ) ) ).
% inf_Sup
thf(fact_750_inf__Sup,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_se4330304633200676677_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ ( comple1686717674086456018_a_nat @ B2 ) )
= ( comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ ( inf_in3249246906714053971_a_nat @ A ) @ B2 ) ) ) ).
% inf_Sup
thf(fact_751_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_752_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_753_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_754_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_755_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_756_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_757_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_758_image__ident,axiom,
! [Y4: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : X
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_759_map__ident,axiom,
( ( map_Su2790769393171190532_a_nat
@ ^ [X: sum_sum_a_nat] : X )
= ( ^ [Xs2: list_Sum_sum_a_nat] : Xs2 ) ) ).
% map_ident
thf(fact_760_map__ident,axiom,
( ( map_nat_nat
@ ^ [X: nat] : X )
= ( ^ [Xs2: list_nat] : Xs2 ) ) ).
% map_ident
thf(fact_761_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
@ ^ [Y: nat] : ( P @ ( F @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_762_le__sup__iff,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X3 @ Z2 )
& ( ord_less_eq_set_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_763_le__sup__iff,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat @ X3 @ Z2 )
& ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_764_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_765_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_766_le__inf__iff,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X3 @ Y2 )
& ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_767_le__inf__iff,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_nat @ X3 @ Y2 )
& ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_768_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_769_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_770_SUP__identity__eq,axiom,
! [A2: set_se4330304633200676677_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_3472601871771700037_a_nat
@ ^ [X: set_li6526943997496501093_a_nat] : X
@ A2 ) )
= ( comple1686717674086456018_a_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_771_SUP__identity__eq,axiom,
! [A2: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( complete_Sup_Sup_nat @ A2 ) ) ).
% SUP_identity_eq
thf(fact_772_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat,B2: nat > set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_773_UN__I,axiom,
! [A: nat,A2: set_nat,B: nat > sum_sum_a_nat,B2: nat > set_na3699693778330250182_a_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member8690443509505302927_a_nat @ B @ ( B2 @ A ) )
=> ( member8690443509505302927_a_nat @ B @ ( comple8643769897048643123_a_nat @ ( image_7901234764482687663_a_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_774_UN__I,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: nat,B2: ( nat > sum_sum_a_nat ) > set_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_775_UN__I,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: nat > sum_sum_a_nat,B2: ( nat > sum_sum_a_nat ) > set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( member8690443509505302927_a_nat @ B @ ( B2 @ A ) )
=> ( member8690443509505302927_a_nat @ B @ ( comple8643769897048643123_a_nat @ ( image_2968167050796371173_a_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_776_UN__I,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: nat,B2: list_Sum_sum_a_nat > set_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( member_nat @ B @ ( B2 @ A ) )
=> ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_777_UN__I,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: nat > sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_na3699693778330250182_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( member8690443509505302927_a_nat @ B @ ( B2 @ A ) )
=> ( member8690443509505302927_a_nat @ B @ ( comple8643769897048643123_a_nat @ ( image_2625623354069934982_a_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_778_UN__I,axiom,
! [A: nat,A2: set_nat,B: list_Sum_sum_a_nat,B2: nat > set_li6526943997496501093_a_nat] :
( ( member_nat @ A @ A2 )
=> ( ( member408289922725080238_a_nat @ B @ ( B2 @ A ) )
=> ( member408289922725080238_a_nat @ B @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_779_UN__I,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: list_Sum_sum_a_nat,B2: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( member408289922725080238_a_nat @ B @ ( B2 @ A ) )
=> ( member408289922725080238_a_nat @ B @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_780_UN__I,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( member408289922725080238_a_nat @ B @ ( B2 @ A ) )
=> ( member408289922725080238_a_nat @ B @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_781_UN__iff,axiom,
! [B: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ( member408289922725080238_a_nat @ B @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_782_SUP__bot,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : bot_bo1033123847703346641_a_nat
@ A2 ) )
= bot_bo1033123847703346641_a_nat ) ).
% SUP_bot
thf(fact_783_SUP__bot__conv_I1_J,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( B2 @ X )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_784_SUP__bot__conv_I2_J,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( B2 @ X )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_785_SUP__const,axiom,
! [A2: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [I4: list_Sum_sum_a_nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_786_SUP__const,axiom,
! [A2: set_nat,F: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [I4: nat] : F
@ A2 ) )
= F ) ) ).
% SUP_const
thf(fact_787_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_788_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_789_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_790_UN__constant,axiom,
! [A2: set_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A2 ) )
= C ) ) ) ).
% UN_constant
thf(fact_791_UN__constant,axiom,
! [A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ( A2 = bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [Y: list_Sum_sum_a_nat] : C
@ A2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [Y: list_Sum_sum_a_nat] : C
@ A2 ) )
= C ) ) ) ).
% UN_constant
thf(fact_792_UN__constant,axiom,
! [A2: set_nat,C: set_li6526943997496501093_a_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [Y: nat] : C
@ A2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [Y: nat] : C
@ A2 ) )
= C ) ) ) ).
% UN_constant
thf(fact_793_UN__Un,axiom,
! [M: nat > set_nat,A2: set_nat,B2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A2 @ B2 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ B2 ) ) ) ) ).
% UN_Un
thf(fact_794_UN__Un,axiom,
! [M: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ M @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ M @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ M @ B2 ) ) ) ) ).
% UN_Un
thf(fact_795_UN__Un,axiom,
! [M: nat > set_li6526943997496501093_a_nat,A2: set_nat,B2: set_nat] :
( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ M @ ( sup_sup_set_nat @ A2 @ B2 ) ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ M @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ M @ B2 ) ) ) ) ).
% UN_Un
thf(fact_796_UN__simps_I2_J,axiom,
! [C2: set_nat,A2: nat > set_nat,B2: set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A2 @ X ) @ B2 )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A2 @ X ) @ B2 )
@ C2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_797_UN__simps_I2_J,axiom,
! [C2: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ ( A2 @ X ) @ B2 )
@ C2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C2 != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ ( A2 @ X ) @ B2 )
@ C2 ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ C2 ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_798_UN__simps_I2_J,axiom,
! [C2: set_nat,A2: nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ ( A2 @ X ) @ B2 )
@ C2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ ( A2 @ X ) @ B2 )
@ C2 ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ A2 @ C2 ) ) @ B2 ) ) ) ) ).
% UN_simps(2)
thf(fact_799_UN__simps_I3_J,axiom,
! [C2: set_nat,A2: set_nat,B2: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A2 @ ( B2 @ X ) )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A2 @ ( B2 @ X ) )
@ C2 ) )
= ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_800_UN__simps_I3_J,axiom,
! [C2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ A2 @ ( B2 @ X ) )
@ C2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C2 != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ A2 @ ( B2 @ X ) )
@ C2 ) )
= ( sup_su4083067149120280889_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_801_UN__simps_I3_J,axiom,
! [C2: set_nat,A2: set_li6526943997496501093_a_nat,B2: nat > set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ A2 @ ( B2 @ X ) )
@ C2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ A2 @ ( B2 @ X ) )
@ C2 ) )
= ( sup_su4083067149120280889_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_802_assms_I1_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ fv_sub )
& ( distinct_nat @ fv_sub ) ) ).
% assms(1)
thf(fact_803_assms_I2_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ fv_sub_comp )
& ( distinct_nat @ fv_sub_comp ) ) ).
% assms(2)
thf(fact_804_assms_I3_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ fv_all )
& ( distinct_nat @ fv_all ) ) ).
% assms(3)
thf(fact_805_Un__def,axiom,
( sup_su3329769938372955546_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A3 )
| ( member8690443509505302927_a_nat @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_806_Un__def,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A3 )
| ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_807_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
| ( member_nat @ X @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_808_Int__def,axiom,
( inf_in8399021836546144180_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A3 )
& ( member8690443509505302927_a_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_809_Int__def,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A3 )
& ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_810_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( member_nat @ X @ B3 ) ) ) ) ) ).
% Int_def
thf(fact_811_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% Set.empty_def
thf(fact_812_Int__Collect,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ X3 @ ( inf_in8399021836546144180_a_nat @ A2 @ ( collec5629555741568564177_a_nat @ P ) ) )
= ( ( member8690443509505302927_a_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_813_Int__Collect,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X3 @ ( inf_in3249246906714053971_a_nat @ A2 @ ( collec7555443234367654128_a_nat @ P ) ) )
= ( ( member408289922725080238_a_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_814_Int__Collect,axiom,
! [X3: nat,A2: set_nat,P: nat > $o] :
( ( member_nat @ X3 @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) ) ) ).
% Int_Collect
thf(fact_815_inf__set__def,axiom,
( inf_in8399021836546144180_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ( inf_in48813113380384809_nat_o
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A3 )
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_816_inf__set__def,axiom,
( inf_in3249246906714053971_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( inf_in954358986474102090_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A3 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_817_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).
