TPTP Problem File: SLH0098^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Eval_FO/0005_Ailamazyan/prob_02112_079998__15790080_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1560 ( 526 unt; 277 typ; 0 def)
% Number of atoms : 3817 (1354 equ; 0 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 12119 ( 409 ~; 38 |; 268 &;9608 @)
% ( 0 <=>;1796 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg)
% Number of types : 27 ( 26 usr)
% Number of type conns : 1397 (1397 >; 0 *; 0 +; 0 <<)
% Number of symbols : 254 ( 251 usr; 21 con; 0-6 aty)
% Number of variables : 3656 ( 130 ^;3409 !; 117 ?;3656 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:07:30.980
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
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% Explicit typings (251)
thf(sy_c_Ailamazyan_Oad__agr__list_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Ailamazyan_Oad__agr__sets_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Ailamazyan_Oad__equiv__list_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Ailamazyan_Oall__tuples_001t__Nat__Onat,type,
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thf(sy_c_Ailamazyan_Oext__tuple__set_001tf__a,type,
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thf(sy_c_Ailamazyan_Ofo__nmlz_001tf__a,type,
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thf(sy_c_Ailamazyan_Ofo__nmlzd_001tf__a,type,
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thf(sy_c_Ailamazyan_Onall__tuples_001t__Nat__Onat,type,
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thf(sy_c_Ailamazyan_Onall__tuples_001tf__a,type,
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thf(sy_c_Ailamazyan_Oproj__tuple_001tf__a,type,
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thf(sy_c_Ailamazyan_Osp__equiv_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Ailamazyan_Osp__equiv__list_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
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thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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fun_up7925228223850514735et_nat: ( ( nat > sum_sum_a_nat ) > set_nat ) > ( nat > sum_sum_a_nat ) > set_nat > ( nat > sum_sum_a_nat ) > set_nat ).
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thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Nat__Onat,type,
fun_upd_nat_nat: ( nat > nat ) > nat > nat > nat > nat ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
member3060896489619847151_a_nat: set_na3699693778330250182_a_nat > set_se5822283258546872870_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_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_ass,type,
ass: set_li6526943997496501093_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_thesis____,type,
thesis: $o ).
thf(sy_v_xs,type,
xs: list_Sum_sum_a_nat ).
% Relevant facts (1276)
thf(fact_0_assms_I3_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ fv_all )
& ( distinct_nat @ fv_all ) ) ).
% assms(3)
thf(fact_1_assms_I9_J,axiom,
( ( size_s5283204784079214577_a_nat @ xs )
= ( size_size_list_nat @ fv_all ) ) ).
% assms(9)
thf(fact_2__092_060open_062_092_060lbrakk_062length_Afv__all_A_061_Alength_Axs_059_Adistinct_Afv__all_092_060rbrakk_062_A_092_060Longrightarrow_062_A_092_060exists_062f_O_Axs_A_061_Amap_Af_Afv__all_092_060close_062,axiom,
( ( ( size_size_list_nat @ fv_all )
= ( size_s5283204784079214577_a_nat @ xs ) )
=> ( ( distinct_nat @ fv_all )
=> ? [F: nat > sum_sum_a_nat] :
( xs
= ( map_na823391071729141993_a_nat @ F @ fv_all ) ) ) ) ).
% \<open>\<lbrakk>length fv_all = length xs; distinct fv_all\<rbrakk> \<Longrightarrow> \<exists>f. xs = map f fv_all\<close>
thf(fact_3_assms_I8_J,axiom,
( ( fo_nmlz_a @ ad @ xs )
= xs ) ).
% assms(8)
thf(fact_4_length__map,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_5_length__map,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_na823391071729141993_a_nat @ F2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_6_length__map,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( size_size_list_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_7_length__map,axiom,
! [F2: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_8_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 )
=> ? [F: sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( map_Su2790769393171190532_a_nat @ F @ Xs ) ) ) ) ).
% exists_map
thf(fact_9_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 )
=> ? [F: sum_sum_a_nat > nat] :
( Ys
= ( map_Su5227373005390213643at_nat @ F @ Xs ) ) ) ) ).
% exists_map
thf(fact_10_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 )
=> ? [F: nat > sum_sum_a_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F @ Xs ) ) ) ) ).
% exists_map
thf(fact_11_exists__map,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ? [F: nat > nat] :
( Ys
= ( map_nat_nat @ F @ Xs ) ) ) ) ).
% exists_map
thf(fact_12_map__eq__imp__length__eq,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Su5227373005390213643at_nat @ F2 @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_13_map__eq__imp__length__eq,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F2 @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_14_map__eq__imp__length__eq,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_Su5227373005390213643at_nat @ F2 @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_15_map__eq__imp__length__eq,axiom,
! [F2: nat > nat,Xs: list_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_16_map__eq__imp__length__eq,axiom,
! [F2: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_17_map__eq__imp__length__eq,axiom,
! [F2: 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 @ F2 @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_18_map__eq__imp__length__eq,axiom,
! [F2: 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 @ F2 @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_19_map__eq__imp__length__eq,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F2 @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_20_fo__nmlz__map,axiom,
! [AD: set_a,Sigma: sum_sum_a_nat > sum_sum_a_nat,Ns: list_Sum_sum_a_nat] :
? [Tau: sum_sum_a_nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_Su2790769393171190532_a_nat @ Sigma @ Ns ) )
= ( map_Su2790769393171190532_a_nat @ Tau @ Ns ) ) ).
% fo_nmlz_map
thf(fact_21_fo__nmlz__map,axiom,
! [AD: set_a,Sigma: nat > sum_sum_a_nat,Ns: list_nat] :
? [Tau: nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) )
= ( map_na823391071729141993_a_nat @ Tau @ Ns ) ) ).
% fo_nmlz_map
thf(fact_22_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_23_map__eq__map__tailrec,axiom,
map_Su5227373005390213643at_nat = map_ta5540980089885783105at_nat ).
% map_eq_map_tailrec
thf(fact_24_map__eq__map__tailrec,axiom,
map_Su2790769393171190532_a_nat = map_ta197872344851190606_a_nat ).
% map_eq_map_tailrec
thf(fact_25_map__eq__map__tailrec,axiom,
map_na823391071729141993_a_nat = map_ta1136998156224711455_a_nat ).
% map_eq_map_tailrec
thf(fact_26_assms_I2_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ fv_sub_comp )
& ( distinct_nat @ fv_sub_comp ) ) ).
% assms(2)
thf(fact_27_assms_I10_J,axiom,
member408289922725080238_a_nat @ ( fo_nmlz_a @ ad @ ( proj_tuple_a @ fv_sub @ ( zip_na2013496608136855606_a_nat @ fv_all @ xs ) ) ) @ ass ).
% assms(10)
thf(fact_28_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_29_rotate1__map,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat] :
( ( rotate2765497868024679250_a_nat @ ( map_na823391071729141993_a_nat @ F2 @ Xs ) )
= ( map_na823391071729141993_a_nat @ F2 @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_30_rotate1__map,axiom,
! [F2: nat > nat,Xs: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F2 @ Xs ) )
= ( map_nat_nat @ F2 @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_31_rotate1__map,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( rotate1_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
= ( map_Su5227373005390213643at_nat @ F2 @ ( rotate2765497868024679250_a_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_32_rotate1__map,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( rotate2765497868024679250_a_nat @ ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) )
= ( map_Su2790769393171190532_a_nat @ F2 @ ( rotate2765497868024679250_a_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_33_list_Omap__disc__iff,axiom,
! [F2: nat > sum_sum_a_nat,A: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F2 @ A )
= nil_Sum_sum_a_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_34_list_Omap__disc__iff,axiom,
! [F2: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F2 @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_35_list_Omap__disc__iff,axiom,
! [F2: sum_sum_a_nat > nat,A: list_Sum_sum_a_nat] :
( ( ( map_Su5227373005390213643at_nat @ F2 @ A )
= nil_nat )
= ( A = nil_Sum_sum_a_nat ) ) ).
% list.map_disc_iff
thf(fact_36_list_Omap__disc__iff,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F2 @ A )
= nil_Sum_sum_a_nat )
= ( A = nil_Sum_sum_a_nat ) ) ).
% list.map_disc_iff
thf(fact_37_Nil__is__map__conv,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat] :
( ( nil_Sum_sum_a_nat
= ( map_na823391071729141993_a_nat @ F2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_38_Nil__is__map__conv,axiom,
! [F2: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_39_Nil__is__map__conv,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( nil_nat
= ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% Nil_is_map_conv
thf(fact_40_Nil__is__map__conv,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( nil_Sum_sum_a_nat
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% Nil_is_map_conv
thf(fact_41_assms_I1_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ fv_sub )
& ( distinct_nat @ fv_sub ) ) ).
% assms(1)
thf(fact_42_fo__nmlz__Nil,axiom,
! [AD: set_a] :
( ( fo_nmlz_a @ AD @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% fo_nmlz_Nil
thf(fact_43_map__is__Nil__conv,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F2 @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_44_map__is__Nil__conv,axiom,
! [F2: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_45_map__is__Nil__conv,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( ( map_Su5227373005390213643at_nat @ F2 @ Xs )
= nil_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% map_is_Nil_conv
thf(fact_46_map__is__Nil__conv,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F2 @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% map_is_Nil_conv
thf(fact_47_map__eq__conv,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > nat] :
( ( ( map_Su5227373005390213643at_nat @ F2 @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Xs ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F2 @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_48_map__eq__conv,axiom,
! [F2: 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 @ F2 @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Xs ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F2 @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_49_map__eq__conv,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F2 @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F2 @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_50_map__eq__conv,axiom,
! [F2: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F2 @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_51_zip__Nil,axiom,
! [Ys: list_Sum_sum_a_nat] :
( ( zip_na2013496608136855606_a_nat @ nil_nat @ Ys )
= nil_Pr237480997409426078_a_nat ) ).
% zip_Nil
thf(fact_52_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_53_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_54_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_55_length__rotate1,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( rotate2765497868024679250_a_nat @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_rotate1
thf(fact_56_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_57_distinct1__rotate,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( rotate1_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct1_rotate
thf(fact_58_distinct1__rotate,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( rotate2765497868024679250_a_nat @ Xs ) )
= ( distin2701893636801681144_a_nat @ Xs ) ) ).
% distinct1_rotate
thf(fact_59_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_60_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_61_distinct_Osimps_I1_J,axiom,
distin2701893636801681144_a_nat @ nil_Sum_sum_a_nat ).
% distinct.simps(1)
thf(fact_62_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_63_proj__tuple_Osimps_I1_J,axiom,
! [Mys: list_P5056861408695629236_a_nat] :
( ( proj_tuple_a @ nil_nat @ Mys )
= nil_Sum_sum_a_nat ) ).
% proj_tuple.simps(1)
thf(fact_64_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_65_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_66_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_67_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_68_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_69_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_70_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_71_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_72_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_73_sorted__wrt__mono__rel,axiom,
! [Xs: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [X2: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( member408289922725080238_a_nat @ Y @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( P @ X2 @ Y )
=> ( Q @ X2 @ Y ) ) ) )
=> ( ( sorted1521004414301201522_a_nat @ P @ Xs )
=> ( sorted1521004414301201522_a_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_74_sorted__wrt__mono__rel,axiom,
! [Xs: list_n989787106983797996_a_nat,P: ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o] :
( ! [X2: nat > sum_sum_a_nat,Y: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( member8690443509505302927_a_nat @ Y @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( P @ X2 @ Y )
=> ( Q @ X2 @ Y ) ) ) )
=> ( ( sorted5637561102903432531_a_nat @ P @ Xs )
=> ( sorted5637561102903432531_a_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_75_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X2: nat,Y: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y @ ( set_nat2 @ Xs ) )
=> ( ( P @ X2 @ Y )
=> ( Q @ X2 @ Y ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_76_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_77_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F2: nat > sum_sum_a_nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F2 @ Xs3 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X
= ( F2 @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_78_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat] :
( ( ? [Xs3: list_Sum_sum_a_nat] :
( Ys
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs3 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y2: sum_sum_a_nat] :
( X
= ( F2 @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_79_ex__map__conv,axiom,
! [Ys: list_nat,F2: nat > nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_nat_nat @ F2 @ Xs3 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X
= ( F2 @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_80_ex__map__conv,axiom,
! [Ys: list_nat,F2: sum_sum_a_nat > nat] :
( ( ? [Xs3: list_Sum_sum_a_nat] :
( Ys
= ( map_Su5227373005390213643at_nat @ F2 @ Xs3 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ? [Y2: sum_sum_a_nat] :
( X
= ( F2 @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_81_map__cong,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,G: sum_sum_a_nat > nat] :
( ( Xs = Ys )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Su5227373005390213643at_nat @ F2 @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_82_map__cong,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,F2: 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 ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F2 @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_83_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F2: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F2 @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_84_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F2: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F2 @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_85_map__idI,axiom,
! [Xs: list_l4703314356710769291_a_nat,F2: 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 ) )
=> ( ( F2 @ X2 )
= X2 ) )
=> ( ( map_li6507455427659069316_a_nat @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_86_map__idI,axiom,
! [Xs: list_n989787106983797996_a_nat,F2: ( 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 ) )
=> ( ( F2 @ X2 )
= X2 ) )
=> ( ( map_na722425985163074756_a_nat @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_87_map__idI,axiom,
! [Xs: list_Sum_sum_a_nat,F2: 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 ) )
=> ( ( F2 @ X2 )
= X2 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_88_map__idI,axiom,
! [Xs: list_nat,F2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F2 @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_89_map__ext,axiom,
! [Xs: list_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,G: sum_sum_a_nat > nat] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Su5227373005390213643at_nat @ F2 @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_90_map__ext,axiom,
! [Xs: list_Sum_sum_a_nat,F2: 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 ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F2 @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_91_map__ext,axiom,
! [Xs: list_nat,F2: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F2 @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_92_map__ext,axiom,
! [Xs: list_nat,F2: nat > nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F2 @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_93_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_94_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_95_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_96_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_97_list_Omap__ident__strong,axiom,
! [T: list_l4703314356710769291_a_nat,F2: 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 ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_li6507455427659069316_a_nat @ F2 @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_98_list_Omap__ident__strong,axiom,
! [T: list_n989787106983797996_a_nat,F2: ( 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 ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_na722425985163074756_a_nat @ F2 @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_99_list_Omap__ident__strong,axiom,
! [T: list_Sum_sum_a_nat,F2: 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 ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_Su2790769393171190532_a_nat @ F2 @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_100_list_Omap__ident__strong,axiom,
! [T: list_nat,F2: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ T ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_nat_nat @ F2 @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_101_list_Oinj__map__strong,axiom,
! [X3: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,Fa: sum_sum_a_nat > 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 ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Su5227373005390213643at_nat @ F2 @ X3 )
= ( map_Su5227373005390213643at_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_102_list_Oinj__map__strong,axiom,
! [X3: list_Sum_sum_a_nat,Xa: list_Sum_sum_a_nat,F2: 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 ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Su2790769393171190532_a_nat @ F2 @ X3 )
= ( map_Su2790769393171190532_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_103_list_Oinj__map__strong,axiom,
! [X3: list_nat,Xa: list_nat,F2: 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 ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_na823391071729141993_a_nat @ F2 @ X3 )
= ( map_na823391071729141993_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_104_list_Oinj__map__strong,axiom,
! [X3: list_nat,Xa: list_nat,F2: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X3 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F2 @ X3 )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_105_list_Omap__cong0,axiom,
! [X3: list_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,G: sum_sum_a_nat > nat] :
( ! [Z: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( set_Sum_sum_a_nat2 @ X3 ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_Su5227373005390213643at_nat @ F2 @ X3 )
= ( map_Su5227373005390213643at_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_106_list_Omap__cong0,axiom,
! [X3: list_Sum_sum_a_nat,F2: 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 ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F2 @ X3 )
= ( map_Su2790769393171190532_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_107_list_Omap__cong0,axiom,
! [X3: list_nat,F2: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X3 ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_na823391071729141993_a_nat @ F2 @ X3 )
= ( map_na823391071729141993_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_108_list_Omap__cong0,axiom,
! [X3: list_nat,F2: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X3 ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F2 @ X3 )
= ( map_nat_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_109_list_Omap__cong,axiom,
! [X3: list_Sum_sum_a_nat,Ya: list_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,G: sum_sum_a_nat > nat] :
( ( X3 = Ya )
=> ( ! [Z: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( set_Sum_sum_a_nat2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_Su5227373005390213643at_nat @ F2 @ X3 )
= ( map_Su5227373005390213643at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_110_list_Omap__cong,axiom,
! [X3: list_Sum_sum_a_nat,Ya: list_Sum_sum_a_nat,F2: 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 ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F2 @ X3 )
= ( map_Su2790769393171190532_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_111_list_Omap__cong,axiom,
! [X3: list_nat,Ya: list_nat,F2: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_na823391071729141993_a_nat @ F2 @ X3 )
= ( map_na823391071729141993_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_112_list_Omap__cong,axiom,
! [X3: list_nat,Ya: list_nat,F2: nat > nat,G: nat > nat] :
( ( X3 = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F2 @ X3 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_113_list_Osimps_I8_J,axiom,
! [F2: nat > sum_sum_a_nat] :
( ( map_na823391071729141993_a_nat @ F2 @ nil_nat )
= nil_Sum_sum_a_nat ) ).
% list.simps(8)
thf(fact_114_list_Osimps_I8_J,axiom,
! [F2: nat > nat] :
( ( map_nat_nat @ F2 @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_115_list_Osimps_I8_J,axiom,
! [F2: sum_sum_a_nat > nat] :
( ( map_Su5227373005390213643at_nat @ F2 @ nil_Sum_sum_a_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_116_list_Osimps_I8_J,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat] :
( ( map_Su2790769393171190532_a_nat @ F2 @ nil_Sum_sum_a_nat )
= nil_Sum_sum_a_nat ) ).
% list.simps(8)
thf(fact_117_le__sup__iff,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X3 @ Z2 )
& ( ord_less_eq_set_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_118_le__sup__iff,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ Z2 )
= ( ( ord_less_eq_nat @ X3 @ Z2 )
& ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_119_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_120_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_121_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_122_assms_I6_J,axiom,
( ass
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( proj_v3643391342904276326_a_nat @ r @ fv_sub ) ) ) ).
% assms(6)
thf(fact_123_proj__tuple__length,axiom,
! [Ns: list_nat,Ms: list_nat,Xs: list_Sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
=> ( ( ( size_size_list_nat @ Ms )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( size_s5283204784079214577_a_nat @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Ms @ Xs ) ) )
= ( size_size_list_nat @ Ns ) ) ) ) ) ) ).
% proj_tuple_length
thf(fact_124_assms_I7_J,axiom,
! [Sigma: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ fv_sub ) @ ( set_nat2 @ fv_sub ) @ ad @ Sigma @ Tau2 )
=> ( ( member8690443509505302927_a_nat @ Sigma @ r )
= ( member8690443509505302927_a_nat @ Tau2 @ r ) ) ) ).
% assms(7)
thf(fact_125_proj__tuple__map,axiom,
! [Ns: list_nat,Ms: list_nat,Sigma: nat > sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
=> ( ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Ms @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) ) )
= ( map_na823391071729141993_a_nat @ Sigma @ Ns ) ) ) ) ) ).
% proj_tuple_map
thf(fact_126_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_127_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_128_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_129_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_130_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_131_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_132_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_133_sup__idem,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_134_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_135_image__eqI,axiom,
! [B: nat > sum_sum_a_nat,F2: list_Sum_sum_a_nat > nat > sum_sum_a_nat,X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ B @ ( image_701559317304863014_a_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_136_image__eqI,axiom,
! [B: list_Sum_sum_a_nat,F2: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ B @ ( image_6721470456781115300_a_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_137_image__eqI,axiom,
! [B: nat > sum_sum_a_nat,F2: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( B
= ( F2 @ X3 ) )
=> ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ B @ ( image_6222892899998961285_a_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_138_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_139_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_140_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_141_sup__left__idem,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% sup_left_idem
thf(fact_142_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_143_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_144_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_145_distinct__union,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( union_Sum_sum_a_nat @ Xs @ Ys ) )
= ( distin2701893636801681144_a_nat @ Ys ) ) ).
% distinct_union
thf(fact_146_list_Oset__map,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F2 @ V ) )
= ( image_5081948215111134021_a_nat @ F2 @ ( set_li2392974972034027290_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_147_list_Oset__map,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,V: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F2 @ V ) )
= ( image_7142520692256960453_a_nat @ F2 @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_148_list_Oset__map,axiom,
! [F2: nat > sum_sum_a_nat,V: list_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F2 @ V ) )
= ( image_7293268710728258664_a_nat @ F2 @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_149_list_Oset__map,axiom,
! [F2: sum_sum_a_nat > nat,V: list_Sum_sum_a_nat] :
( ( set_nat2 @ ( map_Su5227373005390213643at_nat @ F2 @ V ) )
= ( image_2473878607534554506at_nat @ F2 @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_150_list_Oset__map,axiom,
! [F2: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F2 @ V ) )
= ( image_nat_nat @ F2 @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_151_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_152_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_153_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_154_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_155_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_156_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_157_image__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ ( image_5081948215111134021_a_nat @ F2 @ B2 ) ) ) ).