% inf_set_def
thf(fact_818_sup__set__def,axiom,
( sup_su3329769938372955546_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ( sup_su3226716170639437251_nat_o
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A3 )
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_819_sup__set__def,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( sup_su1334248866174809316_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A3 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_820_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_821_sup__Un__eq,axiom,
! [R3: set_na3699693778330250182_a_nat,S: set_na3699693778330250182_a_nat] :
( ( sup_su3226716170639437251_nat_o
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ R3 )
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ S ) )
= ( ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ ( sup_su3329769938372955546_a_nat @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_822_sup__Un__eq,axiom,
! [R3: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( sup_su1334248866174809316_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ R3 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ S ) )
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( sup_su4083067149120280889_a_nat @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_823_sup__Un__eq,axiom,
! [R3: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R3 )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( sup_sup_set_nat @ R3 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_824_inf__Int__eq,axiom,
! [R3: set_na3699693778330250182_a_nat,S: set_na3699693778330250182_a_nat] :
( ( inf_in48813113380384809_nat_o
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ R3 )
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ S ) )
= ( ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ ( inf_in8399021836546144180_a_nat @ R3 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_825_inf__Int__eq,axiom,
! [R3: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( inf_in954358986474102090_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ R3 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ S ) )
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( inf_in3249246906714053971_a_nat @ R3 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_826_inf__Int__eq,axiom,
! [R3: set_nat,S: set_nat] :
( ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R3 )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( inf_inf_set_nat @ R3 @ S ) ) ) ) ).
% inf_Int_eq
thf(fact_827_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_828_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_829_Sup__set__def,axiom,
( comple7399068483239264473et_nat
= ( ^ [A3: set_set_nat] :
( collect_nat
@ ^ [X: nat] : ( complete_Sup_Sup_o @ ( image_set_nat_o @ ( member_nat @ X ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_830_Sup__set__def,axiom,
( comple8643769897048643123_a_nat
= ( ^ [A3: set_se5822283258546872870_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( complete_Sup_Sup_o @ ( image_4108232345474965207_nat_o @ ( member8690443509505302927_a_nat @ X ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_831_Sup__set__def,axiom,
( comple1686717674086456018_a_nat
= ( ^ [A3: set_se4330304633200676677_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( complete_Sup_Sup_o @ ( image_6425633437027283832_nat_o @ ( member408289922725080238_a_nat @ X ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_832_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S2: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_833_Sup__SUP__eq,axiom,
( comple8250499579560596074_nat_o
= ( ^ [S2: set_na8448764090737828173_nat_o,X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ ( comple8643769897048643123_a_nat @ ( image_1911308356169199528_a_nat @ collec5629555741568564177_a_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_834_Sup__SUP__eq,axiom,
( comple1334631452232999051_nat_o
= ( ^ [S2: set_li4526012430949197550_nat_o,X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ ( image_2943553480580064358_a_nat @ collec7555443234367654128_a_nat @ S2 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_835_SUP__Sup__eq,axiom,
! [S: set_set_nat] :
( ( comple8317665133742190828_nat_o
@ ( image_set_nat_nat_o
@ ^ [I4: set_nat,X: nat] : ( member_nat @ X @ I4 )
@ S ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_836_SUP__Sup__eq,axiom,
! [S: set_se5822283258546872870_a_nat] :
( ( comple8250499579560596074_nat_o
@ ( image_5989390580295776408_nat_o
@ ^ [I4: set_na3699693778330250182_a_nat,X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ I4 )
@ S ) )
= ( ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ ( comple8643769897048643123_a_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_837_SUP__Sup__eq,axiom,
! [S: set_se4330304633200676677_a_nat] :
( ( comple1334631452232999051_nat_o
@ ( image_7111272113017043416_nat_o
@ ^ [I4: set_li6526943997496501093_a_nat,X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ I4 )
@ S ) )
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_838_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( collec7528627406912015568_a_nat
@ ^ [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_7676043921908783909_a_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_839_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_840_Compr__image__eq,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat,A2: set_na3699693778330250182_a_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_5786201776793816049at_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_5786201776793816049at_nat @ F
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_841_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( image_2535339886381165584at_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_2535339886381165584at_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_842_Compr__image__eq,axiom,
! [F: nat > nat > sum_sum_a_nat,A2: set_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ ( image_1051037728736664655_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_1051037728736664655_a_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_843_Compr__image__eq,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ ( image_6222892899998961285_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_6222892899998961285_a_nat @ F
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_844_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ ( image_701559317304863014_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_701559317304863014_a_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_845_Compr__image__eq,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: set_nat,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( image_6262589752765146990_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_6262589752765146990_a_nat @ F
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_846_Compr__image__eq,axiom,
! [F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( image_6721470456781115300_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_6721470456781115300_a_nat @ F
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_847_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
& ( P @ X ) ) )
= ( image_5081948215111134021_a_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_848_SUP__UN__eq,axiom,
! [R: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( comple1334631452232999051_nat_o
@ ( image_8108004184916853240_nat_o
@ ^ [I4: list_Sum_sum_a_nat,X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( R @ I4 ) )
@ S ) )
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ R @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_849_image__image,axiom,
! [F: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( image_1711104591301794661_a_nat @ F @ ( image_7676043921908783909_a_nat @ G @ A2 ) )
= ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_850_image__image,axiom,
! [F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( image_3472601871771700037_a_nat @ F @ ( image_7676043921908783909_a_nat @ G @ A2 ) )
= ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_851_image__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ A2 ) )
= ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_852_image__image,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ A2 ) )
= ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_853_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X: nat] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_854_imageE,axiom,
! [B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ B @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
=> ~ ! [X2: list_Sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member408289922725080238_a_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_855_imageE,axiom,
! [B: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_856_imageE,axiom,
! [B: nat,F: ( nat > sum_sum_a_nat ) > nat,A2: set_na3699693778330250182_a_nat] :
( ( member_nat @ B @ ( image_5786201776793816049at_nat @ F @ A2 ) )
=> ~ ! [X2: nat > sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member8690443509505302927_a_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_857_imageE,axiom,
! [B: nat,F: list_Sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat] :
( ( member_nat @ B @ ( image_2535339886381165584at_nat @ F @ A2 ) )
=> ~ ! [X2: list_Sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member408289922725080238_a_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_858_imageE,axiom,
! [B: nat > sum_sum_a_nat,F: nat > nat > sum_sum_a_nat,A2: set_nat] :
( ( member8690443509505302927_a_nat @ B @ ( image_1051037728736664655_a_nat @ F @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_859_imageE,axiom,
! [B: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ B @ ( image_6222892899998961285_a_nat @ F @ A2 ) )
=> ~ ! [X2: nat > sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member8690443509505302927_a_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_860_imageE,axiom,
! [B: nat > sum_sum_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ B @ ( image_701559317304863014_a_nat @ F @ A2 ) )
=> ~ ! [X2: list_Sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member408289922725080238_a_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_861_imageE,axiom,
! [B: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,A2: set_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_6262589752765146990_a_nat @ F @ A2 ) )
=> ~ ! [X2: nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_862_imageE,axiom,
! [B: list_Sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_6721470456781115300_a_nat @ F @ A2 ) )
=> ~ ! [X2: nat > sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member8690443509505302927_a_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_863_imageE,axiom,
! [B: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
=> ~ ! [X2: list_Sum_sum_a_nat] :
( ( B
= ( F @ X2 ) )
=> ~ ( member408289922725080238_a_nat @ X2 @ A2 ) ) ) ).
% imageE
thf(fact_864_SUP__subset__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_li6526943997496501093_a_nat,G: nat > set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_865_SUP__subset__mono,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,G: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ B2 )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_866_SUP__subset__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_867_SUP__upper2,axiom,
! [I3: nat,A2: set_nat,U: set_li6526943997496501093_a_nat,F: nat > set_li6526943997496501093_a_nat] :
( ( member_nat @ I3 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ U @ ( F @ I3 ) )
=> ( ord_le1147066620699065093_a_nat @ U @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_868_SUP__upper2,axiom,
! [I3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,U: set_li6526943997496501093_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ I3 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ U @ ( F @ I3 ) )
=> ( ord_le1147066620699065093_a_nat @ U @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_869_SUP__upper2,axiom,
! [I3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,U: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ I3 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ U @ ( F @ I3 ) )
=> ( ord_le1147066620699065093_a_nat @ U @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) ) ) ) ).
% SUP_upper2
thf(fact_870_SUP__le__iff,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,U: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) @ U )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X ) @ U ) ) ) ) ).
% SUP_le_iff
thf(fact_871_SUP__upper,axiom,
! [I3: nat,A2: set_nat,F: nat > set_li6526943997496501093_a_nat] :
( ( member_nat @ I3 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I3 ) @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_872_SUP__upper,axiom,
! [I3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ I3 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I3 ) @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_873_SUP__upper,axiom,
! [I3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ I3 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I3 ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) ) ) ).
% SUP_upper
thf(fact_874_SUP__mono_H,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ! [X2: list_Sum_sum_a_nat] : ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ A2 ) ) ) ) ).
% SUP_mono'
thf(fact_875_SUP__least,axiom,
! [A2: set_nat,F: nat > set_li6526943997496501093_a_nat,U: set_li6526943997496501093_a_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ U ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_876_SUP__least,axiom,
! [A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,U: set_li6526943997496501093_a_nat] :
( ! [I2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ U ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_877_SUP__least,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,U: set_li6526943997496501093_a_nat] :
( ! [I2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ U ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) @ U ) ) ).
% SUP_least
thf(fact_878_SUP__mono,axiom,
! [A2: set_nat,B2: set_li6526943997496501093_a_nat,F: nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ! [N4: nat] :
( ( member_nat @ N4 @ A2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ N4 ) @ ( G @ X4 ) ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_879_SUP__mono,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_li6526943997496501093_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ! [N4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ N4 @ A2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ N4 ) @ ( G @ X4 ) ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_880_SUP__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ! [N4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ N4 @ A2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ N4 ) @ ( G @ X4 ) ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ B2 ) ) ) ) ).
% SUP_mono
thf(fact_881_SUP__eqI,axiom,
! [A2: set_nat,F: nat > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_li6526943997496501093_a_nat] :
( ! [I5: nat] :
( ( member_nat @ I5 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I5 ) @ Y3 ) )
=> ( ord_le1147066620699065093_a_nat @ X3 @ Y3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_882_SUP__eqI,axiom,
! [A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ! [I2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_li6526943997496501093_a_nat] :
( ! [I5: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I5 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I5 ) @ Y3 ) )
=> ( ord_le1147066620699065093_a_nat @ X3 @ Y3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_883_SUP__eqI,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ! [I2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ X3 ) )
=> ( ! [Y3: set_li6526943997496501093_a_nat] :
( ! [I5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I5 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ I5 ) @ Y3 ) )
=> ( ord_le1147066620699065093_a_nat @ X3 @ Y3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
= X3 ) ) ) ).
% SUP_eqI
thf(fact_884_Sup_OSUP__identity__eq,axiom,
! [Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( Sup
@ ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_885_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_886_Inf_OINF__identity__eq,axiom,
! [Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( Inf
@ ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_887_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_888_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_889_list_Omap__ident,axiom,
! [T: list_Sum_sum_a_nat] :
( ( map_Su2790769393171190532_a_nat
@ ^ [X: sum_sum_a_nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_890_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_891_SUP__commute,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [I4: list_Sum_sum_a_nat] : ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ ( F @ I4 ) @ B2 ) )
@ A2 ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [J: list_Sum_sum_a_nat] :
( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [I4: list_Sum_sum_a_nat] : ( F @ I4 @ J )
@ A2 ) )
@ B2 ) ) ) ).
% SUP_commute
thf(fact_892_Union__natural,axiom,
! [F: nat > nat] :
( ( comp_s174380336271864291et_nat @ comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) ) )
= ( comp_s6993074788030935341et_nat @ ( image_nat_nat @ F ) @ comple7399068483239264473et_nat ) ) ).
% Union_natural
thf(fact_893_Union__natural,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( comp_s2212010409885379394_a_nat @ comple1778571073001984690_a_nat @ ( image_7557436503106877221_a_nat @ ( image_7676043921908783909_a_nat @ F ) ) )
= ( comp_s7798250529893408642_a_nat @ ( image_7676043921908783909_a_nat @ F ) @ comple1686717674086456018_a_nat ) ) ).
% Union_natural
thf(fact_894_Union__natural,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( comp_s2882954424060012354_a_nat @ comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ ( image_5081948215111134021_a_nat @ F ) ) )
= ( comp_s4805735529744850786_a_nat @ ( image_5081948215111134021_a_nat @ F ) @ comple1686717674086456018_a_nat ) ) ).
% Union_natural
thf(fact_895_image__Union,axiom,
! [F: nat > nat,S: set_set_nat] :
( ( image_nat_nat @ F @ ( comple7399068483239264473et_nat @ S ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_896_image__Union,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,S: set_se4330304633200676677_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ ( comple1686717674086456018_a_nat @ S ) )
= ( comple1778571073001984690_a_nat @ ( image_7557436503106877221_a_nat @ ( image_7676043921908783909_a_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_897_image__Union,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,S: set_se4330304633200676677_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( comple1686717674086456018_a_nat @ S ) )
= ( comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ ( image_5081948215111134021_a_nat @ F ) @ S ) ) ) ).
% image_Union
thf(fact_898_Int__Union2,axiom,
! [B2: set_set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B2 ) @ A2 )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [C3: set_nat] : ( inf_inf_set_nat @ C3 @ A2 )
@ B2 ) ) ) ).
% Int_Union2
thf(fact_899_Int__Union2,axiom,
! [B2: set_se4330304633200676677_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ B2 ) @ A2 )
= ( comple1686717674086456018_a_nat
@ ( image_3472601871771700037_a_nat
@ ^ [C3: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ C3 @ A2 )
@ B2 ) ) ) ).
% Int_Union2
thf(fact_900_Int__Union,axiom,
! [A2: set_nat,B2: set_set_nat] :
( ( inf_inf_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B2 ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A2 ) @ B2 ) ) ) ).
% Int_Union
thf(fact_901_Int__Union,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_se4330304633200676677_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ B2 ) )
= ( comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 ) @ B2 ) ) ) ).
% Int_Union
thf(fact_902_UN__extend__simps_I9_J,axiom,
! [C2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ C2 @ ( B2 @ X ) ) )
@ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ C2 @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_903_UN__extend__simps_I8_J,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_se4330304633200676677_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_3472601871771700037_a_nat
@ ^ [Y: set_li6526943997496501093_a_nat] : ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ Y ) )
@ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ ( comple1686717674086456018_a_nat @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_904_UN__E,axiom,
! [B: nat,B2: nat > set_nat,A2: set_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_905_UN__E,axiom,
! [B: nat,B2: ( nat > sum_sum_a_nat ) > set_nat,A2: set_na3699693778330250182_a_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ B2 @ A2 ) ) )
=> ~ ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_906_UN__E,axiom,
! [B: nat,B2: list_Sum_sum_a_nat > set_nat,A2: set_li6526943997496501093_a_nat] :
( ( member_nat @ B @ ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ B2 @ A2 ) ) )
=> ~ ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ~ ( member_nat @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_907_UN__E,axiom,
! [B: nat > sum_sum_a_nat,B2: nat > set_na3699693778330250182_a_nat,A2: set_nat] :
( ( member8690443509505302927_a_nat @ B @ ( comple8643769897048643123_a_nat @ ( image_7901234764482687663_a_nat @ B2 @ A2 ) ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member8690443509505302927_a_nat @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_908_UN__E,axiom,
! [B: nat > sum_sum_a_nat,B2: ( nat > sum_sum_a_nat ) > set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ B @ ( comple8643769897048643123_a_nat @ ( image_2968167050796371173_a_nat @ B2 @ A2 ) ) )
=> ~ ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ~ ( member8690443509505302927_a_nat @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_909_UN__E,axiom,
! [B: nat > sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_na3699693778330250182_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ B @ ( comple8643769897048643123_a_nat @ ( image_2625623354069934982_a_nat @ B2 @ A2 ) ) )
=> ~ ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ~ ( member8690443509505302927_a_nat @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_910_UN__E,axiom,
! [B: list_Sum_sum_a_nat,B2: nat > set_li6526943997496501093_a_nat,A2: set_nat] :
( ( member408289922725080238_a_nat @ B @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ A2 ) ) )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member408289922725080238_a_nat @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_911_UN__E,axiom,
! [B: list_Sum_sum_a_nat,B2: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ B2 @ A2 ) ) )
=> ~ ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ~ ( member408289922725080238_a_nat @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_912_UN__E,axiom,
! [B: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) )
=> ~ ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ~ ( member408289922725080238_a_nat @ B @ ( B2 @ X2 ) ) ) ) ).
% UN_E
thf(fact_913_UN__UN__flatten,axiom,
! [C2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ C2 @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [Y: list_Sum_sum_a_nat] : ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ C2 @ ( B2 @ Y ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_914_vimage__Union,axiom,
! [F: nat > sum_sum_a_nat,A2: set_se4904748513628223167_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( comple1247738100258233164_a_nat @ A2 ) )
= ( comple7399068483239264473et_nat @ ( image_1313604448114346656et_nat @ ( vimage3040984495076556338_a_nat @ F ) @ A2 ) ) ) ).
% vimage_Union
thf(fact_915_vimage__Union,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_se4330304633200676677_a_nat] :
( ( vimage7538740559230589179_a_nat @ F @ ( comple1686717674086456018_a_nat @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ ( vimage7538740559230589179_a_nat @ F ) @ A2 ) ) ) ).
% vimage_Union
thf(fact_916_SUP__inf__distrib2,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ B2 ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [A4: list_Sum_sum_a_nat] :
( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [B4: list_Sum_sum_a_nat] : ( inf_in3249246906714053971_a_nat @ ( F @ A4 ) @ ( G @ B4 ) )
@ B2 ) )
@ A2 ) ) ) ).