% image_mono
thf(fact_158_subset__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
! [T2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ T2 @ A3 )
=> ( member408289922725080238_a_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_159_subset__iff,axiom,
( ord_le8108555184339247974_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
! [T2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ T2 @ A3 )
=> ( member8690443509505302927_a_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_160_image__subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( member408289922725080238_a_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_161_image__subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le8108555184339247974_a_nat @ ( image_701559317304863014_a_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_162_image__subsetI,axiom,
! [A2: set_na3699693778330250182_a_nat,F2: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( member408289922725080238_a_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6721470456781115300_a_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_163_image__subsetI,axiom,
! [A2: set_na3699693778330250182_a_nat,F2: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F2 @ X2 ) @ B2 ) )
=> ( ord_le8108555184339247974_a_nat @ ( image_6222892899998961285_a_nat @ F2 @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_164_subset__imageE,axiom,
! [B2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
=> ~ ! [C3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C3 @ A2 )
=> ( B2
!= ( image_5081948215111134021_a_nat @ F2 @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_165_image__subset__iff,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ B2 )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
=> ( member408289922725080238_a_nat @ ( F2 @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_166_subset__image__iff,axiom,
! [B2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( ? [AA: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ AA @ A2 )
& ( B2
= ( image_5081948215111134021_a_nat @ F2 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_167_ad__agr__sets__mono,axiom,
! [X4: set_a,Y4: set_a,FV: set_nat,S: set_nat,Sigma: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ad_agr_sets_a_nat @ FV @ S @ Y4 @ Sigma @ Tau2 )
=> ( ad_agr_sets_a_nat @ FV @ S @ X4 @ Sigma @ Tau2 ) ) ) ).
% ad_agr_sets_mono
thf(fact_168_ad__agr__sets__mono_H,axiom,
! [S: set_nat,S2: set_nat,FV: set_nat,X4: set_a,Sigma: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ S @ S2 )
=> ( ( ad_agr_sets_a_nat @ FV @ S2 @ X4 @ Sigma @ Tau2 )
=> ( ad_agr_sets_a_nat @ FV @ S @ X4 @ Sigma @ Tau2 ) ) ) ).
% ad_agr_sets_mono'
thf(fact_169_rev__image__eqI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_170_rev__image__eqI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: nat > sum_sum_a_nat,F2: list_Sum_sum_a_nat > nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member8690443509505302927_a_nat @ B @ ( image_701559317304863014_a_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_171_rev__image__eqI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: list_Sum_sum_a_nat,F2: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member408289922725080238_a_nat @ B @ ( image_6721470456781115300_a_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_172_rev__image__eqI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: nat > sum_sum_a_nat,F2: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( B
= ( F2 @ X3 ) )
=> ( member8690443509505302927_a_nat @ B @ ( image_6222892899998961285_a_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_173_ball__imageD,axiom,
! [F2: 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 @ F2 @ A2 ) )
=> ( P @ X2 ) )
=> ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A2 )
=> ( P @ ( F2 @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_174_image__cong,axiom,
! [M: set_li6526943997496501093_a_nat,N2: set_li6526943997496501093_a_nat,F2: 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 )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_5081948215111134021_a_nat @ F2 @ M )
= ( image_5081948215111134021_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_175_bex__imageD,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ? [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
& ( P @ X5 ) )
=> ? [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_176_image__iff,axiom,
! [Z2: list_Sum_sum_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ Z2 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A2 )
& ( Z2
= ( F2 @ X ) ) ) ) ) ).
% image_iff
thf(fact_177_imageI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ ( F2 @ X3 ) @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_178_imageI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F2 @ X3 ) @ ( image_701559317304863014_a_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_179_imageI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F2: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ ( F2 @ X3 ) @ ( image_6721470456781115300_a_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_180_imageI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F2: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F2 @ X3 ) @ ( image_6222892899998961285_a_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_181_ad__agr__sets__comm,axiom,
! [FV: set_nat,S: set_nat,X4: set_a,Sigma: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ FV @ S @ X4 @ Sigma @ Tau2 )
=> ( ad_agr_sets_a_nat @ FV @ S @ X4 @ Tau2 @ Sigma ) ) ).
% ad_agr_sets_comm
thf(fact_182_image__Un,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ ( image_5081948215111134021_a_nat @ F2 @ B2 ) ) ) ).
% image_Un
thf(fact_183_image__Un,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( image_nat_nat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ ( image_nat_nat @ F2 @ B2 ) ) ) ).
% image_Un
thf(fact_184_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_185_subset__UnE,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ~ ! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ A2 )
=> ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ B2 )
=> ( C2
!= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_186_Un__absorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_187_Un__absorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_188_Un__upper2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_189_Un__upper1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_190_Un__least,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_191_Un__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_192_subset__code_I1_J,axiom,
! [Xs: list_l4703314356710769291_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ B2 )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( member408289922725080238_a_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_193_subset__code_I1_J,axiom,
! [Xs: list_n989787106983797996_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ ( set_na645604395003041787_a_nat @ Xs ) @ B2 )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( member8690443509505302927_a_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_194_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_195_image__set,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( image_5081948215111134021_a_nat @ F2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
= ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F2 @ Xs ) ) ) ).
% image_set
thf(fact_196_image__set,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) ) ) ).
% image_set
thf(fact_197_image__set,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( image_2473878607534554506at_nat @ F2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( set_nat2 @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) ) ) ).
% image_set
thf(fact_198_image__set,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat] :
( ( image_7293268710728258664_a_nat @ F2 @ ( set_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F2 @ Xs ) ) ) ).
% image_set
thf(fact_199_image__set,axiom,
! [F2: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F2 @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F2 @ Xs ) ) ) ).
% image_set
thf(fact_200_sup__left__commute,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_201_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_202_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X ) ) ) ).
% sup_commute
thf(fact_203_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_204_sup__assoc,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_205_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_206_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_207_inf__sup__aci_I6_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_208_inf__sup__aci_I7_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_209_inf__sup__aci_I8_J,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_210_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_211_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_212_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_213_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_214_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_215_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_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_231_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_232_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_233_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_234_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_235_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_236_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_237_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_238_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( A5
= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_239_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_240_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_241_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_242_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_243_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_244_sup__absorb2,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( sup_sup_set_nat @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_245_sup__absorb2,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( sup_sup_nat @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_246_sup__absorb1,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( sup_sup_set_nat @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_247_sup__absorb1,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_248_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_249_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_250_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_251_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_252_sup__unique,axiom,
! [F2: set_nat > set_nat > set_nat,X3: set_nat,Y3: set_nat] :
( ! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F2 @ X2 @ Y ) )
=> ( ! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ Y @ ( F2 @ X2 @ Y ) )
=> ( ! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ Z @ X2 )
=> ( ord_less_eq_set_nat @ ( F2 @ Y @ Z ) @ X2 ) ) )
=> ( ( sup_sup_set_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_253_sup__unique,axiom,
! [F2: nat > nat > nat,X3: nat,Y3: nat] :
( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( F2 @ X2 @ Y ) )
=> ( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ ( F2 @ X2 @ Y ) )
=> ( ! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ ( F2 @ Y @ Z ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_254_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_255_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_256_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_257_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_258_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_259_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y2: nat] :
( ( sup_sup_nat @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_260_sup__least,axiom,
! [Y3: set_nat,X3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_261_sup__least,axiom,
! [Y3: nat,X3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_262_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_263_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_264_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_265_sup_Omono,axiom,
! [C: nat,A: nat,D2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D2 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_266_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_267_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_268_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_269_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_270_sup__ge2,axiom,
! [Y3: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_271_sup__ge2,axiom,
! [Y3: nat,X3: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_272_sup__ge1,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_273_sup__ge1,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_274_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_275_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_276_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_277_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_278_inf__sup__ord_I3_J,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_279_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_280_inf__sup__ord_I4_J,axiom,
! [Y3: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_281_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X3: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_282_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_283_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_284_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 )
=> ? [F: sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_285_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 )
=> ? [F: nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ F @ Xs ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_286_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_287_all__subset__image,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ A2 )
=> ( P @ ( image_5081948215111134021_a_nat @ F2 @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_288_in__set__simps_I3_J,axiom,
! [X3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ nil_li1906260230833442699_a_nat ) ) ).
% in_set_simps(3)
thf(fact_289_in__set__simps_I3_J,axiom,
! [X3: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ nil_na5178701105474258796_a_nat ) ) ).
% in_set_simps(3)
thf(fact_290_in__set__simps_I3_J,axiom,
! [X3: nat] :
~ ( member_nat @ X3 @ ( set_nat2 @ nil_nat ) ) ).
% in_set_simps(3)
thf(fact_291_sp__equiv__list__subset,axiom,
! [Ms: list_Sum_sum_a_nat,Ns: list_Sum_sum_a_nat,Sigma: sum_sum_a_nat > sum_sum_a_nat,Sigma2: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Ms ) @ ( set_Sum_sum_a_nat2 @ Ns ) )
=> ( ( sp_equiv_list_a_nat @ ( map_Su2790769393171190532_a_nat @ Sigma @ Ns ) @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ns ) )
=> ( sp_equiv_list_a_nat @ ( map_Su2790769393171190532_a_nat @ Sigma @ Ms ) @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_292_sp__equiv__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,Sigma: nat > sum_sum_a_nat,Sigma2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) )
=> ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_293_map__sorted__distinct__set__unique,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( inj_on8752143810983750942at_nat @ F2 @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
=> ( ( distinct_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Ys ) )
=> ( ( distinct_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Ys ) )
=> ( ( ( set_Sum_sum_a_nat2 @ Xs )
= ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_294_map__sorted__distinct__set__unique,axiom,
! [F2: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_nat @ F2 @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F2 @ Xs ) )
=> ( ( distinct_nat @ ( map_nat_nat @ F2 @ Xs ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F2 @ Ys ) )
=> ( ( distinct_nat @ ( map_nat_nat @ F2 @ Ys ) )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_295_sorted__insort__insert__key,axiom,
! [F2: nat > nat,Xs: list_nat,X3: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F2 @ Xs ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F2 @ ( linord1921536354676448932at_nat @ F2 @ X3 @ Xs ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_296_sorted__insort__insert__key,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Su5227373005390213643at_nat @ F2 @ ( linord458367490959630669at_nat @ F2 @ X3 @ Xs ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_297_ad__agr__list__link,axiom,
! [Ns: list_nat,AD: set_a,Sigma: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ns ) @ AD @ Sigma @ Tau2 )
= ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau2 @ Ns ) ) ) ).
% ad_agr_list_link
thf(fact_298_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_299_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_300_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_301_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_302_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_303_empty__iff,axiom,
! [C: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_304_empty__iff,axiom,
! [C: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ C @ bot_bo6441361344521902642_a_nat ) ).
% empty_iff
thf(fact_305_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_306_inf__right__idem,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Y3 )
= ( inf_inf_set_nat @ X3 @ Y3 ) ) ).
% inf_right_idem
thf(fact_307_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_308_inf__left__idem,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
= ( inf_inf_set_nat @ X3 @ Y3 ) ) ).
% inf_left_idem
thf(fact_309_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_310_inf__idem,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_311_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_312_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_313_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_314_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_315_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_316_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_317_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_318_le__inf__iff,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_319_le__inf__iff,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_320_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_321_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_322_image__empty,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat @ F2 @ bot_bo1033123847703346641_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% image_empty
thf(fact_323_image__empty,axiom,
! [F2: nat > nat] :
( ( image_nat_nat @ F2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_324_empty__is__image,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% empty_is_image
thf(fact_325_empty__is__image,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_326_image__is__empty,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F2 @ A2 )
= bot_bo1033123847703346641_a_nat )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% image_is_empty
thf(fact_327_image__is__empty,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F2 @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_328_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_329_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_330_sup__bot__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_331_sup__bot__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_332_bot__eq__sup__iff,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_333_sup__eq__bot__iff,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y3 )
= bot_bot_set_nat )
= ( ( X3 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_334_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_335_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_336_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_337_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_338_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_339_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_340_inf__sup__absorb,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_341_sup__inf__absorb,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_342_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_343_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_344_Int__Un__eq_I4_J,axiom,
! [T3: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T3 @ ( inf_inf_set_nat @ S @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_345_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T3 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_346_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_347_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T3 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_348_Un__Int__eq_I4_J,axiom,
! [T3: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T3 @ ( sup_sup_set_nat @ S @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_349_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T3 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_350_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_351_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T3 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_352_List_Oset__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% List.set_empty
thf(fact_353_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_354_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 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
=> ( X != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_363_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_364_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_365_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_366_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_367_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_368_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_369_Int__absorb,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_370_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_371_equals0I,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ! [Y: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y @ A2 )
=> ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_372_equals0I,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ! [Y: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ Y @ A2 )
=> ( A2 = bot_bo6441361344521902642_a_nat ) ) ).
% equals0I
thf(fact_373_equals0I,axiom,
! [A2: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_374_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_375_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_376_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_377_emptyE,axiom,
! [A: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_378_emptyE,axiom,
! [A: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ).
% emptyE
thf(fact_379_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_380_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_381_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_382_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_383_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_384_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_385_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_386_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_387_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_388_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_389_inf__left__commute,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_390_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_391_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X ) ) ) ).
% inf_commute
thf(fact_392_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_393_inf__assoc,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_394_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_395_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_396_inf__sup__aci_I2_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_397_inf__sup__aci_I3_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_398_inf__sup__aci_I4_J,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
= ( inf_inf_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_399_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_400_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_401_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_402_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_403_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_404_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_405_inf__sup__ord_I2_J,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_406_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_407_inf__sup__ord_I1_J,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_408_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_409_inf__le1,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_410_inf__le1,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_411_inf__le2,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_412_inf__le2,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_413_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_414_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_415_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_416_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_417_inf__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_418_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_419_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_420_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_421_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_422_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_423_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_424_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_425_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_426_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_427_inf__unique,axiom,
! [F2: set_nat > set_nat > set_nat,X3: set_nat,Y3: set_nat] :
( ! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X2 @ Y ) @ X2 )
=> ( ! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X2 @ Y ) @ Y )
=> ( ! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ X2 @ Z )
=> ( ord_less_eq_set_nat @ X2 @ ( F2 @ Y @ Z ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_428_inf__unique,axiom,
! [F2: nat > nat > nat,X3: nat,Y3: nat] :
( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( F2 @ X2 @ Y ) @ X2 )
=> ( ! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( F2 @ X2 @ Y ) @ Y )
=> ( ! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ ( F2 @ Y @ Z ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y3 )
= ( F2 @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_429_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_430_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y2: nat] :
( ( inf_inf_nat @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_431_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_432_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_433_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_434_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_435_inf__absorb1,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( inf_inf_set_nat @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_436_inf__absorb1,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_437_inf__absorb2,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( inf_inf_set_nat @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_438_inf__absorb2,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_439_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_440_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_441_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_442_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_443_inf__greatest,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_444_inf__greatest,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_445_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( A5
= ( inf_inf_set_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_446_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_447_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_448_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_449_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_450_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_451_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_452_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_453_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_454_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_455_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_456_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_457_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_458_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_459_distrib__imp1,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y ) @ ( inf_inf_set_nat @ X2 @ Z ) ) )
=> ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_460_distrib__imp2,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ ( sup_sup_set_nat @ X2 @ Z ) ) )
=> ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_461_inf__sup__distrib1,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_462_inf__sup__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_463_sup__inf__distrib1,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_464_sup__inf__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_465_Int__Collect__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 @ ( collec7555443234367654128_a_nat @ P ) ) @ ( inf_in3249246906714053971_a_nat @ B2 @ ( collec7555443234367654128_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_466_Int__Collect__mono,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ A2 @ B2 )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le8108555184339247974_a_nat @ ( inf_in8399021836546144180_a_nat @ A2 @ ( collec5629555741568564177_a_nat @ P ) ) @ ( inf_in8399021836546144180_a_nat @ B2 @ ( collec5629555741568564177_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_467_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_468_Int__greatest,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_469_Int__absorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_470_Int__absorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_471_Int__lower2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_472_Int__lower1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_473_Int__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_474_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_475_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_476_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_477_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_478_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_479_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_480_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_481_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_482_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_483_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_484_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_485_ad__agr__list__length,axiom,
! [X4: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X4 @ Xs @ Ys )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% ad_agr_list_length
thf(fact_486_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_487_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_488_fo__nmlzd__code,axiom,
( fo_nmlzd_a
= ( ^ [AD2: set_a,Xs3: list_Sum_sum_a_nat] :
( ( fo_nmlz_a @ AD2 @ Xs3 )
= Xs3 ) ) ) ).
% fo_nmlzd_code
thf(fact_489_distrib__inf__le,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) @ ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_490_distrib__inf__le,axiom,
! [X3: nat,Y3: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ ( inf_inf_nat @ X3 @ Z2 ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_491_distrib__sup__le,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_492_distrib__sup__le,axiom,
! [X3: nat,Y3: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ ( sup_sup_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_493_image__Int__subset,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) @ ( inf_in3249246906714053971_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ ( image_5081948215111134021_a_nat @ F2 @ B2 ) ) ) ).
% image_Int_subset
thf(fact_494_image__Int__subset,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ ( image_nat_nat @ F2 @ B2 ) ) ) ).
% image_Int_subset
thf(fact_495_Un__Int__assoc__eq,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_496_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_497_map__inj__on,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Su5227373005390213643at_nat @ F2 @ Xs )
= ( map_Su5227373005390213643at_nat @ F2 @ Ys ) )
=> ( ( inj_on8752143810983750942at_nat @ F2 @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_498_map__inj__on,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F2 @ Xs )
= ( map_Su2790769393171190532_a_nat @ F2 @ Ys ) )
=> ( ( inj_on6255688694610590513_a_nat @ F2 @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_499_map__inj__on,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F2 @ Xs )
= ( map_na823391071729141993_a_nat @ F2 @ Ys ) )
=> ( ( inj_on4348161877322679292_a_nat @ F2 @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_500_map__inj__on,axiom,
! [F2: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( map_nat_nat @ F2 @ Ys ) )
=> ( ( inj_on_nat_nat @ F2 @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_501_inj__on__map__eq__map,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( inj_on8752143810983750942at_nat @ F2 @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) )
=> ( ( ( map_Su5227373005390213643at_nat @ F2 @ Xs )
= ( map_Su5227373005390213643at_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_502_inj__on__map__eq__map,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( inj_on6255688694610590513_a_nat @ F2 @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) )
=> ( ( ( map_Su2790769393171190532_a_nat @ F2 @ Xs )
= ( map_Su2790769393171190532_a_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_503_inj__on__map__eq__map,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on4348161877322679292_a_nat @ F2 @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( ( map_na823391071729141993_a_nat @ F2 @ Xs )
= ( map_na823391071729141993_a_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_504_inj__on__map__eq__map,axiom,
! [F2: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_nat @ F2 @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( ( map_nat_nat @ F2 @ Xs )
= ( map_nat_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_505_distinct__map,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( distinct_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
= ( ( distin2701893636801681144_a_nat @ Xs )
& ( inj_on8752143810983750942at_nat @ F2 @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_506_distinct__map,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) )
= ( ( distin2701893636801681144_a_nat @ Xs )
& ( inj_on6255688694610590513_a_nat @ F2 @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_507_distinct__map,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat] :
( ( distin2701893636801681144_a_nat @ ( map_na823391071729141993_a_nat @ F2 @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on4348161877322679292_a_nat @ F2 @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_508_distinct__map,axiom,
! [F2: nat > nat,Xs: list_nat] :
( ( distinct_nat @ ( map_nat_nat @ F2 @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on_nat_nat @ F2 @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_509_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_510_le__cases3,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_511_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_512_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_513_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_514_order__antisym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_515_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_516_order__trans,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_517_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_518_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_519_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_520_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_521_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_522_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_523_order__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_524_order__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_525_order__eq__refl,axiom,
! [X3: nat,Y3: nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_526_linorder__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_527_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_528_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_529_linorder__le__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_530_order__antisym__conv,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_531_ad__agr__list__subset,axiom,
! [Ms: list_Sum_sum_a_nat,Ns: list_Sum_sum_a_nat,X4: set_a,Sigma: sum_sum_a_nat > sum_sum_a_nat,Sigma2: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Ms ) @ ( set_Sum_sum_a_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X4 @ ( map_Su2790769393171190532_a_nat @ Sigma @ Ns ) @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X4 @ ( map_Su2790769393171190532_a_nat @ Sigma @ Ms ) @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_532_ad__agr__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,X4: set_a,Sigma: nat > sum_sum_a_nat,Sigma2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X4 @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) )
=> ( ad_agr_list_a_nat @ X4 @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_533_ext__tuple__correct,axiom,
! [Fv_sub: list_nat,Fv_sub_comp: list_nat,Fv_all: list_nat,Ass: set_li6526943997496501093_a_nat,AD: set_a,R: set_na3699693778330250182_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_sub )
& ( distinct_nat @ Fv_sub ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_sub_comp )
& ( distinct_nat @ Fv_sub_comp ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_all )
& ( distinct_nat @ Fv_all ) )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Fv_sub ) @ ( set_nat2 @ Fv_sub_comp ) )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ ( set_nat2 @ Fv_sub ) @ ( set_nat2 @ Fv_sub_comp ) )
= ( set_nat2 @ Fv_all ) )
=> ( ( Ass
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R @ Fv_sub ) ) )
=> ( ! [Sigma3: 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 @ Sigma3 @ Tau )
=> ( ( member8690443509505302927_a_nat @ Sigma3 @ R )
= ( member8690443509505302927_a_nat @ Tau @ R ) ) )
=> ( ( ext_tuple_set_a @ AD @ Fv_sub @ Fv_sub_comp @ Ass )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R @ Fv_all ) ) ) ) ) ) ) ) ) ) ).