% SUP_inf_distrib2
thf(fact_917_inf__SUP,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ B2 ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [B4: list_Sum_sum_a_nat] : ( inf_in3249246906714053971_a_nat @ A @ ( F @ B4 ) )
@ B2 ) ) ) ).
% inf_SUP
thf(fact_918_Sup__inf,axiom,
! [B2: set_set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B2 ) @ A )
= ( comple7399068483239264473et_nat
@ ( image_7916887816326733075et_nat
@ ^ [B4: set_nat] : ( inf_inf_set_nat @ B4 @ A )
@ B2 ) ) ) ).
% Sup_inf
thf(fact_919_Sup__inf,axiom,
! [B2: set_se4330304633200676677_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ B2 ) @ A )
= ( comple1686717674086456018_a_nat
@ ( image_3472601871771700037_a_nat
@ ^ [B4: set_li6526943997496501093_a_nat] : ( inf_in3249246906714053971_a_nat @ B4 @ A )
@ B2 ) ) ) ).
% Sup_inf
thf(fact_920_SUP__inf,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ B2 ) ) @ A )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [B4: list_Sum_sum_a_nat] : ( inf_in3249246906714053971_a_nat @ ( F @ B4 ) @ A )
@ B2 ) ) ) ).
% SUP_inf
thf(fact_921_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_922_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_923_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_924_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_925_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_926_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_927_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_928_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X3: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_929_inf__sup__ord_I3_J,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_930_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_931_le__supE,axiom,
! [A: set_nat,B: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X3 )
=> ~ ( ord_less_eq_set_nat @ B @ X3 ) ) ) ).
% le_supE
thf(fact_932_le__supE,axiom,
! [A: nat,B: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A @ X3 )
=> ~ ( ord_less_eq_nat @ B @ X3 ) ) ) ).
% le_supE
thf(fact_933_le__supI,axiom,
! [A: set_nat,X3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X3 )
=> ( ( ord_less_eq_set_nat @ B @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_934_le__supI,axiom,
! [A: nat,X3: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X3 )
=> ( ( ord_less_eq_nat @ B @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_935_sup__ge1,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_936_sup__ge1,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).
% sup_ge1
thf(fact_937_sup__ge2,axiom,
! [Y2: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_938_sup__ge2,axiom,
! [Y2: nat,X3: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X3 @ Y2 ) ) ).
% sup_ge2
thf(fact_939_le__supI1,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_940_le__supI1,axiom,
! [X3: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_941_le__supI2,axiom,
! [X3: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ B )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_942_le__supI2,axiom,
! [X3: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X3 @ B )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_943_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_944_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_945_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_946_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_947_sup__least,axiom,
! [Y2: set_nat,X3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_948_sup__least,axiom,
! [Y2: nat,X3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_949_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_950_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_951_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_952_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_953_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_954_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_955_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y2: set_nat] :
( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z @ X2 )
=> ( ord_less_eq_set_nat @ ( F @ Y3 @ Z ) @ X2 ) ) )
=> ( ( sup_sup_set_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_956_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y2: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F @ X2 @ Y3 ) )
=> ( ! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y3 @ Z ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_957_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_958_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_959_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_960_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_961_sup__absorb1,axiom,
! [Y2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( sup_sup_set_nat @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_962_sup__absorb1,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y2 )
= X3 ) ) ).
% sup_absorb1
thf(fact_963_sup__absorb2,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( sup_sup_set_nat @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_964_sup__absorb2,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( sup_sup_nat @ X3 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_965_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_966_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_967_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_968_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_969_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_970_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_971_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_972_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_973_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_974_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_975_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_976_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_977_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_978_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_979_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_980_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_981_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_982_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_983_inf__sup__ord_I2_J,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_984_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_985_inf__sup__ord_I1_J,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_986_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_987_inf__le1,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_988_inf__le1,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ X3 ) ).
% inf_le1
thf(fact_989_inf__le2,axiom,
! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_990_inf__le2,axiom,
! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_991_le__infE,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X3 @ A )
=> ~ ( ord_less_eq_set_nat @ X3 @ B ) ) ) ).
% le_infE
thf(fact_992_le__infE,axiom,
! [X3: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A )
=> ~ ( ord_less_eq_nat @ X3 @ B ) ) ) ).
% le_infE
thf(fact_993_le__infI,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ B )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_994_le__infI,axiom,
! [X3: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ A )
=> ( ( ord_less_eq_nat @ X3 @ B )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_995_inf__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_996_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_997_le__infI1,axiom,
! [A: set_nat,X3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_998_le__infI1,axiom,
! [A: nat,X3: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_999_le__infI2,axiom,
! [B: set_nat,X3: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_1000_le__infI2,axiom,
! [B: nat,X3: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_1001_inf_OorderE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( A
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1002_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_1003_inf_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_1004_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_1005_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y2: set_nat] :
( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ Y3 @ Z ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1006_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y2: nat] :
( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y3 ) @ X2 )
=> ( ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y3 ) @ Y3 )
=> ( ! [X2: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y3 @ Z ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y2 )
= ( F @ X3 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_1007_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_1008_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_1009_inf_Oabsorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1010_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_1011_inf_Oabsorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1012_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_1013_inf__absorb1,axiom,
! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( inf_inf_set_nat @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_1014_inf__absorb1,axiom,
! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( inf_inf_nat @ X3 @ Y2 )
= X3 ) ) ).
% inf_absorb1
thf(fact_1015_inf__absorb2,axiom,
! [Y2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X3 )
=> ( ( inf_inf_set_nat @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1016_inf__absorb2,axiom,
! [Y2: nat,X3: nat] :
( ( ord_less_eq_nat @ Y2 @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_1017_inf_OboundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1018_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_1019_inf_OboundedI,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1020_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_1021_inf__greatest,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1022_inf__greatest,axiom,
! [X3: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_1023_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_1024_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_1025_inf_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1026_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_1027_inf_Ocobounded2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1028_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_1029_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_1030_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_1031_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_1032_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_1033_inf_OcoboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1034_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_1035_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1036_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_1037_complete__lattice__class_OSUP__sup__distrib,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ A2 ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [A4: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ ( F @ A4 ) @ ( G @ A4 ) )
@ A2 ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_1038_SUP__absorb,axiom,
! [K: nat,I: set_nat,A2: nat > set_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1039_SUP__absorb,axiom,
! [K: nat > sum_sum_a_nat,I: set_na3699693778330250182_a_nat,A2: ( nat > sum_sum_a_nat ) > set_nat] :
( ( member8690443509505302927_a_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1040_SUP__absorb,axiom,
! [K: list_Sum_sum_a_nat,I: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat > set_nat] :
( ( member408289922725080238_a_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1041_SUP__absorb,axiom,
! [K: nat,I: set_nat,A2: nat > set_li6526943997496501093_a_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_su4083067149120280889_a_nat @ ( A2 @ K ) @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ A2 @ I ) ) )
= ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1042_SUP__absorb,axiom,
! [K: nat > sum_sum_a_nat,I: set_na3699693778330250182_a_nat,A2: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ K @ I )
=> ( ( sup_su4083067149120280889_a_nat @ ( A2 @ K ) @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ A2 @ I ) ) )
= ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1043_SUP__absorb,axiom,
! [K: list_Sum_sum_a_nat,I: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ K @ I )
=> ( ( sup_su4083067149120280889_a_nat @ ( A2 @ K ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ I ) ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ I ) ) ) ) ).
% SUP_absorb
thf(fact_1044_UN__extend__simps_I10_J,axiom,
! [B2: nat > set_li6526943997496501093_a_nat,F: nat > nat,A2: set_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [A4: nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1045_UN__extend__simps_I10_J,axiom,
! [B2: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [A4: list_Sum_sum_a_nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ B2 @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1046_UN__extend__simps_I10_J,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [A4: list_Sum_sum_a_nat] : ( B2 @ ( F @ A4 ) )
@ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1047_image__UN,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) )
= ( comple1778571073001984690_a_nat
@ ( image_7324140262315149061_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( image_7676043921908783909_a_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1048_image__UN,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( image_5081948215111134021_a_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_1049_UN__empty2,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : bot_bo1033123847703346641_a_nat
@ A2 ) )
= bot_bo1033123847703346641_a_nat ) ).
% UN_empty2
thf(fact_1050_UN__empty,axiom,
! [B2: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% UN_empty
thf(fact_1051_UN__empty,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) )
= bot_bo1033123847703346641_a_nat ) ).
% UN_empty
thf(fact_1052_UN__empty,axiom,
! [B2: nat > set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ bot_bot_set_nat ) )
= bot_bo1033123847703346641_a_nat ) ).
% UN_empty
thf(fact_1053_UNION__empty__conv_I1_J,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( B2 @ X )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_1054_UNION__empty__conv_I2_J,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( ( B2 @ X )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_1055_proj__vals__def,axiom,
( proj_v2875961069841262333_a_nat
= ( ^ [R4: set_na964695563613990089_a_nat,Ns2: list_nat] :
( image_8844818676099448910_a_nat
@ ^ [Tau3: nat > produc5454855657571582244_a_nat] : ( map_na4983825868277236666_a_nat @ Tau3 @ Ns2 )
@ R4 ) ) ) ).