% ext_tuple_correct
thf(fact_534_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_535_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_536_inj__on__image__Int,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C2 )
=> ( ( image_5081948215111134021_a_nat @ F2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) )
= ( inf_in3249246906714053971_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ ( image_5081948215111134021_a_nat @ F2 @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_537_inj__on__image__Int,axiom,
! [F2: nat > nat,C2: set_nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( image_nat_nat @ F2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ ( image_nat_nat @ F2 @ B2 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_538_inj__on__iff__surj,axiom,
! [A2: set_li6526943997496501093_a_nat,A7: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ( ? [F3: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F3 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F3 @ A2 ) @ A7 ) ) )
= ( ? [G2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat @ G2 @ A7 )
= A2 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_539_ext__tuple__sound_I1_J,axiom,
! [Fv_sub: list_nat,Fv_sub_comp: list_nat,Fv_all: list_nat,Ass: set_li6526943997496501093_a_nat,AD: set_a,R: set_na3699693778330250182_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_sub )
& ( distinct_nat @ Fv_sub ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_sub_comp )
& ( distinct_nat @ Fv_sub_comp ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_all )
& ( distinct_nat @ Fv_all ) )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Fv_sub ) @ ( set_nat2 @ Fv_sub_comp ) )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ ( set_nat2 @ Fv_sub ) @ ( set_nat2 @ Fv_sub_comp ) )
= ( set_nat2 @ Fv_all ) )
=> ( ( Ass
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R @ Fv_sub ) ) )
=> ( ! [Sigma3: 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 @ Sigma3 @ Tau )
=> ( ( member8690443509505302927_a_nat @ Sigma3 @ R )
= ( member8690443509505302927_a_nat @ Tau @ R ) ) )
=> ( ( 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 ) ) ) )
=> ( member408289922725080238_a_nat @ ( fo_nmlz_a @ AD @ ( proj_tuple_a @ Fv_sub @ ( zip_na2013496608136855606_a_nat @ Fv_all @ Xs ) ) ) @ Ass ) ) ) ) ) ) ) ) ) ).
% ext_tuple_sound(1)
thf(fact_540_inj__on__Un__image__eq__iff,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
=> ( ( ( image_5081948215111134021_a_nat @ F2 @ A2 )
= ( image_5081948215111134021_a_nat @ F2 @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_541_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_542_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_543_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_544_ext__tuple__set__def,axiom,
( ext_tuple_set_a
= ( ^ [AD2: set_a,Ns2: list_nat,Ns3: list_nat,X6: set_li6526943997496501093_a_nat] : ( if_set7709265119413304363_a_nat @ ( Ns3 = nil_nat ) @ X6 @ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD2 ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ ( ext_tuple_a @ AD2 @ Ns2 @ Ns3 ) @ X6 ) ) ) ) ) ) ).
% ext_tuple_set_def
thf(fact_545_inj__on__mapI,axiom,
! [F2: sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on8752143810983750942at_nat @ F2 @ ( comple1247738100258233164_a_nat @ ( image_3940260845811589407_a_nat @ set_Sum_sum_a_nat2 @ A2 ) ) )
=> ( inj_on5303670800567886516st_nat @ ( map_Su5227373005390213643at_nat @ F2 ) @ A2 ) ) ).
% inj_on_mapI
thf(fact_546_inj__on__mapI,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on6255688694610590513_a_nat @ F2 @ ( comple1247738100258233164_a_nat @ ( image_3940260845811589407_a_nat @ set_Sum_sum_a_nat2 @ A2 ) ) )
=> ( inj_on6609798167860701873_a_nat @ ( map_Su2790769393171190532_a_nat @ F2 ) @ A2 ) ) ).
% inj_on_mapI
thf(fact_547_inj__on__mapI,axiom,
! [F2: nat > sum_sum_a_nat,A2: set_list_nat] :
( ( inj_on4348161877322679292_a_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_1775855109352712557et_nat @ set_nat2 @ A2 ) ) )
=> ( inj_on1175180479830800114_a_nat @ ( map_na823391071729141993_a_nat @ F2 ) @ A2 ) ) ).
% inj_on_mapI
thf(fact_548_inj__on__mapI,axiom,
! [F2: nat > nat,A2: set_list_nat] :
( ( inj_on_nat_nat @ F2 @ ( comple7399068483239264473et_nat @ ( image_1775855109352712557et_nat @ set_nat2 @ A2 ) ) )
=> ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F2 ) @ A2 ) ) ).
% inj_on_mapI
thf(fact_549_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_550_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_551_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_552_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_553_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_554_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_555_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_556_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_557_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_558_ext__tuple__sound_I2_J,axiom,
! [Fv_sub: list_nat,Fv_sub_comp: list_nat,Fv_all: list_nat,Ass: set_li6526943997496501093_a_nat,AD: set_a,R: set_na3699693778330250182_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_sub )
& ( distinct_nat @ Fv_sub ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_sub_comp )
& ( distinct_nat @ Fv_sub_comp ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_all )
& ( distinct_nat @ Fv_all ) )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Fv_sub ) @ ( set_nat2 @ Fv_sub_comp ) )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ ( set_nat2 @ Fv_sub ) @ ( set_nat2 @ Fv_sub_comp ) )
= ( set_nat2 @ Fv_all ) )
=> ( ( Ass
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R @ Fv_sub ) ) )
=> ( ! [Sigma3: 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 @ Sigma3 @ Tau )
=> ( ( member8690443509505302927_a_nat @ Sigma3 @ R )
= ( member8690443509505302927_a_nat @ Tau @ R ) ) )
=> ( ( 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 ) ) ) )
=> ( member408289922725080238_a_nat @ Xs @ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R @ Fv_all ) ) ) ) ) ) ) ) ) ) ) ).
% ext_tuple_sound(2)
thf(fact_559_ext__tuple__complete,axiom,
! [Fv_sub: list_nat,Fv_sub_comp: list_nat,Fv_all: list_nat,Ass: set_li6526943997496501093_a_nat,AD: set_a,R: set_na3699693778330250182_a_nat,Xs: list_Sum_sum_a_nat,Sigma: nat > sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_sub )
& ( distinct_nat @ Fv_sub ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_sub_comp )
& ( distinct_nat @ Fv_sub_comp ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Fv_all )
& ( distinct_nat @ Fv_all ) )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ Fv_sub ) @ ( set_nat2 @ Fv_sub_comp ) )
= bot_bot_set_nat )
=> ( ( ( sup_sup_set_nat @ ( set_nat2 @ Fv_sub ) @ ( set_nat2 @ Fv_sub_comp ) )
= ( set_nat2 @ Fv_all ) )
=> ( ( Ass
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( proj_v3643391342904276326_a_nat @ R @ Fv_sub ) ) )
=> ( ! [Sigma3: 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 @ Sigma3 @ Tau )
=> ( ( member8690443509505302927_a_nat @ Sigma3 @ R )
= ( member8690443509505302927_a_nat @ Tau @ R ) ) )
=> ( ( Xs
= ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ Sigma @ Fv_all ) ) )
=> ( ( member8690443509505302927_a_nat @ Sigma @ R )
=> ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% ext_tuple_complete
thf(fact_560_inj__on__image__eq__iff,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C2 )
=> ( ( ( image_5081948215111134021_a_nat @ F2 @ A2 )
= ( image_5081948215111134021_a_nat @ F2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_561_inj__on__image__mem__iff,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ B2 )
=> ( ( member408289922725080238_a_nat @ A @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( member408289922725080238_a_nat @ ( F2 @ A ) @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( member408289922725080238_a_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_562_inj__on__image__mem__iff,axiom,
! [F2: list_Sum_sum_a_nat > nat > sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on3806193600625622162_a_nat @ F2 @ B2 )
=> ( ( member408289922725080238_a_nat @ A @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ( member8690443509505302927_a_nat @ ( F2 @ A ) @ ( image_701559317304863014_a_nat @ F2 @ A2 ) )
= ( member408289922725080238_a_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_563_inj__on__image__mem__iff,axiom,
! [F2: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,B2: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( inj_on602732703247098640_a_nat @ F2 @ B2 )
=> ( ( member8690443509505302927_a_nat @ A @ B2 )
=> ( ( ord_le8108555184339247974_a_nat @ A2 @ B2 )
=> ( ( member408289922725080238_a_nat @ ( F2 @ A ) @ ( image_6721470456781115300_a_nat @ F2 @ A2 ) )
= ( member8690443509505302927_a_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_564_inj__on__image__mem__iff,axiom,
! [F2: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( inj_on8496004383624361457_a_nat @ F2 @ B2 )
=> ( ( member8690443509505302927_a_nat @ A @ B2 )
=> ( ( ord_le8108555184339247974_a_nat @ A2 @ B2 )
=> ( ( member8690443509505302927_a_nat @ ( F2 @ A ) @ ( image_6222892899998961285_a_nat @ F2 @ A2 ) )
= ( member8690443509505302927_a_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_565_subset__image__inj,axiom,
! [S: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ S @ ( image_5081948215111134021_a_nat @ F2 @ T3 ) )
= ( ? [U: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ U @ T3 )
& ( inj_on6609798167860701873_a_nat @ F2 @ U )
& ( S
= ( image_5081948215111134021_a_nat @ F2 @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_566_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_567_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_568_UnionI,axiom,
! [X4: set_li6526943997496501093_a_nat,C2: set_se4330304633200676677_a_nat,A2: list_Sum_sum_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ C2 )
=> ( ( member408289922725080238_a_nat @ A2 @ X4 )
=> ( member408289922725080238_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_569_UnionI,axiom,
! [X4: set_na3699693778330250182_a_nat,C2: set_se5822283258546872870_a_nat,A2: nat > sum_sum_a_nat] :
( ( member3060896489619847151_a_nat @ X4 @ C2 )
=> ( ( member8690443509505302927_a_nat @ A2 @ X4 )
=> ( member8690443509505302927_a_nat @ A2 @ ( comple8643769897048643123_a_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_570_Union__iff,axiom,
! [A2: list_Sum_sum_a_nat,C2: set_se4330304633200676677_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ C2 ) )
= ( ? [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ C2 )
& ( member408289922725080238_a_nat @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_571_Union__iff,axiom,
! [A2: nat > sum_sum_a_nat,C2: set_se5822283258546872870_a_nat] :
( ( member8690443509505302927_a_nat @ A2 @ ( comple8643769897048643123_a_nat @ C2 ) )
= ( ? [X: set_na3699693778330250182_a_nat] :
( ( member3060896489619847151_a_nat @ X @ C2 )
& ( member8690443509505302927_a_nat @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_572_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_573_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_574_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,D: 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 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_5081948215111134021_a_nat @ C2 @ A2 ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_575_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,D: 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 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_5081948215111134021_a_nat @ C2 @ A2 ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_576_UnionE,axiom,
! [A2: list_Sum_sum_a_nat,C2: set_se4330304633200676677_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ C2 ) )
=> ~ ! [X7: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A2 @ X7 )
=> ~ ( member5553968465346197646_a_nat @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_577_UnionE,axiom,
! [A2: nat > sum_sum_a_nat,C2: set_se5822283258546872870_a_nat] :
( ( member8690443509505302927_a_nat @ A2 @ ( comple8643769897048643123_a_nat @ C2 ) )
=> ~ ! [X7: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A2 @ X7 )
=> ~ ( member3060896489619847151_a_nat @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_578_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_579_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_580_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_581_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_582_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_583_Union__Int__subset,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 ) ) ) ).
% Union_Int_subset
thf(fact_584_inj__on__image,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_se4330304633200676677_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ ( comple1686717674086456018_a_nat @ A2 ) )
=> ( inj_on561899399213738673_a_nat @ ( image_5081948215111134021_a_nat @ F2 ) @ A2 ) ) ).
% inj_on_image
thf(fact_585_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_586_Union__image__empty,axiom,
! [A2: set_nat,F2: nat > set_nat] :
( ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ bot_bot_set_nat ) ) )
= A2 ) ).
% Union_image_empty
thf(fact_587_cSUP__least,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: 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 @ ( F2 @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ F2 @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_588_cSUP__least,axiom,
! [A2: set_na3699693778330250182_a_nat,F2: ( 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 @ ( F2 @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ F2 @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_589_cSUP__least,axiom,
! [A2: set_nat,F2: nat > nat,M: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ A2 ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_590_cSup__least,axiom,
! [X4: set_nat,Z2: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X4 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_591_cSup__eq__non__empty,axiom,
! [X4: set_nat,A: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ X2 @ A ) )
=> ( ! [Y: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ X4 )
=> ( ord_less_eq_nat @ X5 @ Y ) )
=> ( ord_less_eq_nat @ A @ Y ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= A ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_592_ext__tuple__set__eq,axiom,
! [X4: set_li6526943997496501093_a_nat,AD: set_a,Ns: list_nat,Ns4: list_nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ X4 )
=> ( fo_nmlzd_a @ AD @ X2 ) )
=> ( ( ext_tuple_set_a @ AD @ Ns @ Ns4 @ X4 )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ AD ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ ( ext_tuple_a @ AD @ Ns @ Ns4 ) @ X4 ) ) ) ) ) ).
% ext_tuple_set_eq
thf(fact_593_ball__empty,axiom,
! [P: nat > $o,X5: nat] :
( ( member_nat @ X5 @ bot_bot_set_nat )
=> ( P @ X5 ) ) ).
% ball_empty
thf(fact_594_Ball__def,axiom,
( ball_l5590288159734409519_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,P2: list_Sum_sum_a_nat > $o] :
! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A3 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def
thf(fact_595_Ball__def,axiom,
( ball_n7465829111230741776_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,P2: ( nat > sum_sum_a_nat ) > $o] :
! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A3 )
=> ( P2 @ X ) ) ) ) ).
% Ball_def
thf(fact_596_in__set__simps_I4_J,axiom,
! [P: nat > $o,X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ nil_nat ) )
=> ( P @ X5 ) ) ).
% in_set_simps(4)
thf(fact_597_cSup__eq__maximum,axiom,
! [Z2: nat,X4: set_nat] :
( ( member_nat @ Z2 @ X4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X4 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_598_sp__equiv__list__link,axiom,
! [Sigma: nat > sum_sum_a_nat,Ns: list_nat,Tau2: nat > sum_sum_a_nat] :
( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau2 @ Ns ) )
= ( sp_equiv_a_nat @ Sigma @ Tau2 @ ( set_nat2 @ Ns ) ) ) ).
% sp_equiv_list_link
thf(fact_599_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_600_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_601_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_602_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_603_ad__agr__list__def,axiom,
( ad_agr_list_a_nat
= ( ^ [X6: set_a,Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs3 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
& ( ad_equiv_list_a_nat @ X6 @ Xs3 @ Ys2 )
& ( sp_equiv_list_a_nat @ Xs3 @ Ys2 ) ) ) ) ).
% ad_agr_list_def
thf(fact_604_Sup__SUP__eq,axiom,
( comple1334631452232999051_nat_o
= ( ^ [S3: set_li4526012430949197550_nat_o,X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ ( image_2943553480580064358_a_nat @ collec7555443234367654128_a_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_605_Sup__SUP__eq,axiom,
( comple8250499579560596074_nat_o
= ( ^ [S3: set_na8448764090737828173_nat_o,X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ ( comple8643769897048643123_a_nat @ ( image_1911308356169199528_a_nat @ collec5629555741568564177_a_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_606_inj__on__image__Fpow,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( inj_on561899399213738673_a_nat @ ( image_5081948215111134021_a_nat @ F2 ) @ ( finite3225531020566593003_a_nat @ A2 ) ) ) ).
% inj_on_image_Fpow
thf(fact_607_subset__emptyI,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ! [X2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) ).
% subset_emptyI
thf(fact_608_subset__emptyI,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ! [X2: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( ord_le8108555184339247974_a_nat @ A2 @ bot_bo6441361344521902642_a_nat ) ) ).
% subset_emptyI
thf(fact_609_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_610_nall__tuplesI,axiom,
! [Vs: list_Sum_sum_a_nat,N: nat,AD: set_a] :
( ( ( size_s5283204784079214577_a_nat @ Vs )
= N )
=> ( ( fo_nmlzd_a @ AD @ Vs )
=> ( member408289922725080238_a_nat @ Vs @ ( nall_tuples_a @ AD @ N ) ) ) ) ).
% nall_tuplesI
thf(fact_611_image__Fpow__mono,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ B2 )
=> ( ord_le8138476598237931237_a_nat @ ( image_3472601871771700037_a_nat @ ( image_5081948215111134021_a_nat @ F2 ) @ ( finite3225531020566593003_a_nat @ A2 ) ) @ ( finite3225531020566593003_a_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_612_all__tuplesD,axiom,
! [Vs: list_Sum_sum_a_nat,Xs: set_Sum_sum_a_nat,N: nat] :
( ( member408289922725080238_a_nat @ Vs @ ( all_tu407047557562860027_a_nat @ Xs @ N ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Vs )
= N )
& ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Vs ) @ Xs ) ) ) ).
% all_tuplesD
thf(fact_613_all__tuplesD,axiom,
! [Vs: list_nat,Xs: set_nat,N: nat] :
( ( member_list_nat @ Vs @ ( all_tuples_nat @ Xs @ N ) )
=> ( ( ( size_size_list_nat @ Vs )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Vs ) @ Xs ) ) ) ).
% all_tuplesD
thf(fact_614_empty__in__Fpow,axiom,
! [A2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A2 ) ) ).
% empty_in_Fpow
thf(fact_615_all__tuples__setD,axiom,
! [Vs: list_Sum_sum_a_nat,Xs: set_Sum_sum_a_nat,N: nat] :
( ( member408289922725080238_a_nat @ Vs @ ( all_tu407047557562860027_a_nat @ Xs @ N ) )
=> ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Vs ) @ Xs ) ) ).
% all_tuples_setD
thf(fact_616_all__tuples__setD,axiom,
! [Vs: list_nat,Xs: set_nat,N: nat] :
( ( member_list_nat @ Vs @ ( all_tuples_nat @ Xs @ N ) )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ Vs ) @ Xs ) ) ).
% all_tuples_setD
thf(fact_617_all__tuplesI,axiom,
! [Vs: list_Sum_sum_a_nat,N: nat,Xs: set_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Vs )
= N )
=> ( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Vs ) @ Xs )
=> ( member408289922725080238_a_nat @ Vs @ ( all_tu407047557562860027_a_nat @ Xs @ N ) ) ) ) ).
% all_tuplesI
thf(fact_618_all__tuplesI,axiom,
! [Vs: list_nat,N: nat,Xs: set_nat] :
( ( ( size_size_list_nat @ Vs )
= N )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Vs ) @ Xs )
=> ( member_list_nat @ Vs @ ( all_tuples_nat @ Xs @ N ) ) ) ) ).
% all_tuplesI
thf(fact_619_inj__on__Un,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B2 ) )
= ( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
& ( inj_on6609798167860701873_a_nat @ F2 @ B2 )
& ( ( inf_in3249246906714053971_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) ) @ ( image_5081948215111134021_a_nat @ F2 @ ( minus_7395159227704179404_a_nat @ B2 @ A2 ) ) )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% inj_on_Un
thf(fact_620_inj__on__Un,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( inj_on_nat_nat @ F2 @ A2 )
& ( inj_on_nat_nat @ F2 @ B2 )
& ( ( inf_inf_set_nat @ ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( image_nat_nat @ F2 @ ( minus_minus_set_nat @ B2 @ A2 ) ) )
= bot_bot_set_nat ) ) ) ).
% inj_on_Un
thf(fact_621_subset__subseqs,axiom,
! [X4: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat @ X4 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_622_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_623_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_624_cSUP__union,axiom,
! [A2: set_nat,F2: nat > set_nat,B2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ B2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ B2 ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_625_Diff__iff,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) )
= ( ( member408289922725080238_a_nat @ C @ A2 )
& ~ ( member408289922725080238_a_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_626_Diff__iff,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( minus_5517490076408937517_a_nat @ A2 @ B2 ) )
= ( ( member8690443509505302927_a_nat @ C @ A2 )
& ~ ( member8690443509505302927_a_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_627_DiffI,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 @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_628_DiffI,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 @ ( minus_5517490076408937517_a_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_629_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_630_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_631_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_632_Un__Diff__cancel,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_633_Un__Diff__cancel2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_634_bdd__above_OI,axiom,
! [A2: set_nat,M: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ( condit2214826472909112428ve_nat @ A2 ) ) ).
% bdd_above.I
thf(fact_635_bdd__above__empty,axiom,
condit2214826472909112428ve_nat @ bot_bot_set_nat ).
% bdd_above_empty
thf(fact_636_bdd__above__Un,axiom,
! [A2: set_nat,B2: set_nat] :
( ( condit2214826472909112428ve_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( condit2214826472909112428ve_nat @ A2 )
& ( condit2214826472909112428ve_nat @ B2 ) ) ) ).
% bdd_above_Un
thf(fact_637_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_638_Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_639_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
= nil_nat ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_640_bdd__above_OE,axiom,
! [A2: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ~ ! [M2: nat] :
~ ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( ord_less_eq_nat @ X5 @ M2 ) ) ) ).
% bdd_above.E
thf(fact_641_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A3: set_nat] :
? [M3: nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ord_less_eq_nat @ X @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_642_bdd__above__Int2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( condit2214826472909112428ve_nat @ B2 )
=> ( condit2214826472909112428ve_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% bdd_above_Int2
thf(fact_643_bdd__above__Int1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ( condit2214826472909112428ve_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% bdd_above_Int1
thf(fact_644_Int__Diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_645_Diff__Int2,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_646_Diff__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_647_Diff__Int__distrib,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A2 ) @ ( inf_inf_set_nat @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_648_Diff__Int__distrib2,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_649_Un__Diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_650_subseqs__refl,axiom,
! [Xs: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ Xs @ ( set_li2392974972034027290_a_nat @ ( subseq8414445098004693972_a_nat @ Xs ) ) ) ).
% subseqs_refl
thf(fact_651_DiffD2,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) )
=> ~ ( member408289922725080238_a_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_652_DiffD2,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( minus_5517490076408937517_a_nat @ A2 @ B2 ) )
=> ~ ( member8690443509505302927_a_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_653_DiffD1,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) )
=> ( member408289922725080238_a_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_654_DiffD1,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( minus_5517490076408937517_a_nat @ A2 @ B2 ) )
=> ( member8690443509505302927_a_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_655_DiffE,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) )
=> ~ ( ( member408289922725080238_a_nat @ C @ A2 )
=> ( member408289922725080238_a_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_656_DiffE,axiom,
! [C: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C @ ( minus_5517490076408937517_a_nat @ A2 @ B2 ) )
=> ~ ( ( member8690443509505302927_a_nat @ C @ A2 )
=> ( member8690443509505302927_a_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_657_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( distinct_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_658_subseqs__distinctD,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( ( distinct_nat @ Xs )
=> ( distinct_nat @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_659_subseqs__distinctD,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Ys @ ( set_li2392974972034027290_a_nat @ ( subseq8414445098004693972_a_nat @ Xs ) ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( distin2701893636801681144_a_nat @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_660_bdd__above_OI2,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > nat,M: nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_2535339886381165584at_nat @ F2 @ A2 ) ) ) ).
% bdd_above.I2
thf(fact_661_bdd__above_OI2,axiom,
! [A2: set_na3699693778330250182_a_nat,F2: ( nat > sum_sum_a_nat ) > nat,M: nat] :
( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ M ) )
=> ( condit2214826472909112428ve_nat @ ( image_5786201776793816049at_nat @ F2 @ A2 ) ) ) ).
% bdd_above.I2
thf(fact_662_cSup__upper2,axiom,
! [X3: nat,X4: set_nat,Y3: nat] :
( ( member_nat @ X3 @ X4 )
=> ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ord_less_eq_nat @ Y3 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).
% cSup_upper2
thf(fact_663_cSup__upper,axiom,
! [X3: nat,X4: set_nat] :
( ( member_nat @ X3 @ X4 )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ord_less_eq_nat @ X3 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ).
% cSup_upper
thf(fact_664_diff__shunt__var,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ( minus_minus_set_nat @ X3 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_665_image__diff__subset,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ ( image_5081948215111134021_a_nat @ F2 @ B2 ) ) @ ( image_5081948215111134021_a_nat @ F2 @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_666_Int__Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_667_Diff__triv,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_668_Diff__subset__conv,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_669_Diff__partition,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_670_Un__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_671_Int__Diff__Un,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_672_Diff__Int,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_673_Diff__Un,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_674_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_675_cSUP__upper,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( condit2214826472909112428ve_nat @ ( image_2535339886381165584at_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ F2 @ A2 ) ) ) ) ) ).
% cSUP_upper
thf(fact_676_cSUP__upper,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F2: ( nat > sum_sum_a_nat ) > nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( condit2214826472909112428ve_nat @ ( image_5786201776793816049at_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ F2 @ A2 ) ) ) ) ) ).
% cSUP_upper
thf(fact_677_cSUP__upper2,axiom,
! [F2: list_Sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat,U2: nat] :
( ( condit2214826472909112428ve_nat @ ( image_2535339886381165584at_nat @ F2 @ A2 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ U2 @ ( F2 @ X3 ) )
=> ( ord_less_eq_nat @ U2 @ ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ F2 @ A2 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_678_cSUP__upper2,axiom,
! [F2: ( nat > sum_sum_a_nat ) > nat,A2: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat,U2: nat] :
( ( condit2214826472909112428ve_nat @ ( image_5786201776793816049at_nat @ F2 @ A2 ) )
=> ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ U2 @ ( F2 @ X3 ) )
=> ( ord_less_eq_nat @ U2 @ ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ F2 @ A2 ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_679_cSup__le__iff,axiom,
! [S: set_nat,A: nat] :
( ( S != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ S )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ S ) @ A )
= ( ! [X: nat] :
( ( member_nat @ X @ S )
=> ( ord_less_eq_nat @ X @ A ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_680_cSup__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( B2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A2 )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ B2 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ord_less_eq_nat @ B6 @ X5 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ B2 ) @ ( complete_Sup_Sup_nat @ A2 ) ) ) ) ) ).
% cSup_mono
thf(fact_681_inj__on__image__set__diff,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) @ C2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ C2 )
=> ( ( image_5081948215111134021_a_nat @ F2 @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) )
= ( minus_7395159227704179404_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ ( image_5081948215111134021_a_nat @ F2 @ B2 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_682_distinct__set__subseqs,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( distin4821671304206132312_a_nat @ ( map_li9079814933684165854_a_nat @ set_Sum_sum_a_nat2 @ ( subseq8414445098004693972_a_nat @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_683_distinct__set__subseqs,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_set_nat @ ( map_list_nat_set_nat @ set_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_684_subset__code_I2_J,axiom,
! [A2: set_li6526943997496501093_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( coset_2342676519794954744_a_nat @ Ys ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Ys ) )
=> ~ ( member408289922725080238_a_nat @ X @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_685_subset__code_I2_J,axiom,
! [A2: set_na3699693778330250182_a_nat,Ys: list_n989787106983797996_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ ( coset_6458349769144696025_a_nat @ Ys ) )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ ( set_na645604395003041787_a_nat @ Ys ) )
=> ~ ( member8690443509505302927_a_nat @ X @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_686_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_687_cSUP__le__iff,axiom,
! [A2: set_nat,F2: nat > nat,U2: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ F2 @ A2 ) )
=> ( ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ A2 ) ) @ U2 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X ) @ U2 ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_688_cSup__subset__mono,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B2 ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_689_cSup__union__distrib,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ A2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ B2 )
=> ( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_690_cSup__union__distrib,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ B2 )
=> ( ( complete_Sup_Sup_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B2 ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_691_cSUP__subset__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > nat,B2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > nat] :
( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_2535339886381165584at_nat @ G @ B2 ) )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ B2 )
=> ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ F2 @ A2 ) ) @ ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_692_cSUP__subset__mono,axiom,
! [A2: set_na3699693778330250182_a_nat,G: ( nat > sum_sum_a_nat ) > nat,B2: set_na3699693778330250182_a_nat,F2: ( nat > sum_sum_a_nat ) > nat] :
( ( A2 != bot_bo6441361344521902642_a_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_5786201776793816049at_nat @ G @ B2 ) )
=> ( ( ord_le8108555184339247974_a_nat @ A2 @ B2 )
=> ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ F2 @ A2 ) ) @ ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_693_cSUP__subset__mono,axiom,
! [A2: set_nat,G: nat > nat,B2: set_nat,F2: nat > nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ ( image_nat_nat @ G @ B2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F2 @ A2 ) ) @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ B2 ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_694_cSup__inter__less__eq,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( condit5477540289124974626et_nat @ A2 )
=> ( ( condit5477540289124974626et_nat @ B2 )
=> ( ( ( inf_inf_set_set_nat @ A2 @ B2 )
!= bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) @ ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ ( comple7399068483239264473et_nat @ B2 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_695_cSup__inter__less__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ( ( condit2214826472909112428ve_nat @ B2 )
=> ( ( ( inf_inf_set_nat @ A2 @ B2 )
!= bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( sup_sup_nat @ ( complete_Sup_Sup_nat @ A2 ) @ ( complete_Sup_Sup_nat @ B2 ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_696_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_697_distinct__concat,axiom,
! [Xs: list_l4703314356710769291_a_nat] :
( ( distin811021574259663358_a_nat @ Xs )
=> ( ! [Ys3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Ys3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( distin2701893636801681144_a_nat @ Ys3 ) )
=> ( ! [Ys3: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Ys3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( member408289922725080238_a_nat @ Zs @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( Ys3 != Zs )
=> ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Ys3 ) @ ( set_Sum_sum_a_nat2 @ Zs ) )
= bot_bo3438331934148233675_a_nat ) ) ) )
=> ( distin2701893636801681144_a_nat @ ( concat_Sum_sum_a_nat @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_698_distinct__concat,axiom,
! [Xs: list_list_nat] :
( ( distinct_list_nat @ Xs )
=> ( ! [Ys3: list_nat] :
( ( member_list_nat @ Ys3 @ ( set_list_nat2 @ Xs ) )
=> ( distinct_nat @ Ys3 ) )
=> ( ! [Ys3: list_nat,Zs: list_nat] :
( ( member_list_nat @ Ys3 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat @ Zs @ ( set_list_nat2 @ Xs ) )
=> ( ( Ys3 != Zs )
=> ( ( inf_inf_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Zs ) )
= bot_bot_set_nat ) ) ) )
=> ( distinct_nat @ ( concat_nat @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_699_cSUP__insert,axiom,
! [A2: set_nat,F2: nat > set_nat,A: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ( condit5477540289124974626et_nat @ ( image_nat_set_nat @ F2 @ A2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ ( insert_nat @ A @ A2 ) ) )
= ( sup_sup_set_nat @ ( F2 @ A ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F2 @ A2 ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_700_subseqs__powset,axiom,
! [Xs: list_nat] :
( ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
= ( pow_nat @ ( set_nat2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_701_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_702_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_703_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_704_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_705_insert__image,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( insert2950094090816004437_a_nat @ ( F2 @ X3 ) @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( image_5081948215111134021_a_nat @ F2 @ A2 ) ) ) ).
% insert_image
thf(fact_706_image__insert,axiom,
! [F2: 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 @ F2 @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( insert2950094090816004437_a_nat @ ( F2 @ A ) @ ( image_5081948215111134021_a_nat @ F2 @ B2 ) ) ) ).
% image_insert
thf(fact_707_singletonI,axiom,
! [A: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% singletonI
thf(fact_708_singletonI,axiom,
! [A: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) ).
% singletonI
thf(fact_709_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_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_722_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_723_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_724_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_725_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_726_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_727_Diff__insert0,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ B2 ) )
= ( minus_7395159227704179404_a_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_728_Diff__insert0,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ B2 ) )
= ( minus_5517490076408937517_a_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_729_insert__Diff1,axiom,
! [X3: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X3 @ A2 ) @ B2 )
= ( minus_7395159227704179404_a_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_730_insert__Diff1,axiom,
! [X3: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ B2 )
=> ( ( minus_5517490076408937517_a_nat @ ( insert5265011953798106934_a_nat @ X3 @ A2 ) @ B2 )
= ( minus_5517490076408937517_a_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_731_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_732_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_733_cSup__singleton,axiom,
! [X3: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% cSup_singleton
thf(fact_734_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_735_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_736_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_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_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_746_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_747_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_748_Pow__singleton__iff,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( ( pow_nat @ X4 )
= ( insert_set_nat @ Y4 @ bot_bot_set_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_749_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_750_Pow__Int__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( ( pow_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B2 ) ) ) ).
% Pow_Int_eq
thf(fact_751_inj__on__insert,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ ( insert2950094090816004437_a_nat @ A @ A2 ) )
= ( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
& ~ ( member408289922725080238_a_nat @ ( F2 @ A ) @ ( image_5081948215111134021_a_nat @ F2 @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_752_inj__on__insert,axiom,
! [F2: nat > list_Sum_sum_a_nat,A: nat,A2: set_nat] :
( ( inj_on901614105087147266_a_nat @ F2 @ ( insert_nat @ A @ A2 ) )
= ( ( inj_on901614105087147266_a_nat @ F2 @ A2 )
& ~ ( member408289922725080238_a_nat @ ( F2 @ A ) @ ( image_6262589752765146990_a_nat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_753_inj__on__insert,axiom,
! [F2: nat > nat > sum_sum_a_nat,A: nat,A2: set_nat] :
( ( inj_on7629884641133321699_a_nat @ F2 @ ( insert_nat @ A @ A2 ) )
= ( ( inj_on7629884641133321699_a_nat @ F2 @ A2 )
& ~ ( member8690443509505302927_a_nat @ ( F2 @ A ) @ ( image_1051037728736664655_a_nat @ F2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_754_insert__Diff__if,axiom,
! [X3: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X3 @ A2 ) @ B2 )
= ( minus_7395159227704179404_a_nat @ A2 @ B2 ) ) )
& ( ~ ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X3 @ A2 ) @ B2 )
= ( insert2950094090816004437_a_nat @ X3 @ ( minus_7395159227704179404_a_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_755_insert__Diff__if,axiom,
! [X3: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( ( member8690443509505302927_a_nat @ X3 @ B2 )
=> ( ( minus_5517490076408937517_a_nat @ ( insert5265011953798106934_a_nat @ X3 @ A2 ) @ B2 )
= ( minus_5517490076408937517_a_nat @ A2 @ B2 ) ) )
& ( ~ ( member8690443509505302927_a_nat @ X3 @ B2 )
=> ( ( minus_5517490076408937517_a_nat @ ( insert5265011953798106934_a_nat @ X3 @ A2 ) @ B2 )
= ( insert5265011953798106934_a_nat @ X3 @ ( minus_5517490076408937517_a_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_756_map__concat,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_list_nat] :
( ( map_na823391071729141993_a_nat @ F2 @ ( concat_nat @ Xs ) )
= ( concat_Sum_sum_a_nat @ ( map_li4577266612483868383_a_nat @ ( map_na823391071729141993_a_nat @ F2 ) @ Xs ) ) ) ).
% map_concat
thf(fact_757_map__concat,axiom,
! [F2: nat > nat,Xs: list_list_nat] :
( ( map_nat_nat @ F2 @ ( concat_nat @ Xs ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F2 ) @ Xs ) ) ) ).
% map_concat
thf(fact_758_map__concat,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_l4703314356710769291_a_nat] :
( ( map_Su5227373005390213643at_nat @ F2 @ ( concat_Sum_sum_a_nat @ Xs ) )
= ( concat_nat @ ( map_li8705756933220954785st_nat @ ( map_Su5227373005390213643at_nat @ F2 ) @ Xs ) ) ) ).
% map_concat
thf(fact_759_map__concat,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( map_Su2790769393171190532_a_nat @ F2 @ ( concat_Sum_sum_a_nat @ Xs ) )
= ( concat_Sum_sum_a_nat @ ( map_li6507455427659069316_a_nat @ ( map_Su2790769393171190532_a_nat @ F2 ) @ Xs ) ) ) ).
% map_concat
thf(fact_760_insert__subsetI,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,X4: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ X4 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ X3 @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_761_insert__subsetI,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,X4: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( ord_le8108555184339247974_a_nat @ X4 @ A2 )
=> ( ord_le8108555184339247974_a_nat @ ( insert5265011953798106934_a_nat @ X3 @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_762_singletonD,axiom,
! [B: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_763_singletonD,axiom,
! [B: nat > sum_sum_a_nat,A: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ B @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_764_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_765_singleton__iff,axiom,
! [B: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_766_singleton__iff,axiom,
! [B: nat > sum_sum_a_nat,A: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ B @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_767_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_768_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_769_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_770_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_771_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_772_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_773_Pow__bottom,axiom,
! [B2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_774_Int__insert__left,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 ) ) ) )
& ( ~ ( member408289922725080238_a_nat @ A @ C2 )
=> ( ( inf_in3249246906714053971_a_nat @ ( insert2950094090816004437_a_nat @ A @ B2 ) @ C2 )
= ( inf_in3249246906714053971_a_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_775_Int__insert__left,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 ) ) ) )
& ( ~ ( member8690443509505302927_a_nat @ A @ C2 )
=> ( ( inf_in8399021836546144180_a_nat @ ( insert5265011953798106934_a_nat @ A @ B2 ) @ C2 )
= ( inf_in8399021836546144180_a_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_776_Int__insert__left,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 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_777_Int__insert__right,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 ) ) ) )
& ( ~ ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( inf_in3249246906714053971_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B2 ) )
= ( inf_in3249246906714053971_a_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_778_Int__insert__right,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 ) ) ) )
& ( ~ ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( inf_in8399021836546144180_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ B2 ) )
= ( inf_in8399021836546144180_a_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_779_Int__insert__right,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 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_780_mk__disjoint__insert,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ? [B7: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ A @ B7 ) )
& ~ ( member408289922725080238_a_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_781_mk__disjoint__insert,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ? [B7: set_na3699693778330250182_a_nat] :
( ( A2
= ( insert5265011953798106934_a_nat @ A @ B7 ) )
& ~ ( member8690443509505302927_a_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_782_insert__eq__iff,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ B @ B2 )
=> ( ( ( insert2950094090816004437_a_nat @ A @ A2 )
= ( insert2950094090816004437_a_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ B @ C4 ) )
& ~ ( member408289922725080238_a_nat @ B @ C4 )
& ( B2
= ( insert2950094090816004437_a_nat @ A @ C4 ) )
& ~ ( member408289922725080238_a_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_783_insert__eq__iff,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ~ ( member8690443509505302927_a_nat @ B @ B2 )
=> ( ( ( insert5265011953798106934_a_nat @ A @ A2 )
= ( insert5265011953798106934_a_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_na3699693778330250182_a_nat] :
( ( A2
= ( insert5265011953798106934_a_nat @ B @ C4 ) )
& ~ ( member8690443509505302927_a_nat @ B @ C4 )
& ( B2
= ( insert5265011953798106934_a_nat @ A @ C4 ) )
& ~ ( member8690443509505302927_a_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_784_insert__absorb,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( insert2950094090816004437_a_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_785_insert__absorb,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( insert5265011953798106934_a_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_786_insert__ident,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( ( insert2950094090816004437_a_nat @ X3 @ A2 )
= ( insert2950094090816004437_a_nat @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_787_insert__ident,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ~ ( member8690443509505302927_a_nat @ X3 @ B2 )
=> ( ( ( insert5265011953798106934_a_nat @ X3 @ A2 )
= ( insert5265011953798106934_a_nat @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_788_Set_Oset__insert,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ~ ! [B7: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ X3 @ B7 ) )
=> ( member408289922725080238_a_nat @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_789_Set_Oset__insert,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ~ ! [B7: set_na3699693778330250182_a_nat] :
( ( A2
= ( insert5265011953798106934_a_nat @ X3 @ B7 ) )
=> ( member8690443509505302927_a_nat @ X3 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_790_insertI2,axiom,
! [A: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A @ B2 )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_791_insertI2,axiom,
! [A: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat,B: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ A @ B2 )
=> ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_792_insertI1,axiom,
! [A: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_793_insertI1,axiom,
! [A: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] : ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_794_insertE,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 ) ) ) ).
% insertE
thf(fact_795_insertE,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 ) ) ) ).