% proj_vals_def
thf(fact_1056_proj__vals__def,axiom,
( proj_v7785048252284376108_a_nat
= ( ^ [R4: set_na5864254054690418060_a_nat,Ns2: list_nat] :
( image_5077931504662633508_a_nat
@ ^ [Tau3: nat > option_Sum_sum_a_nat] : ( map_na4778969939102885679_a_nat @ Tau3 @ Ns2 )
@ R4 ) ) ) ).
% proj_vals_def
thf(fact_1057_proj__vals__def,axiom,
( proj_vals_nat
= ( ^ [R4: set_nat_nat,Ns2: list_nat] :
( image_2065990475722762790st_nat
@ ^ [Tau3: nat > nat] : ( map_nat_nat @ Tau3 @ Ns2 )
@ R4 ) ) ) ).
% proj_vals_def
thf(fact_1058_proj__vals__def,axiom,
( proj_v3643391342904276326_a_nat
= ( ^ [R4: set_na3699693778330250182_a_nat,Ns2: list_nat] :
( image_6721470456781115300_a_nat
@ ^ [Tau3: nat > sum_sum_a_nat] : ( map_na823391071729141993_a_nat @ Tau3 @ Ns2 )
@ R4 ) ) ) ).
% proj_vals_def
thf(fact_1059_Int__UN__distrib2,axiom,
! [A2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,I: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,J2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ I ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ J2 ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [I4: list_Sum_sum_a_nat] :
( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [J: list_Sum_sum_a_nat] : ( inf_in3249246906714053971_a_nat @ ( A2 @ I4 ) @ ( B2 @ J ) )
@ J2 ) )
@ I ) ) ) ).
% Int_UN_distrib2
thf(fact_1060_Int__UN__distrib,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,I: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ B2 @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ I ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [I4: list_Sum_sum_a_nat] : ( inf_in3249246906714053971_a_nat @ B2 @ ( A2 @ I4 ) )
@ I ) ) ) ).
% Int_UN_distrib
thf(fact_1061_UN__extend__simps_I4_J,axiom,
! [A2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ C2 ) ) @ B2 )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( inf_in3249246906714053971_a_nat @ ( A2 @ X ) @ B2 )
@ C2 ) ) ) ).
% UN_extend_simps(4)
thf(fact_1062_UN__extend__simps_I5_J,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ C2 ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( inf_in3249246906714053971_a_nat @ A2 @ ( B2 @ X ) )
@ C2 ) ) ) ).
% UN_extend_simps(5)
thf(fact_1063_Un__Union__image,axiom,
! [A2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ ( A2 @ X ) @ ( B2 @ X ) )
@ C2 ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ C2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ C2 ) ) ) ) ).
% Un_Union_image
thf(fact_1064_UN__Un__distrib,axiom,
! [A2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,I: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [I4: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ ( A2 @ I4 ) @ ( B2 @ I4 ) )
@ I ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ I ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ I ) ) ) ) ).
% UN_Un_distrib
thf(fact_1065_UN__absorb,axiom,
! [K: nat,I: set_nat,A2: nat > set_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ I ) ) ) ) ).
% UN_absorb
thf(fact_1066_UN__absorb,axiom,
! [K: nat > sum_sum_a_nat,I: set_na3699693778330250182_a_nat,A2: ( nat > sum_sum_a_nat ) > set_nat] :
( ( member8690443509505302927_a_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ A2 @ I ) ) ) ) ).
% UN_absorb
thf(fact_1067_UN__absorb,axiom,
! [K: list_Sum_sum_a_nat,I: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat > set_nat] :
( ( member408289922725080238_a_nat @ K @ I )
=> ( ( sup_sup_set_nat @ ( A2 @ K ) @ ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ A2 @ I ) ) )
= ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ A2 @ I ) ) ) ) ).
% UN_absorb
thf(fact_1068_UN__absorb,axiom,
! [K: nat,I: set_nat,A2: nat > set_li6526943997496501093_a_nat] :
( ( member_nat @ K @ I )
=> ( ( sup_su4083067149120280889_a_nat @ ( A2 @ K ) @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ A2 @ I ) ) )
= ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ A2 @ I ) ) ) ) ).
% UN_absorb
thf(fact_1069_UN__absorb,axiom,
! [K: nat > sum_sum_a_nat,I: set_na3699693778330250182_a_nat,A2: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ K @ I )
=> ( ( sup_su4083067149120280889_a_nat @ ( A2 @ K ) @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ A2 @ I ) ) )
= ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ A2 @ I ) ) ) ) ).
% UN_absorb
thf(fact_1070_UN__absorb,axiom,
! [K: list_Sum_sum_a_nat,I: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ K @ I )
=> ( ( sup_su4083067149120280889_a_nat @ ( A2 @ K ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ I ) ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ I ) ) ) ) ).
% UN_absorb
thf(fact_1071_vimage__UN,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( vimage7538740559230589179_a_nat @ F @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( vimage7538740559230589179_a_nat @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% vimage_UN
thf(fact_1072_SUP__empty,axiom,
! [F: nat > set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% SUP_empty
thf(fact_1073_SUP__empty,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ bot_bo1033123847703346641_a_nat ) )
= bot_bo1033123847703346641_a_nat ) ).
% SUP_empty
thf(fact_1074_SUP__empty,axiom,
! [F: nat > set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ bot_bot_set_nat ) )
= bot_bo1033123847703346641_a_nat ) ).
% SUP_empty
thf(fact_1075_SUP__constant,axiom,
! [A2: set_nat,C: set_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A2 ) )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y: nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1076_SUP__constant,axiom,
! [A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ( A2 = bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [Y: list_Sum_sum_a_nat] : C
@ A2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [Y: list_Sum_sum_a_nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1077_SUP__constant,axiom,
! [A2: set_nat,C: set_li6526943997496501093_a_nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [Y: nat] : C
@ A2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [Y: nat] : C
@ A2 ) )
= C ) ) ) ).
% SUP_constant
thf(fact_1078_SUP__union,axiom,
! [M: nat > set_nat,A2: set_nat,B2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A2 @ B2 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1079_SUP__union,axiom,
! [M: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ M @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ M @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ M @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1080_SUP__union,axiom,
! [M: nat > set_li6526943997496501093_a_nat,A2: set_nat,B2: set_nat] :
( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ M @ ( sup_sup_set_nat @ A2 @ B2 ) ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ M @ A2 ) ) @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ M @ B2 ) ) ) ) ).
% SUP_union
thf(fact_1081_SUP__UNION,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ A2 ) ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [Y: list_Sum_sum_a_nat] : ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ ( G @ Y ) ) )
@ A2 ) ) ) ).
% SUP_UNION
thf(fact_1082_UN__extend__simps_I2_J,axiom,
! [C2: set_nat,A2: nat > set_nat,B2: set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ B2 )
= B2 ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ C2 ) ) @ B2 )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A2 @ X ) @ B2 )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1083_UN__extend__simps_I2_J,axiom,
! [C2: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bo1033123847703346641_a_nat )
=> ( ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ C2 ) ) @ B2 )
= B2 ) )
& ( ( C2 != bot_bo1033123847703346641_a_nat )
=> ( ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A2 @ C2 ) ) @ B2 )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ ( A2 @ X ) @ B2 )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1084_UN__extend__simps_I2_J,axiom,
! [C2: set_nat,A2: nat > set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ A2 @ C2 ) ) @ B2 )
= B2 ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ A2 @ C2 ) ) @ B2 )
= ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ ( A2 @ X ) @ B2 )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_1085_UN__extend__simps_I3_J,axiom,
! [C2: set_nat,A2: set_nat,B2: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ C2 ) ) )
= A2 ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ C2 ) ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A2 @ ( B2 @ X ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1086_UN__extend__simps_I3_J,axiom,
! [C2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bo1033123847703346641_a_nat )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ C2 ) ) )
= A2 ) )
& ( ( C2 != bot_bo1033123847703346641_a_nat )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ C2 ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ A2 @ ( B2 @ X ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1087_UN__extend__simps_I3_J,axiom,
! [C2: set_nat,A2: set_li6526943997496501093_a_nat,B2: nat > set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ C2 ) ) )
= A2 ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ C2 ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ A2 @ ( B2 @ X ) )
@ C2 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_1088_nall__tuples__set,axiom,
( nall_tuples_a
= ( ^ [AD2: set_a,N: nat] :
( collec7555443234367654128_a_nat
@ ^ [Ys2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys2 )
= N )
& ( fo_nmlzd_a @ AD2 @ Ys2 ) ) ) ) ) ).