% insertE
thf(fact_796_Pow__set_I1_J,axiom,
( ( pow_nat @ ( set_nat2 @ nil_nat ) )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_set(1)
thf(fact_797_subset__singleton__iff,axiom,
! [X4: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
| ( X4
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_798_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_799_singleton__Un__iff,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ( ( insert_nat @ X3 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_800_Un__singleton__iff,axiom,
! [A2: set_nat,B2: set_nat,X3: nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_801_insert__is__Un,axiom,
( insert_nat
= ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_802_Diff__insert__absorb,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X3 @ A2 ) @ ( insert2950094090816004437_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_803_Diff__insert__absorb,axiom,
! [X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( minus_5517490076408937517_a_nat @ ( insert5265011953798106934_a_nat @ X3 @ A2 ) @ ( insert5265011953798106934_a_nat @ X3 @ bot_bo6441361344521902642_a_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_804_Diff__insert__absorb,axiom,
! [X3: nat,A2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_805_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_806_insert__Diff,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( insert2950094090816004437_a_nat @ A @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_807_insert__Diff,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( insert5265011953798106934_a_nat @ A @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_808_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_809_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_810_inj__img__insertE,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( ( insert2950094090816004437_a_nat @ X3 @ B2 )
= ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
=> ~ ! [X8: list_Sum_sum_a_nat,A4: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X8 @ A4 )
=> ( ( A2
= ( insert2950094090816004437_a_nat @ X8 @ A4 ) )
=> ( ( X3
= ( F2 @ X8 ) )
=> ( B2
!= ( image_5081948215111134021_a_nat @ F2 @ A4 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_811_inj__img__insertE,axiom,
! [F2: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,X3: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inj_on602732703247098640_a_nat @ F2 @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( ( insert2950094090816004437_a_nat @ X3 @ B2 )
= ( image_6721470456781115300_a_nat @ F2 @ A2 ) )
=> ~ ! [X8: nat > sum_sum_a_nat,A4: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X8 @ A4 )
=> ( ( A2
= ( insert5265011953798106934_a_nat @ X8 @ A4 ) )
=> ( ( X3
= ( F2 @ X8 ) )
=> ( B2
!= ( image_6721470456781115300_a_nat @ F2 @ A4 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_812_inj__img__insertE,axiom,
! [F2: list_Sum_sum_a_nat > nat > sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,X3: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( inj_on3806193600625622162_a_nat @ F2 @ A2 )
=> ( ~ ( member8690443509505302927_a_nat @ X3 @ B2 )
=> ( ( ( insert5265011953798106934_a_nat @ X3 @ B2 )
= ( image_701559317304863014_a_nat @ F2 @ A2 ) )
=> ~ ! [X8: list_Sum_sum_a_nat,A4: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X8 @ A4 )
=> ( ( A2
= ( insert2950094090816004437_a_nat @ X8 @ A4 ) )
=> ( ( X3
= ( F2 @ X8 ) )
=> ( B2
!= ( image_701559317304863014_a_nat @ F2 @ A4 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_813_inj__img__insertE,axiom,
! [F2: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( inj_on8496004383624361457_a_nat @ F2 @ A2 )
=> ( ~ ( member8690443509505302927_a_nat @ X3 @ B2 )
=> ( ( ( insert5265011953798106934_a_nat @ X3 @ B2 )
= ( image_6222892899998961285_a_nat @ F2 @ A2 ) )
=> ~ ! [X8: nat > sum_sum_a_nat,A4: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X8 @ A4 )
=> ( ( A2
= ( insert5265011953798106934_a_nat @ X8 @ A4 ) )
=> ( ( X3
= ( F2 @ X8 ) )
=> ( B2
!= ( image_6222892899998961285_a_nat @ F2 @ A4 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_814_subset__Diff__insert,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( minus_7395159227704179404_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ X3 @ C2 ) ) )
= ( ( ord_le1147066620699065093_a_nat @ A2 @ ( minus_7395159227704179404_a_nat @ B2 @ C2 ) )
& ~ ( member408289922725080238_a_nat @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_815_subset__Diff__insert,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat,C2: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ ( minus_5517490076408937517_a_nat @ B2 @ ( insert5265011953798106934_a_nat @ X3 @ C2 ) ) )
= ( ( ord_le8108555184339247974_a_nat @ A2 @ ( minus_5517490076408937517_a_nat @ B2 @ C2 ) )
& ~ ( member8690443509505302927_a_nat @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_816_Union__insert,axiom,
! [A: set_nat,B2: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ B2 ) ) ) ).
% Union_insert
thf(fact_817_image__Pow__surj,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F2 @ A2 )
= B2 )
=> ( ( image_3472601871771700037_a_nat @ ( image_5081948215111134021_a_nat @ F2 ) @ ( pow_li8024330045898428450_a_nat @ A2 ) )
= ( pow_li8024330045898428450_a_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_818_Diff__single__insert,axiom,
! [A2: set_nat,X3: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_819_subset__insert__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ B2 ) )
= ( ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) ) @ B2 ) )
& ( ~ ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_820_subset__insert__iff,axiom,
! [A2: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ B2 ) )
= ( ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ord_le8108555184339247974_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ bot_bo6441361344521902642_a_nat ) ) @ B2 ) )
& ( ~ ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ord_le8108555184339247974_a_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_821_subset__insert__iff,axiom,
! [A2: set_nat,X3: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
= ( ( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_822_insert__partition,axiom,
! [X3: set_nat,F4: set_set_nat] :
( ~ ( member_set_nat @ X3 @ F4 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( insert_set_nat @ X3 @ F4 ) )
=> ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ ( insert_set_nat @ X3 @ F4 ) )
=> ( ( X2 != Xa2 )
=> ( ( inf_inf_set_nat @ X2 @ Xa2 )
= bot_bot_set_nat ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ ( comple7399068483239264473et_nat @ F4 ) )
= bot_bot_set_nat ) ) ) ).
% insert_partition
thf(fact_823_in__image__insert__iff,axiom,
! [B2: set_se4330304633200676677_a_nat,X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ! [C3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ C3 @ B2 )
=> ~ ( member408289922725080238_a_nat @ X3 @ C3 ) )
=> ( ( member5553968465346197646_a_nat @ A2 @ ( image_3472601871771700037_a_nat @ ( insert2950094090816004437_a_nat @ X3 ) @ B2 ) )
= ( ( member408289922725080238_a_nat @ X3 @ A2 )
& ( member5553968465346197646_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_824_in__image__insert__iff,axiom,
! [B2: set_se5822283258546872870_a_nat,X3: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ! [C3: set_na3699693778330250182_a_nat] :
( ( member3060896489619847151_a_nat @ C3 @ B2 )
=> ~ ( member8690443509505302927_a_nat @ X3 @ C3 ) )
=> ( ( member3060896489619847151_a_nat @ A2 @ ( image_4398635103182451333_a_nat @ ( insert5265011953798106934_a_nat @ X3 ) @ B2 ) )
= ( ( member8690443509505302927_a_nat @ X3 @ A2 )
& ( member3060896489619847151_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ bot_bo6441361344521902642_a_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_825_in__image__insert__iff,axiom,
! [B2: set_set_nat,X3: nat,A2: set_nat] :
( ! [C3: set_nat] :
( ( member_set_nat @ C3 @ B2 )
=> ~ ( member_nat @ X3 @ C3 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X3 ) @ B2 ) )
= ( ( member_nat @ X3 @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_826_size__neq__size__imp__neq,axiom,
! [X3: list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ X3 )
!= ( size_s5283204784079214577_a_nat @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_827_size__neq__size__imp__neq,axiom,
! [X3: list_nat,Y3: list_nat] :
( ( ( size_size_list_nat @ X3 )
!= ( size_size_list_nat @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_828_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_829_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_830_eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( M4 = N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% eq_imp_le
thf(fact_831_le__antisym,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( ord_less_eq_nat @ N @ M4 )
=> ( M4 = N ) ) ) ).
% le_antisym
thf(fact_832_nat__le__linear,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
| ( ord_less_eq_nat @ N @ M4 ) ) ).
% nat_le_linear
thf(fact_833_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_834_diff__le__mono2,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_835_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_836_diff__le__self,axiom,
! [M4: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N ) @ M4 ) ).
% diff_le_self
thf(fact_837_diff__le__mono,axiom,
! [M4: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_838_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M4 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_839_le__diff__iff,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M4 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_840_eq__diff__iff,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M4 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_841_Un__Pow__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B2 ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_842_inj__on__image__Pow,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( inj_on561899399213738673_a_nat @ ( image_5081948215111134021_a_nat @ F2 ) @ ( pow_li8024330045898428450_a_nat @ A2 ) ) ) ).
% inj_on_image_Pow
thf(fact_843_cSup__insert,axiom,
! [X4: set_set_nat,A: set_nat] :
( ( X4 != bot_bot_set_set_nat )
=> ( ( condit5477540289124974626et_nat @ X4 )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X4 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_844_cSup__insert,axiom,
! [X4: set_nat,A: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X4 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ).
% cSup_insert
thf(fact_845_cSup__insert__If,axiom,
! [X4: set_set_nat,A: set_nat] :
( ( condit5477540289124974626et_nat @ X4 )
=> ( ( ( X4 = bot_bot_set_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X4 ) )
= A ) )
& ( ( X4 != bot_bot_set_set_nat )
=> ( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A @ X4 ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_846_cSup__insert__If,axiom,
! [X4: set_nat,A: nat] :
( ( condit2214826472909112428ve_nat @ X4 )
=> ( ( ( X4 = bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X4 ) )
= A ) )
& ( ( X4 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ ( insert_nat @ A @ X4 ) )
= ( sup_sup_nat @ A @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_847_image__Pow__mono,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ B2 )
=> ( ord_le8138476598237931237_a_nat @ ( image_3472601871771700037_a_nat @ ( image_5081948215111134021_a_nat @ F2 ) @ ( pow_li8024330045898428450_a_nat @ A2 ) ) @ ( pow_li8024330045898428450_a_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_848_distinct__concat__iff,axiom,
! [Xs: list_l4703314356710769291_a_nat] :
( ( distin2701893636801681144_a_nat @ ( concat_Sum_sum_a_nat @ Xs ) )
= ( ( distin811021574259663358_a_nat @ ( remove910890064017026449_a_nat @ nil_Sum_sum_a_nat @ Xs ) )
& ! [Ys2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Ys2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( distin2701893636801681144_a_nat @ Ys2 ) )
& ! [Ys2: list_Sum_sum_a_nat,Zs2: list_Sum_sum_a_nat] :
( ( ( member408289922725080238_a_nat @ Ys2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
& ( member408289922725080238_a_nat @ Zs2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
& ( Ys2 != Zs2 ) )
=> ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Ys2 ) @ ( set_Sum_sum_a_nat2 @ Zs2 ) )
= bot_bo3438331934148233675_a_nat ) ) ) ) ).
% distinct_concat_iff
thf(fact_849_distinct__concat__iff,axiom,
! [Xs: list_list_nat] :
( ( distinct_nat @ ( concat_nat @ Xs ) )
= ( ( distinct_list_nat @ ( removeAll_list_nat @ nil_nat @ Xs ) )
& ! [Ys2: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ Xs ) )
=> ( distinct_nat @ Ys2 ) )
& ! [Ys2: list_nat,Zs2: list_nat] :
( ( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ Xs ) )
& ( member_list_nat @ Zs2 @ ( set_list_nat2 @ Xs ) )
& ( Ys2 != Zs2 ) )
=> ( ( inf_inf_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Zs2 ) )
= bot_bot_set_nat ) ) ) ) ).
% distinct_concat_iff
thf(fact_850_distinct__product__lists,axiom,
! [Xss: list_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss ) )
=> ( distinct_nat @ X2 ) )
=> ( distinct_list_nat @ ( product_lists_nat @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_851_distinct__product__lists,axiom,
! [Xss: list_l4703314356710769291_a_nat] :
( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xss ) )
=> ( distin2701893636801681144_a_nat @ X2 ) )
=> ( distin811021574259663358_a_nat @ ( produc2893206433618375022_a_nat @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_852_removeAll__id,axiom,
! [X3: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ~ ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( remove910890064017026449_a_nat @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_853_removeAll__id,axiom,
! [X3: nat > sum_sum_a_nat,Xs: list_n989787106983797996_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( remove1885113525496864626_a_nat @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_854_removeAll__id,axiom,
! [X3: nat,Xs: list_nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X3 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_855_set__removeAll,axiom,
! [X3: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X3 @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_856_distinct__removeAll,axiom,
! [Xs: list_nat,X3: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( removeAll_nat @ X3 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_857_distinct__removeAll,axiom,
! [Xs: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( distin2701893636801681144_a_nat @ ( remove3909449470355376139_a_nat @ X3 @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_858_length__removeAll__less__eq,axiom,
! [X3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ ( remove3909449470355376139_a_nat @ X3 @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_859_length__removeAll__less__eq,axiom,
! [X3: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X3 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_860_in__set__product__lists__length,axiom,
! [Xs: list_Sum_sum_a_nat,Xss: list_l4703314356710769291_a_nat] :
( ( member408289922725080238_a_nat @ Xs @ ( set_li2392974972034027290_a_nat @ ( produc2893206433618375022_a_nat @ Xss ) ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5212483967078203639_a_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_861_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_862_map__removeAll__inj__on,axiom,
! [F2: sum_sum_a_nat > nat,X3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( inj_on8752143810983750942at_nat @ F2 @ ( insert_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) ) )
=> ( ( map_Su5227373005390213643at_nat @ F2 @ ( remove3909449470355376139_a_nat @ X3 @ Xs ) )
= ( removeAll_nat @ ( F2 @ X3 ) @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_863_map__removeAll__inj__on,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,X3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( inj_on6255688694610590513_a_nat @ F2 @ ( insert_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) ) )
=> ( ( map_Su2790769393171190532_a_nat @ F2 @ ( remove3909449470355376139_a_nat @ X3 @ Xs ) )
= ( remove3909449470355376139_a_nat @ ( F2 @ X3 ) @ ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_864_map__removeAll__inj__on,axiom,
! [F2: nat > sum_sum_a_nat,X3: nat,Xs: list_nat] :
( ( inj_on4348161877322679292_a_nat @ F2 @ ( insert_nat @ X3 @ ( set_nat2 @ Xs ) ) )
=> ( ( map_na823391071729141993_a_nat @ F2 @ ( removeAll_nat @ X3 @ Xs ) )
= ( remove3909449470355376139_a_nat @ ( F2 @ X3 ) @ ( map_na823391071729141993_a_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_865_map__removeAll__inj__on,axiom,
! [F2: nat > nat,X3: nat,Xs: list_nat] :
( ( inj_on_nat_nat @ F2 @ ( insert_nat @ X3 @ ( set_nat2 @ Xs ) ) )
=> ( ( map_nat_nat @ F2 @ ( removeAll_nat @ X3 @ Xs ) )
= ( removeAll_nat @ ( F2 @ X3 ) @ ( map_nat_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_866_is__singletonI,axiom,
! [X3: nat] : ( is_singleton_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_867_set__remove1__eq,axiom,
! [Xs: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( set_Sum_sum_a_nat2 @ ( remove2233709901542100379_a_nat @ X3 @ Xs ) )
= ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( insert_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) ) ).
% set_remove1_eq
thf(fact_868_set__remove1__eq,axiom,
! [Xs: list_nat,X3: nat] :
( ( distinct_nat @ Xs )
=> ( ( set_nat2 @ ( remove1_nat @ X3 @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% set_remove1_eq
thf(fact_869_UNION__fun__upd,axiom,
! [A2: list_Sum_sum_a_nat > set_nat,I: list_Sum_sum_a_nat,B2: set_nat,J2: set_li6526943997496501093_a_nat] :
( ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ ( fun_up9111695906565944654et_nat @ A2 @ I @ B2 ) @ J2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ A2 @ ( minus_7395159227704179404_a_nat @ J2 @ ( insert2950094090816004437_a_nat @ I @ bot_bo1033123847703346641_a_nat ) ) ) ) @ ( if_set_nat @ ( member408289922725080238_a_nat @ I @ J2 ) @ B2 @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_870_UNION__fun__upd,axiom,
! [A2: ( nat > sum_sum_a_nat ) > set_nat,I: nat > sum_sum_a_nat,B2: set_nat,J2: set_na3699693778330250182_a_nat] :
( ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ ( fun_up7925228223850514735et_nat @ A2 @ I @ B2 ) @ J2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ A2 @ ( minus_5517490076408937517_a_nat @ J2 @ ( insert5265011953798106934_a_nat @ I @ bot_bo6441361344521902642_a_nat ) ) ) ) @ ( if_set_nat @ ( member8690443509505302927_a_nat @ I @ J2 ) @ B2 @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_871_UNION__fun__upd,axiom,
! [A2: nat > set_nat,I: nat,B2: set_nat,J2: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ ( fun_upd_nat_set_nat @ A2 @ I @ B2 ) @ J2 ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A2 @ ( minus_minus_set_nat @ J2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) ) ) @ ( if_set_nat @ ( member_nat @ I @ J2 ) @ B2 @ bot_bot_set_nat ) ) ) ).
% UNION_fun_upd
thf(fact_872_in__set__remove1,axiom,
! [A: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( A != B )
=> ( ( member408289922725080238_a_nat @ A @ ( set_li2392974972034027290_a_nat @ ( remove4274251903526005537_a_nat @ B @ Xs ) ) )
= ( member408289922725080238_a_nat @ A @ ( set_li2392974972034027290_a_nat @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_873_in__set__remove1,axiom,
! [A: nat > sum_sum_a_nat,B: nat > sum_sum_a_nat,Xs: list_n989787106983797996_a_nat] :
( ( A != B )
=> ( ( member8690443509505302927_a_nat @ A @ ( set_na645604395003041787_a_nat @ ( remove134331365954440450_a_nat @ B @ Xs ) ) )
= ( member8690443509505302927_a_nat @ A @ ( set_na645604395003041787_a_nat @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_874_in__set__remove1,axiom,
! [A: nat,B: nat,Xs: list_nat] :
( ( A != B )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_875_map__fun__upd,axiom,
! [Y3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,V: nat] :
( ~ ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( map_Su5227373005390213643at_nat @ ( fun_up4584519350643678994at_nat @ F2 @ Y3 @ V ) @ Xs )
= ( map_Su5227373005390213643at_nat @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_876_map__fun__upd,axiom,
! [Y3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat,V: sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ Y3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( map_Su2790769393171190532_a_nat @ ( fun_up6086130847573437501_a_nat @ F2 @ Y3 @ V ) @ Xs )
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_877_map__fun__upd,axiom,
! [Y3: nat,Xs: list_nat,F2: nat > sum_sum_a_nat,V: sum_sum_a_nat] :
( ~ ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( map_na823391071729141993_a_nat @ ( fun_up180537416982607344_a_nat @ F2 @ Y3 @ V ) @ Xs )
= ( map_na823391071729141993_a_nat @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_878_map__fun__upd,axiom,
! [Y3: nat,Xs: list_nat,F2: nat > nat,V: nat] :
( ~ ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( map_nat_nat @ ( fun_upd_nat_nat @ F2 @ Y3 @ V ) @ Xs )
= ( map_nat_nat @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_879_notin__set__remove1,axiom,
! [X3: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,Y3: list_Sum_sum_a_nat] :
( ~ ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ~ ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ ( remove4274251903526005537_a_nat @ Y3 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_880_notin__set__remove1,axiom,
! [X3: nat > sum_sum_a_nat,Xs: list_n989787106983797996_a_nat,Y3: nat > sum_sum_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ~ ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ ( remove134331365954440450_a_nat @ Y3 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_881_notin__set__remove1,axiom,
! [X3: nat,Xs: list_nat,Y3: nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat @ X3 @ ( set_nat2 @ ( remove1_nat @ Y3 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_882_remove1__idem,axiom,
! [X3: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ~ ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( remove4274251903526005537_a_nat @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_883_remove1__idem,axiom,
! [X3: nat > sum_sum_a_nat,Xs: list_n989787106983797996_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( remove134331365954440450_a_nat @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_884_remove1__idem,axiom,
! [X3: nat,Xs: list_nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X3 @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_885_distinct__remove1,axiom,
! [Xs: list_nat,X3: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( remove1_nat @ X3 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_886_distinct__remove1,axiom,
! [Xs: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( distin2701893636801681144_a_nat @ ( remove2233709901542100379_a_nat @ X3 @ Xs ) ) ) ).
% distinct_remove1
thf(fact_887_set__remove1__subset,axiom,
! [X3: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X3 @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_888_sorted__remove1,axiom,
! [Xs: list_nat,A: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs ) ) ) ).
% sorted_remove1
thf(fact_889_inj__on__fun__updI,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,Y3: list_Sum_sum_a_nat,X3: list_Sum_sum_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ Y3 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
=> ( inj_on6609798167860701873_a_nat @ ( fun_up52870354139672509_a_nat @ F2 @ X3 @ Y3 ) @ A2 ) ) ) ).
% inj_on_fun_updI
thf(fact_890_is__singletonI_H,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ! [X2: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ( member408289922725080238_a_nat @ Y @ A2 )
=> ( X2 = Y ) ) )
=> ( is_sin2231188923920309881_a_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_891_is__singletonI_H,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( A2 != bot_bo6441361344521902642_a_nat )
=> ( ! [X2: nat > sum_sum_a_nat,Y: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
=> ( ( member8690443509505302927_a_nat @ Y @ A2 )
=> ( X2 = Y ) ) )
=> ( is_sin884740366827251802_a_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_892_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X2 = Y ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_893_distinct__remove1__removeAll,axiom,
! [Xs: list_nat,X3: nat] :
( ( distinct_nat @ Xs )
=> ( ( remove1_nat @ X3 @ Xs )
= ( removeAll_nat @ X3 @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_894_distinct__remove1__removeAll,axiom,
! [Xs: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( remove2233709901542100379_a_nat @ X3 @ Xs )
= ( remove3909449470355376139_a_nat @ X3 @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_895_sorted__map__remove1,axiom,
! [F2: nat > nat,Xs: list_nat,X3: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F2 @ Xs ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F2 @ ( remove1_nat @ X3 @ Xs ) ) ) ) ).