% nall_tuples_set
thf(fact_1089_SUP__eq,axiom,
! [A2: set_nat,B2: set_nat,F: nat > set_li6526943997496501093_a_nat,G: nat > set_li6526943997496501093_a_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1090_SUP__eq,axiom,
! [A2: set_nat,B2: set_na3699693778330250182_a_nat,F: nat > set_li6526943997496501093_a_nat,G: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ? [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1091_SUP__eq,axiom,
! [A2: set_nat,B2: set_li6526943997496501093_a_nat,F: nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ A2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ J3 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1092_SUP__eq,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,G: nat > set_li6526943997496501093_a_nat] :
( ! [I2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1093_SUP__eq,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,G: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ! [I2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I2 @ A2 )
=> ? [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ J3 @ B2 )
=> ? [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1094_SUP__eq,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_li6526943997496501093_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ! [I2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I2 @ A2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ J3 @ B2 )
=> ? [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1095_SUP__eq,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: nat > set_li6526943997496501093_a_nat] :
( ! [I2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I2 @ A2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat] :
( ( member_nat @ J3 @ B2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1096_SUP__eq,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_na3699693778330250182_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ! [I2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I2 @ A2 )
=> ? [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ J3 @ B2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1097_SUP__eq,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ! [I2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I2 @ A2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B2 )
& ( ord_le1147066620699065093_a_nat @ ( F @ I2 ) @ ( G @ X4 ) ) ) )
=> ( ! [J3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ J3 @ B2 )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( G @ J3 ) @ ( F @ X4 ) ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ B2 ) ) ) ) ) ).
% SUP_eq
thf(fact_1098_less__eq__Sup,axiom,
! [A2: set_se4330304633200676677_a_nat,U: set_li6526943997496501093_a_nat] :
( ! [V3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ V3 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ U @ V3 ) )
=> ( ( A2 != bot_bo3237059034911209905_a_nat )
=> ( ord_le1147066620699065093_a_nat @ U @ ( comple1686717674086456018_a_nat @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_1099_distrib__inf__le,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y2 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) @ ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1100_distrib__inf__le,axiom,
! [X3: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y2 ) @ ( inf_inf_nat @ X3 @ Z2 ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1101_distrib__sup__le,axiom,
! [X3: set_nat,Y2: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y2 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1102_distrib__sup__le,axiom,
! [X3: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y2 @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y2 ) @ ( sup_sup_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1103_SUP__eq__iff,axiom,
! [I: set_na3699693778330250182_a_nat,C: set_li6526943997496501093_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat] :
( ( I != bot_bo6441361344521902642_a_nat )
=> ( ! [I2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I2 @ I )
=> ( ord_le1147066620699065093_a_nat @ C @ ( F @ I2 ) ) )
=> ( ( ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ I ) )
= C )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1104_SUP__eq__iff,axiom,
! [I: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( I != bot_bo1033123847703346641_a_nat )
=> ( ! [I2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I2 @ I )
=> ( ord_le1147066620699065093_a_nat @ C @ ( F @ I2 ) ) )
=> ( ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ I ) )
= C )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1105_SUP__eq__iff,axiom,
! [I: set_nat,C: set_li6526943997496501093_a_nat,F: nat > set_li6526943997496501093_a_nat] :
( ( I != bot_bot_set_nat )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le1147066620699065093_a_nat @ C @ ( F @ I2 ) ) )
=> ( ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ I ) )
= C )
= ( ! [X: nat] :
( ( member_nat @ X @ I )
=> ( ( F @ X )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_1106_Sup__inter__less__eq,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1107_Sup__inter__less__eq,axiom,
! [A2: set_se4330304633200676677_a_nat,B2: set_se4330304633200676677_a_nat] : ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( inf_in5367731912061063475_a_nat @ A2 @ B2 ) ) @ ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ A2 ) @ ( comple1686717674086456018_a_nat @ B2 ) ) ) ).
% Sup_inter_less_eq
thf(fact_1108_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_1109_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_1110_cSUP__const,axiom,
! [A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_1111_cSUP__const,axiom,
! [A2: set_nat,C: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_1112_cSUP__const,axiom,
! [A2: set_nat,C: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X: nat] : C
@ A2 ) )
= C ) ) ).
% cSUP_const
thf(fact_1113_cSUP__least,axiom,
! [A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,M: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bo6441361344521902642_a_nat )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ M ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1114_cSUP__least,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,M: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ M ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1115_cSUP__least,axiom,
! [A2: set_nat,F: nat > set_li6526943997496501093_a_nat,M: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X2 ) @ M ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1116_cSUP__least,axiom,
! [A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > nat,M: nat] :
( ( A2 != bot_bo6441361344521902642_a_nat )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1117_cSUP__least,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat,M: nat] :
( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1118_cSUP__least,axiom,
! [A2: set_nat,F: nat > nat,M: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1119_cSup__eq__non__empty,axiom,
! [X6: set_se4330304633200676677_a_nat,A: set_li6526943997496501093_a_nat] :
( ( X6 != bot_bo3237059034911209905_a_nat )
=> ( ! [X2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X2 @ X6 )
=> ( ord_le1147066620699065093_a_nat @ X2 @ A ) )
=> ( ! [Y3: set_li6526943997496501093_a_nat] :
( ! [X4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ X6 )
=> ( ord_le1147066620699065093_a_nat @ X4 @ Y3 ) )
=> ( ord_le1147066620699065093_a_nat @ A @ Y3 ) )
=> ( ( comple1686717674086456018_a_nat @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1120_cSup__eq__non__empty,axiom,
! [X6: set_nat,A: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ X2 @ A ) )
=> ( ! [Y3: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) )
=> ( ord_less_eq_nat @ A @ Y3 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_1121_cSup__least,axiom,
! [X6: set_se4330304633200676677_a_nat,Z2: set_li6526943997496501093_a_nat] :
( ( X6 != bot_bo3237059034911209905_a_nat )
=> ( ! [X2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X2 @ X6 )
=> ( ord_le1147066620699065093_a_nat @ X2 @ Z2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_1122_cSup__least,axiom,
! [X6: set_nat,Z2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_1123_ext__tuple__def,axiom,
( ext_tuple_a
= ( ^ [AD2: set_a,Fv_sub2: list_nat,Fv_sub_comp2: list_nat,As3: list_Sum_sum_a_nat] :
( if_set7709265119413304363_a_nat @ ( Fv_sub_comp2 = nil_nat ) @ ( insert2950094090816004437_a_nat @ As3 @ bot_bo1033123847703346641_a_nat )
@ ( 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_sub2 @ As3 ) @ ( zip_na2013496608136855606_a_nat @ Fv_sub_comp2 @ Fs2 ) ) )
@ ( nall_tuples_rec_a @ AD2 @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ As3 ) ) ) @ ( size_size_list_nat @ Fv_sub_comp2 ) ) ) ) ) ) ).
% ext_tuple_def
thf(fact_1124_UN__constant__eq,axiom,
! [A: nat,A2: set_nat,F: nat > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( member_nat @ A @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= C ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A2 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1125_UN__constant__eq,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= C ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ A2 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1126_UN__constant__eq,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ( F @ X2 )
= C ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
= C ) ) ) ).
% UN_constant_eq
thf(fact_1127_set__list__bind,axiom,
! [Xs: list_nat,F: nat > list_nat] :
( ( set_nat2 @ ( bind_nat_nat @ Xs @ F ) )
= ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( set_nat2 @ ( F @ X ) )
@ ( set_nat2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_1128_set__list__bind,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_nat] :
( ( set_nat2 @ ( bind_S7309778641578091144at_nat @ Xs @ F ) )
= ( comple7399068483239264473et_nat
@ ( image_4589483402070311232et_nat
@ ^ [X: sum_sum_a_nat] : ( set_nat2 @ ( F @ X ) )
@ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_1129_set__list__bind,axiom,
! [Xs: list_nat,F: nat > list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( bind_n2905796707917019494_a_nat @ Xs @ F ) )
= ( comple1247738100258233164_a_nat
@ ( image_4085361583586468296_a_nat
@ ^ [X: nat] : ( set_Sum_sum_a_nat2 @ ( F @ X ) )
@ ( set_nat2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_1130_set__list__bind,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( bind_S8497345843576593223_a_nat @ Xs @ F ) )
= ( comple1247738100258233164_a_nat
@ ( image_7877458644602423589_a_nat
@ ^ [X: sum_sum_a_nat] : ( set_Sum_sum_a_nat2 @ ( F @ X ) )
@ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_1131_set__list__bind,axiom,
! [Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( bind_l6437164444685190343_a_nat @ Xs @ F ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( set_li2392974972034027290_a_nat @ ( F @ X ) )
@ ( set_li2392974972034027290_a_nat @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_1132_set__list__bind,axiom,
! [Xs: list_nat,F: nat > list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( bind_n431031891494352236_a_nat @ Xs @ F ) )
= ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( set_li2392974972034027290_a_nat @ ( F @ X ) )
@ ( set_nat2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_1133_set__list__bind,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( bind_S7470141391546446541_a_nat @ Xs @ F ) )
= ( comple1686717674086456018_a_nat
@ ( image_7788113019651049515_a_nat
@ ^ [X: sum_sum_a_nat] : ( set_li2392974972034027290_a_nat @ ( F @ X ) )
@ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_1134_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1135_subsetI,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( member8690443509505302927_a_nat @ X2 @ B2 ) )
=> ( ord_le8108555184339247974_a_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1136_subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( member408289922725080238_a_nat @ X2 @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_1137_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1138_insertCI,axiom,
! [A: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat,B: nat > sum_sum_a_nat] :
( ( ~ ( member8690443509505302927_a_nat @ A @ B2 )
=> ( A = B ) )
=> ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1139_insertCI,axiom,
! [A: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ A @ B2 )
=> ( A = B ) )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1140_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1141_insert__iff,axiom,
! [A: nat > sum_sum_a_nat,B: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ B @ A2 ) )
= ( ( A = B )
| ( member8690443509505302927_a_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1142_insert__iff,axiom,
! [A: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ A2 ) )
= ( ( A = B )
| ( member408289922725080238_a_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1143_insert__absorb2,axiom,
! [X3: nat,A2: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A2 ) )
= ( insert_nat @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_1144_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_1145_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_1146_insert__image,axiom,
! [X3: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( insert_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1147_insert__image,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( insert_nat @ ( F @ X3 ) @ ( image_5786201776793816049at_nat @ F @ A2 ) )
= ( image_5786201776793816049at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1148_insert__image,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( insert2950094090816004437_a_nat @ ( F @ X3 ) @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1149_insert__image,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( insert8442372787808178485_a_nat @ ( F @ X3 ) @ ( image_7676043921908783909_a_nat @ F @ A2 ) )
= ( image_7676043921908783909_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1150_insert__image,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( insert_nat @ ( F @ X3 ) @ ( image_2535339886381165584at_nat @ F @ A2 ) )
= ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_1151_image__insert,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( insert2950094090816004437_a_nat @ ( F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1152_image__insert,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( insert8442372787808178485_a_nat @ ( F @ A ) @ ( image_7676043921908783909_a_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1153_image__insert,axiom,
! [F: nat > nat,A: nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1154_list_Omap__disc__iff,axiom,
! [F: nat > sum_sum_a_nat,A: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ A )
= nil_Sum_sum_a_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1155_list_Omap__disc__iff,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,A: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ A )
= nil_Sum_sum_a_nat )
= ( A = nil_Pr237480997409426078_a_nat ) ) ).
% list.map_disc_iff
thf(fact_1156_list_Omap__disc__iff,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,A: list_o6767316136206324043_a_nat] :
( ( ( map_op8672217432457595070_a_nat @ F @ A )
= nil_Sum_sum_a_nat )
= ( A = nil_op6198398268141876043_a_nat ) ) ).
% list.map_disc_iff
thf(fact_1157_list_Omap__disc__iff,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ A )
= nil_Sum_sum_a_nat )
= ( A = nil_Sum_sum_a_nat ) ) ).
% list.map_disc_iff
thf(fact_1158_list_Omap__disc__iff,axiom,
! [F: nat > produc5454855657571582244_a_nat,A: list_nat] :
( ( ( map_na4983825868277236666_a_nat @ F @ A )
= nil_Pr237480997409426078_a_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1159_list_Omap__disc__iff,axiom,
! [F: nat > option_Sum_sum_a_nat,A: list_nat] :
( ( ( map_na4778969939102885679_a_nat @ F @ A )
= nil_op6198398268141876043_a_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1160_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1161_Nil__is__map__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( nil_Sum_sum_a_nat
= ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1162_Nil__is__map__conv,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( nil_Sum_sum_a_nat
= ( map_Pr2515564165894599445_a_nat @ F @ Xs ) )
= ( Xs = nil_Pr237480997409426078_a_nat ) ) ).
% Nil_is_map_conv
thf(fact_1163_Nil__is__map__conv,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_o6767316136206324043_a_nat] :
( ( nil_Sum_sum_a_nat
= ( map_op8672217432457595070_a_nat @ F @ Xs ) )
= ( Xs = nil_op6198398268141876043_a_nat ) ) ).
% Nil_is_map_conv
thf(fact_1164_Nil__is__map__conv,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( nil_Sum_sum_a_nat
= ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% Nil_is_map_conv
thf(fact_1165_Nil__is__map__conv,axiom,
! [F: nat > produc5454855657571582244_a_nat,Xs: list_nat] :
( ( nil_Pr237480997409426078_a_nat
= ( map_na4983825868277236666_a_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1166_Nil__is__map__conv,axiom,
! [F: nat > option_Sum_sum_a_nat,Xs: list_nat] :
( ( nil_op6198398268141876043_a_nat
= ( map_na4778969939102885679_a_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1167_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1168_map__is__Nil__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1169_map__is__Nil__conv,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_Pr237480997409426078_a_nat ) ) ).
% map_is_Nil_conv
thf(fact_1170_map__is__Nil__conv,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,Xs: list_o6767316136206324043_a_nat] :
( ( ( map_op8672217432457595070_a_nat @ F @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_op6198398268141876043_a_nat ) ) ).
% map_is_Nil_conv
thf(fact_1171_map__is__Nil__conv,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% map_is_Nil_conv
thf(fact_1172_map__is__Nil__conv,axiom,
! [F: nat > produc5454855657571582244_a_nat,Xs: list_nat] :
( ( ( map_na4983825868277236666_a_nat @ F @ Xs )
= nil_Pr237480997409426078_a_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1173_map__is__Nil__conv,axiom,
! [F: nat > option_Sum_sum_a_nat,Xs: list_nat] :
( ( ( map_na4778969939102885679_a_nat @ F @ Xs )
= nil_op6198398268141876043_a_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1174_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1175_singletonI,axiom,
! [A: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) ).
% singletonI
thf(fact_1176_singletonI,axiom,
! [A: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% singletonI
thf(fact_1177_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1178_insert__subset,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
= ( ( member_nat @ X3 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1179_insert__subset,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ ( insert5265011953798106934_a_nat @ X3 @ A2 ) @ B2 )
= ( ( member8690443509505302927_a_nat @ X3 @ B2 )
& ( ord_le8108555184339247974_a_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1180_insert__subset,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ X3 @ A2 ) @ B2 )
= ( ( member408289922725080238_a_nat @ X3 @ B2 )
& ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1181_Int__subset__iff,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1182_Un__subset__iff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A2 @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1183_Int__insert__right__if1,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( inf_in8399021836546144180_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ B2 ) )
= ( insert5265011953798106934_a_nat @ A @ ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1184_Int__insert__right__if1,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( insert2950094090816004437_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1185_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1186_Int__insert__right__if0,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( inf_in8399021836546144180_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ B2 ) )
= ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1187_Int__insert__right__if0,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1188_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1189_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_1190_Int__insert__left__if1,axiom,
! [A: nat > sum_sum_a_nat,C2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ C2 )
=> ( ( inf_in8399021836546144180_a_nat @ ( insert5265011953798106934_a_nat @ A @ B2 ) @ C2 )
= ( insert5265011953798106934_a_nat @ A @ ( inf_in8399021836546144180_a_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1191_Int__insert__left__if1,axiom,
! [A: list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ C2 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ B2 ) @ C2 )
= ( insert2950094090816004437_a_nat @ A @ ( inf_in3249246906714053971_a_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1192_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1193_Int__insert__left__if0,axiom,
! [A: nat > sum_sum_a_nat,C2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ A @ C2 )
=> ( ( inf_in8399021836546144180_a_nat @ ( insert5265011953798106934_a_nat @ A @ B2 ) @ C2 )
= ( inf_in8399021836546144180_a_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1194_Int__insert__left__if0,axiom,
! [A: list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A @ C2 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ B2 ) @ C2 )
= ( inf_in3249246906714053971_a_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1195_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1196_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1197_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1198_zip__eq__Nil__iff,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( zip_na2013496608136855606_a_nat @ Xs @ Ys )
= nil_Pr237480997409426078_a_nat )
= ( ( Xs = nil_nat )
| ( Ys = nil_Sum_sum_a_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_1199_zip__eq__Nil__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( zip_nat_nat @ Xs @ Ys )
= nil_Pr5478986624290739719at_nat )
= ( ( Xs = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% zip_eq_Nil_iff
thf(fact_1200_Nil__eq__zip__iff,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( nil_Pr237480997409426078_a_nat
= ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
| ( Ys = nil_Sum_sum_a_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_1201_Nil__eq__zip__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_Pr5478986624290739719at_nat
= ( zip_nat_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
| ( Ys = nil_nat ) ) ) ).
% Nil_eq_zip_iff
thf(fact_1202_zip__Nil,axiom,
! [Ys: list_Sum_sum_a_nat] :
( ( zip_na2013496608136855606_a_nat @ nil_nat @ Ys )
= nil_Pr237480997409426078_a_nat ) ).