% sorted_map_remove1
thf(fact_896_sorted__map__remove1,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Su5227373005390213643at_nat @ F2 @ ( remove2233709901542100379_a_nat @ X3 @ Xs ) ) ) ) ).
% sorted_map_remove1
thf(fact_897_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X2: nat] :
( A2
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_898_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X: nat] :
( A3
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_899_fun__upd__image,axiom,
! [X3: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] :
( ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( image_5081948215111134021_a_nat @ ( fun_up52870354139672509_a_nat @ F2 @ X3 @ Y3 ) @ A2 )
= ( insert2950094090816004437_a_nat @ Y3 @ ( image_5081948215111134021_a_nat @ F2 @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) ) ) ) ) )
& ( ~ ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( image_5081948215111134021_a_nat @ ( fun_up52870354139672509_a_nat @ F2 @ X3 @ Y3 ) @ A2 )
= ( image_5081948215111134021_a_nat @ F2 @ A2 ) ) ) ) ).
% fun_upd_image
thf(fact_900_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_901_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X3 @ ( linord2614967742042102400et_nat @ A2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_902_distinct__append,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( ( distin2701893636801681144_a_nat @ Xs )
& ( distin2701893636801681144_a_nat @ Ys )
& ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% distinct_append
thf(fact_903_distinct__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( distinct_nat @ Xs )
& ( distinct_nat @ Ys )
& ( ( inf_inf_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
= bot_bot_set_nat ) ) ) ).
% distinct_append
thf(fact_904_finite__imageI,axiom,
! [F4: set_li6526943997496501093_a_nat,H: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ F4 )
=> ( finite1487985464145237934_a_nat @ ( image_5081948215111134021_a_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_905_finite__imageI,axiom,
! [F4: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_906_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_907_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_908_finite__Int,axiom,
! [F4: set_nat,G3: set_nat] :
( ( ( finite_finite_nat @ F4 )
| ( finite_finite_nat @ G3 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F4 @ G3 ) ) ) ).
% finite_Int
thf(fact_909_finite__Un,axiom,
! [F4: set_nat,G3: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G3 ) )
= ( ( finite_finite_nat @ F4 )
& ( finite_finite_nat @ G3 ) ) ) ).
% finite_Un
thf(fact_910_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_911_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_912_finite__Union,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ! [M2: set_nat] :
( ( member_set_nat @ M2 @ A2 )
=> ( finite_finite_nat @ M2 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) ) ) ) ).
% finite_Union
thf(fact_913_append__eq__append__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
| ( ( size_s5283204784079214577_a_nat @ Us )
= ( size_s5283204784079214577_a_nat @ Vs ) ) )
=> ( ( ( append_Sum_sum_a_nat @ Xs @ Us )
= ( append_Sum_sum_a_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_914_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_915_map__append,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_nat] :
( ( map_na823391071729141993_a_nat @ F2 @ ( append_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( map_na823391071729141993_a_nat @ F2 @ Xs ) @ ( map_na823391071729141993_a_nat @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_916_map__append,axiom,
! [F2: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F2 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F2 @ Xs ) @ ( map_nat_nat @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_917_map__append,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( map_Su5227373005390213643at_nat @ F2 @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) @ ( map_Su5227373005390213643at_nat @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_918_map__append,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( map_Su2790769393171190532_a_nat @ F2 @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) @ ( map_Su2790769393171190532_a_nat @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_919_finite__Pow__iff,axiom,
! [A2: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_Pow_iff
thf(fact_920_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_921_set__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_922_finite__UN,axiom,
! [A2: set_nat,B2: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( finite_finite_nat @ ( B2 @ X ) ) ) ) ) ) ).
% finite_UN
thf(fact_923_zip__append,axiom,
! [Xs: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Us ) )
=> ( ( zip_Su7355543910597222519_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us @ Vs ) )
= ( append1996214168388709506_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Us ) @ ( zip_Su7355543910597222519_a_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_924_zip__append,axiom,
! [Xs: list_Sum_sum_a_nat,Us: list_nat,Ys: list_Sum_sum_a_nat,Vs: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_Su6417478541797927256at_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append2142653904031976739at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Us ) @ ( zip_Su6417478541797927256at_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_925_zip__append,axiom,
! [Xs: list_nat,Us: list_Sum_sum_a_nat,Ys: list_nat,Vs: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Us ) )
=> ( ( zip_na2013496608136855606_a_nat @ ( append_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us @ Vs ) )
= ( append338925788367110473_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Us ) @ ( zip_na2013496608136855606_a_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_926_zip__append,axiom,
! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_927_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_928_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A2 ) )
= A2 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_929_the__elem__eq,axiom,
! [X3: nat] :
( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% the_elem_eq
thf(fact_930_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_931_map__eq__append__conv,axiom,
! [F2: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F2 @ Xs )
= ( append_Sum_sum_a_nat @ Ys @ Zs3 ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_na823391071729141993_a_nat @ F2 @ Us2 ) )
& ( Zs3
= ( map_na823391071729141993_a_nat @ F2 @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_932_map__eq__append__conv,axiom,
! [F2: nat > nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( append_nat @ Ys @ Zs3 ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_nat_nat @ F2 @ Us2 ) )
& ( Zs3
= ( map_nat_nat @ F2 @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_933_map__eq__append__conv,axiom,
! [F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( map_Su5227373005390213643at_nat @ F2 @ Xs )
= ( append_nat @ Ys @ Zs3 ) )
= ( ? [Us2: list_Sum_sum_a_nat,Vs3: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_Su5227373005390213643at_nat @ F2 @ Us2 ) )
& ( Zs3
= ( map_Su5227373005390213643at_nat @ F2 @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_934_map__eq__append__conv,axiom,
! [F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F2 @ Xs )
= ( append_Sum_sum_a_nat @ Ys @ Zs3 ) )
= ( ? [Us2: list_Sum_sum_a_nat,Vs3: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_Su2790769393171190532_a_nat @ F2 @ Us2 ) )
& ( Zs3
= ( map_Su2790769393171190532_a_nat @ F2 @ Vs3 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_935_append__eq__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat,F2: nat > sum_sum_a_nat,Xs: list_nat] :
( ( ( append_Sum_sum_a_nat @ Ys @ Zs3 )
= ( map_na823391071729141993_a_nat @ F2 @ Xs ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_na823391071729141993_a_nat @ F2 @ Us2 ) )
& ( Zs3
= ( map_na823391071729141993_a_nat @ F2 @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_936_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs3: list_nat,F2: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs3 )
= ( map_nat_nat @ F2 @ Xs ) )
= ( ? [Us2: list_nat,Vs3: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_nat_nat @ F2 @ Us2 ) )
& ( Zs3
= ( map_nat_nat @ F2 @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_937_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs3: list_nat,F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( ( append_nat @ Ys @ Zs3 )
= ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
= ( ? [Us2: list_Sum_sum_a_nat,Vs3: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_Su5227373005390213643at_nat @ F2 @ Us2 ) )
& ( Zs3
= ( map_Su5227373005390213643at_nat @ F2 @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_938_append__eq__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,Zs3: list_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ( append_Sum_sum_a_nat @ Ys @ Zs3 )
= ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) )
= ( ? [Us2: list_Sum_sum_a_nat,Vs3: list_Sum_sum_a_nat] :
( ( Xs
= ( append_Sum_sum_a_nat @ Us2 @ Vs3 ) )
& ( Ys
= ( map_Su2790769393171190532_a_nat @ F2 @ Us2 ) )
& ( Zs3
= ( map_Su2790769393171190532_a_nat @ F2 @ Vs3 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_939_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_940_finite__subset__Union,axiom,
! [A2: set_nat,B8: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B8 ) )
=> ~ ! [F5: set_set_nat] :
( ( finite1152437895449049373et_nat @ F5 )
=> ( ( ord_le6893508408891458716et_nat @ F5 @ B8 )
=> ~ ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ F5 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_941_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( linord2614967742042102400et_nat @ A2 )
= ( linord2614967742042102400et_nat @ B2 ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_942_finite__UnionD,axiom,
! [A2: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A2 ) )
=> ( finite1152437895449049373et_nat @ A2 ) ) ).
% finite_UnionD
thf(fact_943_nall__tuples__finite,axiom,
! [AD: set_nat,N: nat] :
( ( finite_finite_nat @ AD )
=> ( finite5087377988160578214at_nat @ ( nall_tuples_nat @ AD @ N ) ) ) ).
% nall_tuples_finite
thf(fact_944_all__tuples__finite,axiom,
! [Xs: set_nat,N: nat] :
( ( finite_finite_nat @ Xs )
=> ( finite8100373058378681591st_nat @ ( all_tuples_nat @ Xs @ N ) ) ) ).
% all_tuples_finite
thf(fact_945_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa3: nat] :
( ( member_nat @ Xa3 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_946_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa3: nat] :
( ( member_nat @ Xa3 @ A2 )
=> ( ( ord_less_eq_nat @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_947_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_948_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_949_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_950_infinite__super,axiom,
! [S: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S @ T3 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_951_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_952_infinite__Un,axiom,
! [S: set_nat,T3: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T3 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_Un
thf(fact_953_Un__infinite,axiom,
! [S: set_nat,T3: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T3 ) ) ) ).
% Un_infinite
thf(fact_954_finite__UnI,axiom,
! [F4: set_nat,G3: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( finite_finite_nat @ G3 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_955_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_956_Diff__infinite__finite,axiom,
! [T3: set_nat,S: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_957_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa3: nat] :
( ( member_nat @ Xa3 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_958_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa3: nat] :
( ( member_nat @ Xa3 @ A2 )
=> ( ( ord_less_eq_nat @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_959_sorted__wrt__append,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ P @ Xs )
& ( sorted_wrt_nat @ P @ Ys )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
=> ( P @ X @ Y2 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_960_le__cSup__finite,axiom,
! [X4: set_nat,X3: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( member_nat @ X3 @ X4 )
=> ( ord_less_eq_nat @ X3 @ ( complete_Sup_Sup_nat @ X4 ) ) ) ) ).
% le_cSup_finite
thf(fact_961_all__finite__subset__image,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ( finite1487985464145237934_a_nat @ B3 )
& ( ord_le1147066620699065093_a_nat @ B3 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ( finite1487985464145237934_a_nat @ B3 )
& ( ord_le1147066620699065093_a_nat @ B3 @ A2 ) )
=> ( P @ ( image_5081948215111134021_a_nat @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_962_all__finite__subset__image,axiom,
! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P @ ( image_nat_nat @ F2 @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_963_ex__finite__subset__image,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( ? [B3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ B3 )
& ( ord_le1147066620699065093_a_nat @ B3 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ B3 )
& ( ord_le1147066620699065093_a_nat @ B3 @ A2 )
& ( P @ ( image_5081948215111134021_a_nat @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_964_ex__finite__subset__image,axiom,
! [F2: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P @ ( image_nat_nat @ F2 @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_965_finite__subset__image,axiom,
! [B2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
=> ? [C3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C3 @ A2 )
& ( finite1487985464145237934_a_nat @ C3 )
& ( B2
= ( image_5081948215111134021_a_nat @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_966_finite__subset__image,axiom,
! [B2: set_nat,F2: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( finite_finite_nat @ C3 )
& ( B2
= ( image_nat_nat @ F2 @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_967_finite__surj,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
=> ( finite1487985464145237934_a_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_968_finite__surj,axiom,
! [A2: set_nat,B2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_969_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A8: set_nat] :
( ? [A6: nat] :
( A
= ( insert_nat @ A6 @ A8 ) )
=> ~ ( finite_finite_nat @ A8 ) ) ) ) ).
% finite.cases
thf(fact_970_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A5
= ( insert_nat @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_971_finite__induct,axiom,
! [F4: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ F4 )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [X2: list_Sum_sum_a_nat,F6: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F6 )
=> ( ~ ( member408289922725080238_a_nat @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2950094090816004437_a_nat @ X2 @ F6 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_972_finite__induct,axiom,
! [F4: set_na3699693778330250182_a_nat,P: set_na3699693778330250182_a_nat > $o] :
( ( finite785833390020136079_a_nat @ F4 )
=> ( ( P @ bot_bo6441361344521902642_a_nat )
=> ( ! [X2: nat > sum_sum_a_nat,F6: set_na3699693778330250182_a_nat] :
( ( finite785833390020136079_a_nat @ F6 )
=> ( ~ ( member8690443509505302927_a_nat @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert5265011953798106934_a_nat @ X2 @ F6 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_973_finite__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ~ ( member_nat @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ X2 @ F6 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_974_finite__ne__induct,axiom,
! [F4: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ F4 )
=> ( ( F4 != bot_bo1033123847703346641_a_nat )
=> ( ! [X2: list_Sum_sum_a_nat] : ( P @ ( insert2950094090816004437_a_nat @ X2 @ bot_bo1033123847703346641_a_nat ) )
=> ( ! [X2: list_Sum_sum_a_nat,F6: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F6 )
=> ( ( F6 != bot_bo1033123847703346641_a_nat )
=> ( ~ ( member408289922725080238_a_nat @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2950094090816004437_a_nat @ X2 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_975_finite__ne__induct,axiom,
! [F4: set_na3699693778330250182_a_nat,P: set_na3699693778330250182_a_nat > $o] :
( ( finite785833390020136079_a_nat @ F4 )
=> ( ( F4 != bot_bo6441361344521902642_a_nat )
=> ( ! [X2: nat > sum_sum_a_nat] : ( P @ ( insert5265011953798106934_a_nat @ X2 @ bot_bo6441361344521902642_a_nat ) )
=> ( ! [X2: nat > sum_sum_a_nat,F6: set_na3699693778330250182_a_nat] :
( ( finite785833390020136079_a_nat @ F6 )
=> ( ( F6 != bot_bo6441361344521902642_a_nat )
=> ( ~ ( member8690443509505302927_a_nat @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert5265011953798106934_a_nat @ X2 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_976_finite__ne__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ( F6 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ X2 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_977_infinite__finite__induct,axiom,
! [P: set_li6526943997496501093_a_nat > $o,A2: set_li6526943997496501093_a_nat] :
( ! [A8: set_li6526943997496501093_a_nat] :
( ~ ( finite1487985464145237934_a_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [X2: list_Sum_sum_a_nat,F6: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F6 )
=> ( ~ ( member408289922725080238_a_nat @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2950094090816004437_a_nat @ X2 @ F6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_978_infinite__finite__induct,axiom,
! [P: set_na3699693778330250182_a_nat > $o,A2: set_na3699693778330250182_a_nat] :
( ! [A8: set_na3699693778330250182_a_nat] :
( ~ ( finite785833390020136079_a_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bo6441361344521902642_a_nat )
=> ( ! [X2: nat > sum_sum_a_nat,F6: set_na3699693778330250182_a_nat] :
( ( finite785833390020136079_a_nat @ F6 )
=> ( ~ ( member8690443509505302927_a_nat @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert5265011953798106934_a_nat @ X2 @ F6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_979_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A8: set_nat] :
( ~ ( finite_finite_nat @ A8 )
=> ( P @ A8 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ~ ( member_nat @ X2 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ X2 @ F6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_980_finite__image__iff,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( ( finite1487985464145237934_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( finite1487985464145237934_a_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_981_finite__image__iff,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_image_iff
thf(fact_982_finite__imageD,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
=> ( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( finite1487985464145237934_a_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_983_finite__imageD,axiom,
! [F2: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A2 ) )
=> ( ( inj_on_nat_nat @ F2 @ A2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_imageD
thf(fact_984_finite__distinct__list,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A2 )
=> ? [Xs2: list_Sum_sum_a_nat] :
( ( ( set_Sum_sum_a_nat2 @ Xs2 )
= A2 )
& ( distin2701893636801681144_a_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_985_finite__distinct__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= A2 )
& ( distinct_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_986_remove1__append,axiom,
! [X3: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( remove4274251903526005537_a_nat @ X3 @ ( append5415888156905520160_a_nat @ Xs @ Ys ) )
= ( append5415888156905520160_a_nat @ ( remove4274251903526005537_a_nat @ X3 @ Xs ) @ Ys ) ) )
& ( ~ ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( remove4274251903526005537_a_nat @ X3 @ ( append5415888156905520160_a_nat @ Xs @ Ys ) )
= ( append5415888156905520160_a_nat @ Xs @ ( remove4274251903526005537_a_nat @ X3 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_987_remove1__append,axiom,
! [X3: nat > sum_sum_a_nat,Xs: list_n989787106983797996_a_nat,Ys: list_n989787106983797996_a_nat] :
( ( ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( remove134331365954440450_a_nat @ X3 @ ( append7040142359210737409_a_nat @ Xs @ Ys ) )
= ( append7040142359210737409_a_nat @ ( remove134331365954440450_a_nat @ X3 @ Xs ) @ Ys ) ) )
& ( ~ ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( remove134331365954440450_a_nat @ X3 @ ( append7040142359210737409_a_nat @ Xs @ Ys ) )
= ( append7040142359210737409_a_nat @ Xs @ ( remove134331365954440450_a_nat @ X3 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_988_remove1__append,axiom,
! [X3: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X3 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remove1_nat @ X3 @ Xs ) @ Ys ) ) )
& ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X3 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( remove1_nat @ X3 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_989_sorted__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ).
% sorted_append
thf(fact_990_finite__subset__induct_H,axiom,
! [F4: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ F4 )
=> ( ( ord_le1147066620699065093_a_nat @ F4 @ A2 )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [A6: list_Sum_sum_a_nat,F6: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F6 )
=> ( ( member408289922725080238_a_nat @ A6 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ F6 @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2950094090816004437_a_nat @ A6 @ F6 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_991_finite__subset__induct_H,axiom,
! [F4: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat,P: set_na3699693778330250182_a_nat > $o] :
( ( finite785833390020136079_a_nat @ F4 )
=> ( ( ord_le8108555184339247974_a_nat @ F4 @ A2 )
=> ( ( P @ bot_bo6441361344521902642_a_nat )
=> ( ! [A6: nat > sum_sum_a_nat,F6: set_na3699693778330250182_a_nat] :
( ( finite785833390020136079_a_nat @ F6 )
=> ( ( member8690443509505302927_a_nat @ A6 @ A2 )
=> ( ( ord_le8108555184339247974_a_nat @ F6 @ A2 )
=> ( ~ ( member8690443509505302927_a_nat @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert5265011953798106934_a_nat @ A6 @ F6 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_992_finite__subset__induct_H,axiom,
! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ( ord_less_eq_set_nat @ F6 @ A2 )
=> ( ~ ( member_nat @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ A6 @ F6 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_993_finite__subset__induct,axiom,
! [F4: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ F4 )
=> ( ( ord_le1147066620699065093_a_nat @ F4 @ A2 )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [A6: list_Sum_sum_a_nat,F6: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ F6 )
=> ( ( member408289922725080238_a_nat @ A6 @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert2950094090816004437_a_nat @ A6 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_994_finite__subset__induct,axiom,
! [F4: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat,P: set_na3699693778330250182_a_nat > $o] :
( ( finite785833390020136079_a_nat @ F4 )
=> ( ( ord_le8108555184339247974_a_nat @ F4 @ A2 )
=> ( ( P @ bot_bo6441361344521902642_a_nat )
=> ( ! [A6: nat > sum_sum_a_nat,F6: set_na3699693778330250182_a_nat] :
( ( finite785833390020136079_a_nat @ F6 )
=> ( ( member8690443509505302927_a_nat @ A6 @ A2 )
=> ( ~ ( member8690443509505302927_a_nat @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert5265011953798106934_a_nat @ A6 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_995_finite__subset__induct,axiom,
! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F6: set_nat] :
( ( finite_finite_nat @ F6 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ~ ( member_nat @ A6 @ F6 )
=> ( ( P @ F6 )
=> ( P @ ( insert_nat @ A6 @ F6 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_996_finite__Sup__in,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat,Y: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( member_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ A2 ) ) )
=> ( member_set_nat @ ( comple7399068483239264473et_nat @ A2 ) @ A2 ) ) ) ) ).
% finite_Sup_in
thf(fact_997_finite__surj__inj,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
=> ( inj_on6609798167860701873_a_nat @ F2 @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_998_finite__surj__inj,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( inj_on_nat_nat @ F2 @ A2 ) ) ) ).
% finite_surj_inj
thf(fact_999_inj__on__finite,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite1487985464145237934_a_nat @ B2 )
=> ( finite1487985464145237934_a_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1000_inj__on__finite,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ) ).
% inj_on_finite
thf(fact_1001_endo__inj__surj,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ A2 )
=> ( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( ( image_5081948215111134021_a_nat @ F2 @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1002_endo__inj__surj,axiom,
! [A2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ A2 )
=> ( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( image_nat_nat @ F2 @ A2 )
= A2 ) ) ) ) ).