% zip_Nil
thf(fact_1203_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1204_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1205_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_1206_set__empty2,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( bot_bo3438331934148233675_a_nat
= ( set_Sum_sum_a_nat2 @ Xs ) )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% set_empty2
thf(fact_1207_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_1208_List_Oset__empty,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( set_Sum_sum_a_nat2 @ Xs )
= bot_bo3438331934148233675_a_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% List.set_empty
thf(fact_1209_List_Oset__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% List.set_empty
thf(fact_1210_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1211_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1212_cSup__singleton,axiom,
! [X3: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( insert8442372787808178485_a_nat @ X3 @ bot_bo3237059034911209905_a_nat ) )
= X3 ) ).
% cSup_singleton
thf(fact_1213_cSup__singleton,axiom,
! [X3: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% cSup_singleton
thf(fact_1214_insert__disjoint_I1_J,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ( inf_in8399021836546144180_a_nat @ ( insert5265011953798106934_a_nat @ A @ A2 ) @ B2 )
= bot_bo6441361344521902642_a_nat )
= ( ~ ( member8690443509505302927_a_nat @ A @ B2 )
& ( ( inf_in8399021836546144180_a_nat @ A2 @ B2 )
= bot_bo6441361344521902642_a_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1215_insert__disjoint_I1_J,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ A2 ) @ B2 )
= bot_bo1033123847703346641_a_nat )
= ( ~ ( member408289922725080238_a_nat @ A @ B2 )
& ( ( inf_in3249246906714053971_a_nat @ A2 @ B2 )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1216_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1217_insert__disjoint_I2_J,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( bot_bo6441361344521902642_a_nat
= ( inf_in8399021836546144180_a_nat @ ( insert5265011953798106934_a_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member8690443509505302927_a_nat @ A @ B2 )
& ( bot_bo6441361344521902642_a_nat
= ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1218_insert__disjoint_I2_J,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member408289922725080238_a_nat @ A @ B2 )
& ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1219_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat @ A @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1220_disjoint__insert_I1_J,axiom,
! [B2: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( ( inf_in8399021836546144180_a_nat @ B2 @ ( insert5265011953798106934_a_nat @ A @ A2 ) )
= bot_bo6441361344521902642_a_nat )
= ( ~ ( member8690443509505302927_a_nat @ A @ B2 )
& ( ( inf_in8399021836546144180_a_nat @ B2 @ A2 )
= bot_bo6441361344521902642_a_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1221_disjoint__insert_I1_J,axiom,
! [B2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A @ A2 ) )
= bot_bo1033123847703346641_a_nat )
= ( ~ ( member408289922725080238_a_nat @ A @ B2 )
& ( ( inf_in3249246906714053971_a_nat @ B2 @ A2 )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1222_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1223_disjoint__insert_I2_J,axiom,
! [A2: set_na3699693778330250182_a_nat,B: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( bot_bo6441361344521902642_a_nat
= ( inf_in8399021836546144180_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ B @ B2 ) ) )
= ( ~ ( member8690443509505302927_a_nat @ B @ A2 )
& ( bot_bo6441361344521902642_a_nat
= ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1224_disjoint__insert_I2_J,axiom,
! [A2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B @ B2 ) ) )
= ( ~ ( member408289922725080238_a_nat @ B @ A2 )
& ( bot_bo1033123847703346641_a_nat
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1225_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1226_Sup__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ A2 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ A2 ) ) ) ).
% Sup_insert
thf(fact_1227_Sup__insert,axiom,
! [A: set_li6526943997496501093_a_nat,A2: set_se4330304633200676677_a_nat] :
( ( comple1686717674086456018_a_nat @ ( insert8442372787808178485_a_nat @ A @ A2 ) )
= ( sup_su4083067149120280889_a_nat @ A @ ( comple1686717674086456018_a_nat @ A2 ) ) ) ).
% Sup_insert
thf(fact_1228_UN__simps_I1_J,axiom,
! [C2: set_nat,A: nat,B2: nat > set_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( insert_nat @ A @ ( B2 @ X ) )
@ C2 ) )
= bot_bot_set_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( insert_nat @ A @ ( B2 @ X ) )
@ C2 ) )
= ( insert_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1229_UN__simps_I1_J,axiom,
! [C2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( insert2950094090816004437_a_nat @ A @ ( B2 @ X ) )
@ C2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C2 != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( insert2950094090816004437_a_nat @ A @ ( B2 @ X ) )
@ C2 ) )
= ( insert2950094090816004437_a_nat @ A @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1230_UN__simps_I1_J,axiom,
! [C2: set_nat,A: list_Sum_sum_a_nat,B2: nat > set_li6526943997496501093_a_nat] :
( ( ( C2 = bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( insert2950094090816004437_a_nat @ A @ ( B2 @ X ) )
@ C2 ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C2 != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( insert2950094090816004437_a_nat @ A @ ( B2 @ X ) )
@ C2 ) )
= ( insert2950094090816004437_a_nat @ A @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ C2 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_1231_UN__singleton,axiom,
! [A2: set_nat] :
( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( insert_nat @ X @ bot_bot_set_nat )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_1232_UN__singleton,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( insert2950094090816004437_a_nat @ X @ bot_bo1033123847703346641_a_nat )
@ A2 ) )
= A2 ) ).
% UN_singleton
thf(fact_1233_UN__insert,axiom,
! [B2: nat > set_nat,A: nat,A2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ ( insert_nat @ A @ A2 ) ) )
= ( sup_sup_set_nat @ ( B2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1234_UN__insert,axiom,
! [B2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A @ A2 ) ) )
= ( sup_su4083067149120280889_a_nat @ ( B2 @ A ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1235_UN__insert,axiom,
! [B2: nat > set_li6526943997496501093_a_nat,A: nat,A2: set_nat] :
( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ ( insert_nat @ A @ A2 ) ) )
= ( sup_su4083067149120280889_a_nat @ ( B2 @ A ) @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B2 @ A2 ) ) ) ) ).
% UN_insert
thf(fact_1236_subset__singletonD,axiom,
! [A2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1237_subset__singleton__iff,axiom,
! [X6: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X6 = bot_bot_set_nat )
| ( X6
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1238_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_1239_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1240_less__eq__set__def,axiom,
( ord_le8108555184339247974_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( ord_le6982925868732858103_nat_o
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A3 )
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1241_less__eq__set__def,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ord_le8737610411969296920_nat_o
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A3 )
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1242_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_1243_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_1244_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1245_Collect__subset,axiom,
! [A2: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ord_le8108555184339247974_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1246_Collect__subset,axiom,
! [A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_1247_subset__insert,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1248_subset__insert,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( ord_le8108555184339247974_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ B2 ) )
= ( ord_le8108555184339247974_a_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1249_subset__insert,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ B2 ) )
= ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1250_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A4 )
| ( member_nat @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_1251_insert__compr,axiom,
( insert5265011953798106934_a_nat
= ( ^ [A4: nat > sum_sum_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] :
( ( X = A4 )
| ( member8690443509505302927_a_nat @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_1252_insert__compr,axiom,
( insert2950094090816004437_a_nat
= ( ^ [A4: list_Sum_sum_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
( ( X = A4 )
| ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_1253_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1254_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A3 )
=> ( member_nat @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_1255_subset__iff,axiom,
( ord_le8108555184339247974_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
! [T3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ T3 @ A3 )
=> ( member8690443509505302927_a_nat @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_1256_subset__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
! [T3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ T3 @ A3 )
=> ( member408289922725080238_a_nat @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_1257_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_1258_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_1259_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_1260_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_1261_subsetD,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,C: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ B2 )
=> ( ( member8690443509505302927_a_nat @ C @ A2 )
=> ( member8690443509505302927_a_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_1262_subsetD,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( member408289922725080238_a_nat @ C @ A2 )
=> ( member408289922725080238_a_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_1263_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_1264_in__mono,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ B2 )
=> ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_1265_in__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_1266_sorted__wrt__true,axiom,
! [Xs: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_1267_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1268_zip_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( zip_na2013496608136855606_a_nat @ Xs @ nil_Sum_sum_a_nat )
= nil_Pr237480997409426078_a_nat ) ).
% zip.simps(1)
thf(fact_1269_fv__all__sort,axiom,
( fv_all
= ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ fv_sub @ fv_sub_comp ) ) ) ).
% fv_all_sort
thf(fact_1270_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1271_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1272_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_1273_lessThan__Suc__eq__insert__0,axiom,
! [N3: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N3 ) )
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N3 ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_1274_lessThan__empty__iff,axiom,
! [N3: nat] :
( ( ( set_ord_lessThan_nat @ N3 )
= bot_bot_set_nat )
= ( N3 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1275_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y2: nat] :
( ( if_nat @ $true @ X3 @ Y2 )
= X3 ) ).
thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X3: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X3: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $true @ X3 @ Y2 )
= 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,Y2: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $false @ X3 @ Y2 )
= Y2 ) ).
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,Y2: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $true @ X3 @ Y2 )
= 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,Y2: set_li6526943997496501093_a_nat] :
( ( if_set7709265119413304363_a_nat @ $false @ X3 @ Y2 )
= Y2 ) ).
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,Y2: set_li6526943997496501093_a_nat] :
( ( if_set7709265119413304363_a_nat @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( 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 ) ) ).
%------------------------------------------------------------------------------