% endo_inj_surj
thf(fact_1003_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1004_infinite__coinduct,axiom,
! [X4: set_nat > $o,A2: set_nat] :
( ( X4 @ A2 )
=> ( ! [A8: set_nat] :
( ( X4 @ A8 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A8 )
& ( ( X4 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1005_finite__empty__induct,axiom,
! [A2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: list_Sum_sum_a_nat,A8: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A8 )
=> ( ( member408289922725080238_a_nat @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_7395159227704179404_a_nat @ A8 @ ( insert2950094090816004437_a_nat @ A6 @ bot_bo1033123847703346641_a_nat ) ) ) ) ) )
=> ( P @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1006_finite__empty__induct,axiom,
! [A2: set_na3699693778330250182_a_nat,P: set_na3699693778330250182_a_nat > $o] :
( ( finite785833390020136079_a_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: nat > sum_sum_a_nat,A8: set_na3699693778330250182_a_nat] :
( ( finite785833390020136079_a_nat @ A8 )
=> ( ( member8690443509505302927_a_nat @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_5517490076408937517_a_nat @ A8 @ ( insert5265011953798106934_a_nat @ A6 @ bot_bo6441361344521902642_a_nat ) ) ) ) ) )
=> ( P @ bot_bo6441361344521902642_a_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1007_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A6 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1008_the__elem__image__unique,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X3: list_Sum_sum_a_nat] :
( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ! [Y: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y @ A2 )
=> ( ( F2 @ Y )
= ( F2 @ X3 ) ) )
=> ( ( the_el2583442515771010938_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( F2 @ X3 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1009_finite__remove__induct,axiom,
! [B2: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ B2 )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [A8: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A8 )
=> ( ( A8 != bot_bo1033123847703346641_a_nat )
=> ( ( ord_le1147066620699065093_a_nat @ A8 @ B2 )
=> ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A8 )
=> ( P @ ( minus_7395159227704179404_a_nat @ A8 @ ( insert2950094090816004437_a_nat @ X5 @ bot_bo1033123847703346641_a_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1010_finite__remove__induct,axiom,
! [B2: set_na3699693778330250182_a_nat,P: set_na3699693778330250182_a_nat > $o] :
( ( finite785833390020136079_a_nat @ B2 )
=> ( ( P @ bot_bo6441361344521902642_a_nat )
=> ( ! [A8: set_na3699693778330250182_a_nat] :
( ( finite785833390020136079_a_nat @ A8 )
=> ( ( A8 != bot_bo6441361344521902642_a_nat )
=> ( ( ord_le8108555184339247974_a_nat @ A8 @ B2 )
=> ( ! [X5: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X5 @ A8 )
=> ( P @ ( minus_5517490076408937517_a_nat @ A8 @ ( insert5265011953798106934_a_nat @ X5 @ bot_bo6441361344521902642_a_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1011_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1012_remove__induct,axiom,
! [P: set_li6526943997496501093_a_nat > $o,B2: set_li6526943997496501093_a_nat] :
( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ( ~ ( finite1487985464145237934_a_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A8 )
=> ( ( A8 != bot_bo1033123847703346641_a_nat )
=> ( ( ord_le1147066620699065093_a_nat @ A8 @ B2 )
=> ( ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A8 )
=> ( P @ ( minus_7395159227704179404_a_nat @ A8 @ ( insert2950094090816004437_a_nat @ X5 @ bot_bo1033123847703346641_a_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1013_remove__induct,axiom,
! [P: set_na3699693778330250182_a_nat > $o,B2: set_na3699693778330250182_a_nat] :
( ( P @ bot_bo6441361344521902642_a_nat )
=> ( ( ~ ( finite785833390020136079_a_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_na3699693778330250182_a_nat] :
( ( finite785833390020136079_a_nat @ A8 )
=> ( ( A8 != bot_bo6441361344521902642_a_nat )
=> ( ( ord_le8108555184339247974_a_nat @ A8 @ B2 )
=> ( ! [X5: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X5 @ A8 )
=> ( P @ ( minus_5517490076408937517_a_nat @ A8 @ ( insert5265011953798106934_a_nat @ X5 @ bot_bo6441361344521902642_a_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1014_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1015_finite__sorted__distinct__unique,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [X2: list_nat] :
( ( ( set_nat2 @ X2 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X2 )
& ( distinct_nat @ X2 )
& ! [Y6: list_nat] :
( ( ( ( set_nat2 @ Y6 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y6 )
& ( distinct_nat @ Y6 ) )
=> ( Y6 = X2 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_1016_finite__ranking__induct,axiom,
! [S: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o,F2: list_Sum_sum_a_nat > nat] :
( ( finite1487985464145237934_a_nat @ S )
=> ( ( P @ bot_bo1033123847703346641_a_nat )
=> ( ! [X2: list_Sum_sum_a_nat,S4: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ S4 )
=> ( ! [Y6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y6 @ S4 )
=> ( ord_less_eq_nat @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert2950094090816004437_a_nat @ X2 @ S4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1017_finite__ranking__induct,axiom,
! [S: set_na3699693778330250182_a_nat,P: set_na3699693778330250182_a_nat > $o,F2: ( nat > sum_sum_a_nat ) > nat] :
( ( finite785833390020136079_a_nat @ S )
=> ( ( P @ bot_bo6441361344521902642_a_nat )
=> ( ! [X2: nat > sum_sum_a_nat,S4: set_na3699693778330250182_a_nat] :
( ( finite785833390020136079_a_nat @ S4 )
=> ( ! [Y6: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Y6 @ S4 )
=> ( ord_less_eq_nat @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert5265011953798106934_a_nat @ X2 @ S4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1018_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S4: set_nat] :
( ( finite_finite_nat @ S4 )
=> ( ! [Y6: nat] :
( ( member_nat @ Y6 @ S4 )
=> ( ord_less_eq_nat @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert_nat @ X2 @ S4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1019_card__le__inj,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( finite1487985464145237934_a_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite9161971191270313901_a_nat @ A2 ) @ ( finite9161971191270313901_a_nat @ B2 ) )
=> ? [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ B2 )
& ( inj_on6609798167860701873_a_nat @ F @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_1020_card__le__inj,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ? [F: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 )
& ( inj_on_nat_nat @ F @ A2 ) ) ) ) ) ).
% card_le_inj
thf(fact_1021_card__inj__on__le,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ B2 )
=> ( ( finite1487985464145237934_a_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite9161971191270313901_a_nat @ A2 ) @ ( finite9161971191270313901_a_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_1022_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A2 ) )
= ( finite_card_nat @ A2 ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_1023_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_1024_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_1025_card__image,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( ( finite9161971191270313901_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( finite9161971191270313901_a_nat @ A2 ) ) ) ).
% card_image
thf(fact_1026_card__image__le,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite9161971191270313901_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) ) @ ( finite9161971191270313901_a_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_1027_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_nat,C2: nat] :
( ! [G4: set_nat] :
( ( ord_less_eq_set_nat @ G4 @ F4 )
=> ( ( finite_finite_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G4 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_1028_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_1029_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C2 )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_1030_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_1031_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_1032_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_1033_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_1034_inj__on__iff__eq__card,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
= ( ( finite9161971191270313901_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( finite9161971191270313901_a_nat @ A2 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_1035_eq__card__imp__inj__on,axiom,
! [A2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( ( finite9161971191270313901_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
= ( finite9161971191270313901_a_nat @ A2 ) )
=> ( inj_on6609798167860701873_a_nat @ F2 @ A2 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_1036_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1037_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_1038_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_1039_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_1040_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_1041_surj__card__le,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) )
=> ( ord_less_eq_nat @ ( finite9161971191270313901_a_nat @ B2 ) @ ( finite9161971191270313901_a_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_1042_card__bij__eq,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on6609798167860701873_a_nat @ G @ B2 )
=> ( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ G @ B2 ) @ A2 )
=> ( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( finite1487985464145237934_a_nat @ B2 )
=> ( ( finite9161971191270313901_a_nat @ A2 )
= ( finite9161971191270313901_a_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_1043_card__bij__eq,axiom,
! [F2: nat > nat,A2: set_nat,B2: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A2 ) @ B2 )
=> ( ( inj_on_nat_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B2 ) @ A2 )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_1044_surjective__iff__injective__gen,axiom,
! [S: set_li6526943997496501093_a_nat,T3: set_li6526943997496501093_a_nat,F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ S )
=> ( ( finite1487985464145237934_a_nat @ T3 )
=> ( ( ( finite9161971191270313901_a_nat @ S )
= ( finite9161971191270313901_a_nat @ T3 ) )
=> ( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ S ) @ T3 )
=> ( ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ T3 )
=> ? [Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y2 @ S )
& ( ( F2 @ Y2 )
= X ) ) ) )
= ( inj_on6609798167860701873_a_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_1045_surjective__iff__injective__gen,axiom,
! [S: set_nat,T3: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ S ) @ T3 )
=> ( ( ! [X: nat] :
( ( member_nat @ X @ T3 )
=> ? [Y2: nat] :
( ( member_nat @ Y2 @ S )
& ( ( F2 @ Y2 )
= X ) ) ) )
= ( inj_on_nat_nat @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_1046_card__Diff1__le,axiom,
! [A2: set_nat,X3: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1047_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1048_card__set__minus,axiom,
! [Xs: list_Sum_sum_a_nat,X4: set_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( minus_1134630996077396038_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ X4 ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% card_set_minus
thf(fact_1049_card__set__minus,axiom,
! [Xs: list_nat,X4: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ X4 ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_set_minus
thf(fact_1050_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1051_card__Diff__subset__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1052_inj__on__iff__card__le,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( finite1487985464145237934_a_nat @ B2 )
=> ( ( ? [F3: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F3 @ A2 )
& ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite9161971191270313901_a_nat @ A2 ) @ ( finite9161971191270313901_a_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_1053_inj__on__iff__card__le,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A2 ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_1054_arb__finite__subset,axiom,
! [Y4: set_nat,N: nat] :
( ( finite_finite_nat @ Y4 )
=> ? [X7: set_nat] :
( ( ( inf_inf_set_nat @ Y4 @ X7 )
= bot_bot_set_nat )
& ( finite_finite_nat @ X7 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ X7 ) ) ) ) ).
% arb_finite_subset
thf(fact_1055_card__le__if__inj__on__rel,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat,R2: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ B2 )
=> ( ! [A6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A6 @ A2 )
=> ? [B9: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B9 @ B2 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A1: list_Sum_sum_a_nat,A22: list_Sum_sum_a_nat,B6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A1 @ A2 )
=> ( ( member408289922725080238_a_nat @ A22 @ A2 )
=> ( ( member408289922725080238_a_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite9161971191270313901_a_nat @ A2 ) @ ( finite9161971191270313901_a_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1056_card__le__if__inj__on__rel,axiom,
! [B2: set_na3699693778330250182_a_nat,A2: set_li6526943997496501093_a_nat,R2: list_Sum_sum_a_nat > ( nat > sum_sum_a_nat ) > $o] :
( ( finite785833390020136079_a_nat @ B2 )
=> ( ! [A6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A6 @ A2 )
=> ? [B9: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ B9 @ B2 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A1: list_Sum_sum_a_nat,A22: list_Sum_sum_a_nat,B6: nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A1 @ A2 )
=> ( ( member408289922725080238_a_nat @ A22 @ A2 )
=> ( ( member8690443509505302927_a_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite9161971191270313901_a_nat @ A2 ) @ ( finite4005154532989035918_a_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1057_card__le__if__inj__on__rel,axiom,
! [B2: set_li6526943997496501093_a_nat,A2: set_na3699693778330250182_a_nat,R2: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat > $o] :
( ( finite1487985464145237934_a_nat @ B2 )
=> ( ! [A6: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ A6 @ A2 )
=> ? [B9: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B9 @ B2 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A1: nat > sum_sum_a_nat,A22: nat > sum_sum_a_nat,B6: list_Sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ A1 @ A2 )
=> ( ( member8690443509505302927_a_nat @ A22 @ A2 )
=> ( ( member408289922725080238_a_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite4005154532989035918_a_nat @ A2 ) @ ( finite9161971191270313901_a_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1058_card__le__if__inj__on__rel,axiom,
! [B2: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat,R2: ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o] :
( ( finite785833390020136079_a_nat @ B2 )
=> ( ! [A6: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ A6 @ A2 )
=> ? [B9: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ B9 @ B2 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A1: nat > sum_sum_a_nat,A22: nat > sum_sum_a_nat,B6: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ A1 @ A2 )
=> ( ( member8690443509505302927_a_nat @ A22 @ A2 )
=> ( ( member8690443509505302927_a_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite4005154532989035918_a_nat @ A2 ) @ ( finite4005154532989035918_a_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1059_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_li6526943997496501093_a_nat,R2: list_Sum_sum_a_nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A6: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A6 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A1: list_Sum_sum_a_nat,A22: list_Sum_sum_a_nat,B6: nat] :
( ( member408289922725080238_a_nat @ A1 @ A2 )
=> ( ( member408289922725080238_a_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite9161971191270313901_a_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1060_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A2: set_na3699693778330250182_a_nat,R2: ( nat > sum_sum_a_nat ) > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A6: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ A6 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B2 )
& ( R2 @ A6 @ B9 ) ) )
=> ( ! [A1: nat > sum_sum_a_nat,A22: nat > sum_sum_a_nat,B6: nat] :
( ( member8690443509505302927_a_nat @ A1 @ A2 )
=> ( ( member8690443509505302927_a_nat @ A22 @ A2 )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite4005154532989035918_a_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_1061_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M5: nat] :
! [X: nat] :
( ( member_nat @ X @ N3 )
=> ( ord_less_eq_nat @ X @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1062_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M: nat] :
( ( P @ X3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1063_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( foldin7929645199718362341_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( ( finite502105017643426984_a_nat @ A2 )
=> ~ ! [L2: list_Sum_sum_a_nat] :
( ( sorted6245805940552704876_a_nat @ Less @ ( map_Su2790769393171190532_a_nat @ F2 @ L2 ) )
=> ( ( ( set_Sum_sum_a_nat2 @ L2 )
= A2 )
=> ( ( size_s5283204784079214577_a_nat @ L2 )
!= ( finite6080979521523705895_a_nat @ A2 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_1064_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,A2: set_Sum_sum_a_nat] :
( ( foldin2124782302959918920_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( ( finite502105017643426984_a_nat @ A2 )
=> ~ ! [L2: list_Sum_sum_a_nat] :
( ( sorted_wrt_nat @ Less @ ( map_Su5227373005390213643at_nat @ F2 @ L2 ) )
=> ( ( ( set_Sum_sum_a_nat2 @ L2 )
= A2 )
=> ( ( size_s5283204784079214577_a_nat @ L2 )
!= ( finite6080979521523705895_a_nat @ A2 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_1065_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_nat,F2: nat > sum_sum_a_nat,A2: set_nat] :
( ( foldin6528764236620990570at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( ( finite_finite_nat @ A2 )
=> ~ ! [L2: list_nat] :
( ( sorted6245805940552704876_a_nat @ Less @ ( map_na823391071729141993_a_nat @ F2 @ L2 ) )
=> ( ( ( set_nat2 @ L2 )
= A2 )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A2 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_1066_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_nat,F2: nat > nat,A2: set_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( ( finite_finite_nat @ A2 )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ Less @ ( map_nat_nat @ F2 @ L2 ) )
=> ( ( ( set_nat2 @ L2 )
= A2 )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A2 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_1067_arg__min__least,axiom,
! [S: set_li6526943997496501093_a_nat,Y3: list_Sum_sum_a_nat,F2: list_Sum_sum_a_nat > nat] :
( ( finite1487985464145237934_a_nat @ S )
=> ( ( S != bot_bo1033123847703346641_a_nat )
=> ( ( member408289922725080238_a_nat @ Y3 @ S )
=> ( ord_less_eq_nat @ ( F2 @ ( lattic7694009417247404284at_nat @ F2 @ S ) ) @ ( F2 @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1068_arg__min__least,axiom,
! [S: set_na3699693778330250182_a_nat,Y3: nat > sum_sum_a_nat,F2: ( nat > sum_sum_a_nat ) > nat] :
( ( finite785833390020136079_a_nat @ S )
=> ( ( S != bot_bo6441361344521902642_a_nat )
=> ( ( member8690443509505302927_a_nat @ Y3 @ S )
=> ( ord_less_eq_nat @ ( F2 @ ( lattic7010757105285391581at_nat @ F2 @ S ) ) @ ( F2 @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1069_arg__min__least,axiom,
! [S: set_nat,Y3: nat,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y3 @ S )
=> ( ord_less_eq_nat @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S ) ) @ ( F2 @ Y3 ) ) ) ) ) ).
% arg_min_least
thf(fact_1070_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_nat,F2: nat > sum_sum_a_nat,Xs: list_nat] :
( ( foldin6528764236620990570at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( distin2701893636801681144_a_nat @ ( map_na823391071729141993_a_nat @ F2 @ Xs ) )
=> ( distinct_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_1071_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_nat,F2: nat > nat,Xs: list_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( distinct_nat @ ( map_nat_nat @ F2 @ Xs ) )
=> ( distinct_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_1072_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( foldin2124782302959918920_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( distinct_nat @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
=> ( distin2701893636801681144_a_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_1073_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( foldin7929645199718362341_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( distin2701893636801681144_a_nat @ ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) )
=> ( distin2701893636801681144_a_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_1074_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat,L: list_Sum_sum_a_nat] :
( ( foldin7929645199718362341_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( ( finite502105017643426984_a_nat @ A2 )
=> ( ( ( sorted6245805940552704876_a_nat @ Less @ ( map_Su2790769393171190532_a_nat @ F2 @ L ) )
& ( ( set_Sum_sum_a_nat2 @ L )
= A2 )
& ( ( size_s5283204784079214577_a_nat @ L )
= ( finite6080979521523705895_a_nat @ A2 ) ) )
= ( ( sorted1513701980552311874_a_nat @ Less_eq @ F2 @ A2 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_1075_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,A2: set_Sum_sum_a_nat,L: list_Sum_sum_a_nat] :
( ( foldin2124782302959918920_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( ( finite502105017643426984_a_nat @ A2 )
=> ( ( ( sorted_wrt_nat @ Less @ ( map_Su5227373005390213643at_nat @ F2 @ L ) )
& ( ( set_Sum_sum_a_nat2 @ L )
= A2 )
& ( ( size_s5283204784079214577_a_nat @ L )
= ( finite6080979521523705895_a_nat @ A2 ) ) )
= ( ( sorted6158257153526155051_a_nat @ Less_eq @ F2 @ A2 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_1076_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_nat,F2: nat > sum_sum_a_nat,A2: set_nat,L: list_nat] :
( ( foldin6528764236620990570at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( ( finite_finite_nat @ A2 )
=> ( ( ( sorted6245805940552704876_a_nat @ Less @ ( map_na823391071729141993_a_nat @ F2 @ L ) )
& ( ( set_nat2 @ L )
= A2 )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A2 ) ) )
= ( ( sorted1338867050332450893at_nat @ Less_eq @ F2 @ A2 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_1077_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_nat,F2: nat > nat,A2: set_nat,L: list_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( ( finite_finite_nat @ A2 )
=> ( ( ( sorted_wrt_nat @ Less @ ( map_nat_nat @ F2 @ L ) )
& ( ( set_nat2 @ L )
= A2 )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A2 ) ) )
= ( ( sorted5905597674102116260at_nat @ Less_eq @ F2 @ A2 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_1078_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_nat,F2: nat > sum_sum_a_nat,A2: set_nat] :
( ( foldin6528764236620990570at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( distin2701893636801681144_a_nat @ ( map_na823391071729141993_a_nat @ F2 @ ( sorted1338867050332450893at_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_1079_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_nat,F2: nat > nat,A2: set_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( distinct_nat @ ( map_nat_nat @ F2 @ ( sorted5905597674102116260at_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_1080_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,A2: set_Sum_sum_a_nat] :
( ( foldin2124782302959918920_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( distinct_nat @ ( map_Su5227373005390213643at_nat @ F2 @ ( sorted6158257153526155051_a_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_1081_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( foldin7929645199718362341_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( distin2701893636801681144_a_nat @ ( map_Su2790769393171190532_a_nat @ F2 @ ( sorted1513701980552311874_a_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_1082_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_nat,F2: nat > sum_sum_a_nat,A2: set_nat] :
( ( foldin6528764236620990570at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( sorted6245805940552704876_a_nat @ Less @ ( map_na823391071729141993_a_nat @ F2 @ ( sorted1338867050332450893at_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_1083_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( foldin7929645199718362341_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( sorted6245805940552704876_a_nat @ Less @ ( map_Su2790769393171190532_a_nat @ F2 @ ( sorted1513701980552311874_a_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_1084_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_nat,F2: nat > nat,A2: set_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( sorted_wrt_nat @ Less @ ( map_nat_nat @ F2 @ ( sorted5905597674102116260at_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_1085_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,A2: set_Sum_sum_a_nat] :
( ( foldin2124782302959918920_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( sorted_wrt_nat @ Less @ ( map_Su5227373005390213643at_nat @ F2 @ ( sorted6158257153526155051_a_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_1086_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_nat,F2: nat > sum_sum_a_nat,A2: set_nat] :
( ( foldin6528764236620990570at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( sorted6245805940552704876_a_nat @ Less_eq @ ( map_na823391071729141993_a_nat @ F2 @ ( sorted1338867050332450893at_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_1087_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( foldin7929645199718362341_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( sorted6245805940552704876_a_nat @ Less_eq @ ( map_Su2790769393171190532_a_nat @ F2 @ ( sorted1513701980552311874_a_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_1088_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_nat,F2: nat > nat,A2: set_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ A2 @ S )
=> ( sorted_wrt_nat @ Less_eq @ ( map_nat_nat @ F2 @ ( sorted5905597674102116260at_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_1089_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,A2: set_Sum_sum_a_nat] :
( ( foldin2124782302959918920_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ S )
=> ( sorted_wrt_nat @ Less_eq @ ( map_Su5227373005390213643at_nat @ F2 @ ( sorted6158257153526155051_a_nat @ Less_eq @ F2 @ A2 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_1090_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( foldin7929645199718362341_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ S )
=> ( ( sorted6245805940552704876_a_nat @ Less_eq @ ( map_Su2790769393171190532_a_nat @ F2 @ Xs ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( sorted1513701980552311874_a_nat @ Less_eq @ F2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_1091_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [Less_eq: sum_sum_a_nat > sum_sum_a_nat > $o,Less: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_nat,F2: nat > sum_sum_a_nat,Xs: list_nat] :
( ( foldin6528764236620990570at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ S )
=> ( ( sorted6245805940552704876_a_nat @ Less_eq @ ( map_na823391071729141993_a_nat @ F2 @ Xs ) )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted1338867050332450893at_nat @ Less_eq @ F2 @ ( set_nat2 @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_1092_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_Sum_sum_a_nat,F2: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( foldin2124782302959918920_a_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ S )
=> ( ( sorted_wrt_nat @ Less_eq @ ( map_Su5227373005390213643at_nat @ F2 @ Xs ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( sorted6158257153526155051_a_nat @ Less_eq @ F2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_1093_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S: set_nat,F2: nat > nat,Xs: list_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S @ F2 )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ S )
=> ( ( sorted_wrt_nat @ Less_eq @ ( map_nat_nat @ F2 @ Xs ) )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted5905597674102116260at_nat @ Less_eq @ F2 @ ( set_nat2 @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_1094_card__partition,axiom,
! [C2: set_set_nat,K: nat] :
( ( finite1152437895449049373et_nat @ C2 )
=> ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ C2 ) )
=> ( ! [C5: set_nat] :
( ( member_set_nat @ C5 @ C2 )
=> ( ( finite_card_nat @ C5 )
= K ) )
=> ( ! [C1: set_nat,C22: set_nat] :
( ( member_set_nat @ C1 @ C2 )
=> ( ( member_set_nat @ C22 @ C2 )
=> ( ( C1 != C22 )
=> ( ( inf_inf_set_nat @ C1 @ C22 )
= bot_bot_set_nat ) ) ) )
=> ( ( times_times_nat @ K @ ( finite_card_set_nat @ C2 ) )
= ( finite_card_nat @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_1095_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A2: set_nat,L: list_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A2 )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A2 ) ) )
= ( ( linord2614967742042102400et_nat @ A2 )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_1096_finite__maxlen,axiom,
! [M: set_li6526943997496501093_a_nat] :
( ( finite1487985464145237934_a_nat @ M )
=> ? [N4: nat] :
! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ M )
=> ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ X5 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_1097_finite__maxlen,axiom,
! [M: set_list_nat] :
( ( finite8100373058378681591st_nat @ M )
=> ? [N4: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ M )
=> ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_1098_finite__imp__Sup__less,axiom,
! [X4: set_nat,X3: nat,A: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( member_nat @ X3 @ X4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ( ord_less_nat @ X2 @ A ) )
=> ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_1099_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_1100_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [B10: set_nat] :
( ( ord_less_set_nat @ B10 @ A8 )
=> ( P @ B10 ) )
=> ( P @ A8 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_1101_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ~ ? [Xa3: nat] :
( ( member_nat @ Xa3 @ S )
& ( ord_less_nat @ Xa3 @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1102_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ X4 )
& ( ord_less_nat @ X2 @ Xa3 ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_1103_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_1104_less__cSupD,axiom,
! [X4: set_nat,Z2: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ Z2 @ ( complete_Sup_Sup_nat @ X4 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ X4 )
& ( ord_less_nat @ Z2 @ X2 ) ) ) ) ).
% less_cSupD
thf(fact_1105_less__cSupE,axiom,
! [Y3: nat,X4: set_nat] :
( ( ord_less_nat @ Y3 @ ( complete_Sup_Sup_nat @ X4 ) )
=> ( ( X4 != bot_bot_set_nat )
=> ~ ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ) ) ).
% less_cSupE
thf(fact_1106_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_1107_image__strict__mono,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ B2 )
=> ( ( ord_le5291801191193052689_a_nat @ A2 @ B2 )
=> ( ord_le5291801191193052689_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) @ ( image_5081948215111134021_a_nat @ F2 @ B2 ) ) ) ) ).
% image_strict_mono
thf(fact_1108_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1109_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1110_less__diff__iff,axiom,
! [K: nat,M4: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M4 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1111_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_1112_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A @ C5 )
& ( ord_less_eq_nat @ C5 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C5 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1113_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1114_linorder__le__less__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_nat @ Y3 @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_1115_order__less__le__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1116_order__less__le__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1117_order__le__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1118_order__le__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1119_order__less__le__trans,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1120_order__le__less__trans,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1121_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1122_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1123_order__less__imp__le,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1124_linorder__not__less,axiom,
! [X3: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_not_less
thf(fact_1125_linorder__not__le,axiom,
! [X3: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X3 ) ) ).
% linorder_not_le
thf(fact_1126_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ( X != Y2 ) ) ) ) ).
% order_less_le
thf(fact_1127_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_less
thf(fact_1128_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1129_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1130_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1131_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1132_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1133_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1134_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1135_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1136_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1137_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1138_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1139_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1140_not__le__imp__less,axiom,
! [Y3: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ord_less_nat @ X3 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_1141_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_1142_antisym__conv2,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_1143_antisym__conv1,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_1144_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1145_leI,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% leI
thf(fact_1146_leD,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y3 ) ) ).
% leD
thf(fact_1147_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1148_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_1149_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1150_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1151_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1152_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1153_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1154_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1155_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1156_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( A5
= ( sup_sup_set_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1157_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1158_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1159_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1160_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1161_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1162_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1163_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1164_less__supI2,axiom,
! [X3: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X3 @ B )
=> ( ord_less_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_1165_less__supI2,axiom,
! [X3: nat,B: nat,A: nat] :
( ( ord_less_nat @ X3 @ B )
=> ( ord_less_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_1166_less__supI1,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X3 @ A )
=> ( ord_less_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_1167_less__supI1,axiom,
! [X3: nat,A: nat,B: nat] :
( ( ord_less_nat @ X3 @ A )
=> ( ord_less_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_1168_less__infI1,axiom,
! [A: set_nat,X3: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ X3 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).
% less_infI1
thf(fact_1169_less__infI1,axiom,
! [A: nat,X3: nat,B: nat] :
( ( ord_less_nat @ A @ X3 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).
% less_infI1
thf(fact_1170_less__infI2,axiom,
! [B: set_nat,X3: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ X3 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).
% less_infI2
thf(fact_1171_less__infI2,axiom,
! [B: nat,X3: nat,A: nat] :
( ( ord_less_nat @ B @ X3 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).
% less_infI2
thf(fact_1172_inf_Oabsorb3,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_1173_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_1174_inf_Oabsorb4,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1175_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1176_inf_Ostrict__boundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1177_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1178_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( A5
= ( inf_inf_set_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1179_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( A5
= ( inf_inf_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1180_inf_Ostrict__coboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1181_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1182_inf_Ostrict__coboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1183_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1184_order__less__imp__not__less,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_1185_order__less__imp__not__eq2,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_1186_order__less__imp__not__eq,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_1187_linorder__less__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_less_nat @ Y3 @ X3 ) ) ).
% linorder_less_linear
thf(fact_1188_order__less__imp__triv,axiom,
! [X3: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_1189_order__less__not__sym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% order_less_not_sym
thf(fact_1190_order__less__subst2,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1191_order__less__subst1,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1192_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_1193_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1194_ord__eq__less__subst,axiom,
! [A: nat,F2: nat > nat,B: nat,C: nat] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1195_order__less__trans,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_1196_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_1197_linorder__neq__iff,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
= ( ( ord_less_nat @ X3 @ Y3 )
| ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_1198_order__less__asym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% order_less_asym
thf(fact_1199_linorder__neqE,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
=> ( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_neqE
thf(fact_1200_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1201_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_1202_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_1203_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1204_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_1205_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_1206_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X9: nat] : ( P3 @ X9 ) )
= ( ^ [P2: nat > $o] :
? [N5: nat] :
( ( P2 @ N5 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ~ ( P2 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1207_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_1208_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_1209_linorder__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_cases
thf(fact_1210_antisym__conv3,axiom,
! [Y3: nat,X3: nat] :
( ~ ( ord_less_nat @ Y3 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_1211_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X2 )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_1212_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1213_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1214_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_1215_less__imp__neq,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_neq
thf(fact_1216_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_1217_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1218_le__neq__implies__less,axiom,
! [M4: nat,N: nat] :
( ( ord_less_eq_nat @ M4 @ N )
=> ( ( M4 != N )
=> ( ord_less_nat @ M4 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1219_less__or__eq__imp__le,axiom,
! [M4: nat,N: nat] :
( ( ( ord_less_nat @ M4 @ N )
| ( M4 = N ) )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1220_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
| ( M5 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1221_less__imp__le__nat,axiom,
! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_eq_nat @ M4 @ N ) ) ).
% less_imp_le_nat
thf(fact_1222_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_1223_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1224_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1225_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1226_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1227_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1228_psubset__imp__ex__mem,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A2 @ B2 )
=> ? [B6: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ B6 @ ( minus_7395159227704179404_a_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1229_psubset__imp__ex__mem,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ord_le1422622543498322546_a_nat @ A2 @ B2 )
=> ? [B6: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ B6 @ ( minus_5517490076408937517_a_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1230_length__induct,axiom,
! [P: list_Sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
( ! [Xs2: list_Sum_sum_a_nat] :
( ! [Ys4: list_Sum_sum_a_nat] :
( ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ Ys4 ) @ ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1231_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys4: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys4 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1232_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_1233_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ X5 @ B6 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1234_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B6: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A8 )
=> ( ord_less_nat @ B6 @ X5 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B6 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1235_finite__Sup__less__iff,axiom,
! [X4: set_nat,A: nat] :
( ( finite_finite_nat @ X4 )
=> ( ( X4 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X4 ) @ A )
= ( ! [X: nat] :
( ( member_nat @ X @ X4 )
=> ( ord_less_nat @ X @ A ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_1236_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_1237_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1238_pigeonhole,axiom,
! [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_less_nat @ ( finite9161971191270313901_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A2 ) ) @ ( finite9161971191270313901_a_nat @ A2 ) )
=> ~ ( inj_on6609798167860701873_a_nat @ F2 @ A2 ) ) ).
% pigeonhole
thf(fact_1239_less__cSup__iff,axiom,
! [X4: set_nat,Y3: nat] :
( ( X4 != bot_bot_set_nat )
=> ( ( condit2214826472909112428ve_nat @ X4 )
=> ( ( ord_less_nat @ Y3 @ ( complete_Sup_Sup_nat @ X4 ) )
= ( ? [X: nat] :
( ( member_nat @ X @ X4 )
& ( ord_less_nat @ Y3 @ X ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_1240_length__removeAll__less,axiom,
! [X3: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ord_less_nat @ ( size_s5212483967078203639_a_nat @ ( remove910890064017026449_a_nat @ X3 @ Xs ) ) @ ( size_s5212483967078203639_a_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1241_length__removeAll__less,axiom,
! [X3: nat > sum_sum_a_nat,Xs: list_n989787106983797996_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ord_less_nat @ ( size_s2119148862884149080_a_nat @ ( remove1885113525496864626_a_nat @ X3 @ Xs ) ) @ ( size_s2119148862884149080_a_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1242_length__removeAll__less,axiom,
! [X3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ ( remove3909449470355376139_a_nat @ X3 @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1243_length__removeAll__less,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X3 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1244_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S )
& ( ord_less_nat @ ( F2 @ X5 ) @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1245_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T4: set_nat] :
( ( ord_less_set_nat @ T4 @ S )
=> ( ( P @ T4 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ ( minus_minus_set_nat @ S @ T4 ) )
& ( P @ ( insert_nat @ X5 @ T4 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1246_psubset__insert__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ B2 ) )
= ( ( ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ord_le5291801191193052689_a_nat @ A2 @ B2 ) )
& ( ~ ( member408289922725080238_a_nat @ X3 @ B2 )
=> ( ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ord_le5291801191193052689_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) ) @ B2 ) )
& ( ~ ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1247_psubset__insert__iff,axiom,
! [A2: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ord_le1422622543498322546_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ B2 ) )
= ( ( ( member8690443509505302927_a_nat @ X3 @ B2 )
=> ( ord_le1422622543498322546_a_nat @ A2 @ B2 ) )
& ( ~ ( member8690443509505302927_a_nat @ X3 @ B2 )
=> ( ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ord_le1422622543498322546_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ bot_bo6441361344521902642_a_nat ) ) @ B2 ) )
& ( ~ ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ord_le8108555184339247974_a_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1248_psubset__insert__iff,axiom,
! [A2: set_nat,X3: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
= ( ( ( member_nat @ X3 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X3 @ B2 )
=> ( ( ( member_nat @ X3 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1249_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A2 )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A2 ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1250_card__Diff1__less__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat] :
( ( ord_less_nat @ ( finite9161971191270313901_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) ) ) @ ( finite9161971191270313901_a_nat @ A2 ) )
= ( ( finite1487985464145237934_a_nat @ A2 )
& ( member408289922725080238_a_nat @ X3 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1251_card__Diff1__less__iff,axiom,
! [A2: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat] :
( ( ord_less_nat @ ( finite4005154532989035918_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ bot_bo6441361344521902642_a_nat ) ) ) @ ( finite4005154532989035918_a_nat @ A2 ) )
= ( ( finite785833390020136079_a_nat @ A2 )
& ( member8690443509505302927_a_nat @ X3 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1252_card__Diff1__less__iff,axiom,
! [A2: set_nat,X3: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X3 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1253_card__Diff2__less,axiom,
! [A2: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ( member408289922725080238_a_nat @ Y3 @ A2 )
=> ( ord_less_nat @ ( finite9161971191270313901_a_nat @ ( minus_7395159227704179404_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) ) @ ( insert2950094090816004437_a_nat @ Y3 @ bot_bo1033123847703346641_a_nat ) ) ) @ ( finite9161971191270313901_a_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1254_card__Diff2__less,axiom,
! [A2: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat,Y3: nat > sum_sum_a_nat] :
( ( finite785833390020136079_a_nat @ A2 )
=> ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ( member8690443509505302927_a_nat @ Y3 @ A2 )
=> ( ord_less_nat @ ( finite4005154532989035918_a_nat @ ( minus_5517490076408937517_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ bot_bo6441361344521902642_a_nat ) ) @ ( insert5265011953798106934_a_nat @ Y3 @ bot_bo6441361344521902642_a_nat ) ) ) @ ( finite4005154532989035918_a_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1255_card__Diff2__less,axiom,
! [A2: set_nat,X3: nat,Y3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1256_card__Diff1__less,axiom,
! [A2: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat] :
( ( finite1487985464145237934_a_nat @ A2 )
=> ( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ord_less_nat @ ( finite9161971191270313901_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) ) ) @ ( finite9161971191270313901_a_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1257_card__Diff1__less,axiom,
! [A2: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat] :
( ( finite785833390020136079_a_nat @ A2 )
=> ( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ord_less_nat @ ( finite4005154532989035918_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X3 @ bot_bo6441361344521902642_a_nat ) ) ) @ ( finite4005154532989035918_a_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1258_card__Diff1__less,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1259_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_1260_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_1261_psubsetD,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat] :
( ( ord_le5291801191193052689_a_nat @ A2 @ B2 )
=> ( ( member408289922725080238_a_nat @ C @ A2 )
=> ( member408289922725080238_a_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1262_psubsetD,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat,C: nat > sum_sum_a_nat] :
( ( ord_le1422622543498322546_a_nat @ A2 @ B2 )
=> ( ( member8690443509505302927_a_nat @ C @ A2 )
=> ( member8690443509505302927_a_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1263_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1264_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1265_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M4: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_1266_verit__comp__simplify1_I3_J,axiom,
! [B11: nat,A9: nat] :
( ( ~ ( ord_less_eq_nat @ B11 @ A9 ) )
= ( ord_less_nat @ A9 @ B11 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1267_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_1268_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1269_sorted__quicksort__part,axiom,
! [Ac: list_nat,Lts: list_nat,X3: nat,Eqs: list_nat,Gts: list_nat,Zs3: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Ac )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( set_nat2 @ Lts ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ ( set_nat2 @ Eqs ) ) @ ( set_nat2 @ Gts ) ) @ ( set_nat2 @ Zs3 ) ) )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ac ) )
=> ( ord_less_nat @ X2 @ Xa2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Lts ) )
=> ( ord_less_nat @ X2 @ X3 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Eqs ) )
=> ( X2 = X3 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Gts ) )
=> ( ord_less_nat @ X3 @ X2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or1804217446461887602rt_nat @ Ac @ X3 @ Lts @ Eqs @ Gts @ Zs3 ) ) ) ) ) ) ) ).
% sorted_quicksort_part
thf(fact_1270_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_1271_set__quicksort__part,axiom,
! [Ac: list_nat,X3: nat,Lts: list_nat,Eqs: list_nat,Gts: list_nat,Zs3: list_nat] :
( ( set_nat2 @ ( set_or1804217446461887602rt_nat @ Ac @ X3 @ Lts @ Eqs @ Gts @ Zs3 ) )
= ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( set_nat2 @ Ac ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ ( set_nat2 @ Lts ) ) @ ( set_nat2 @ Eqs ) ) @ ( set_nat2 @ Gts ) ) @ ( set_nat2 @ Zs3 ) ) ) ).
% set_quicksort_part
thf(fact_1272_remove1__split,axiom,
! [A: list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( ( remove4274251903526005537_a_nat @ A @ Xs )
= Ys )
= ( ? [Ls: list_l4703314356710769291_a_nat,Rs: list_l4703314356710769291_a_nat] :
( ( Xs
= ( append5415888156905520160_a_nat @ Ls @ ( cons_l6604326339930385211_a_nat @ A @ Rs ) ) )
& ~ ( member408289922725080238_a_nat @ A @ ( set_li2392974972034027290_a_nat @ Ls ) )
& ( Ys
= ( append5415888156905520160_a_nat @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1273_remove1__split,axiom,
! [A: nat > sum_sum_a_nat,Xs: list_n989787106983797996_a_nat,Ys: list_n989787106983797996_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( ( remove134331365954440450_a_nat @ A @ Xs )
= Ys )
= ( ? [Ls: list_n989787106983797996_a_nat,Rs: list_n989787106983797996_a_nat] :
( ( Xs
= ( append7040142359210737409_a_nat @ Ls @ ( cons_n1358282950952874780_a_nat @ A @ Rs ) ) )
& ~ ( member8690443509505302927_a_nat @ A @ ( set_na645604395003041787_a_nat @ Ls ) )
& ( Ys
= ( append7040142359210737409_a_nat @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1274_remove1__split,axiom,
! [A: nat,Xs: list_nat,Ys: list_nat] :
( ( member_nat @ A @ ( set_nat2 @ Xs ) )
=> ( ( ( remove1_nat @ A @ Xs )
= Ys )
= ( ? [Ls: list_nat,Rs: list_nat] :
( ( Xs
= ( append_nat @ Ls @ ( cons_nat @ A @ Rs ) ) )
& ~ ( member_nat @ A @ ( set_nat2 @ Ls ) )
& ( Ys
= ( append_nat @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1275_distinct__length__2__or__more,axiom,
! [A: sum_sum_a_nat,B: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ A @ ( cons_Sum_sum_a_nat @ B @ Xs ) ) )
= ( ( A != B )
& ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ A @ Xs ) )
& ( distin2701893636801681144_a_nat @ ( cons_Sum_sum_a_nat @ B @ Xs ) ) ) ) ).
% distinct_length_2_or_more
% Helper facts (5)
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( if_set_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( if_set_nat @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_T,axiom,
! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( if_set7709265119413304363_a_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_T,axiom,
! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( if_set7709265119413304363_a_nat @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [Sigma4: nat > sum_sum_a_nat] :
( ( xs
= ( map_na823391071729141993_a_nat @ Sigma4 @ fv_all ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------