TPTP Problem File: SLH0632^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_01688_064254__16394726_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1482 ( 665 unt; 207 typ; 0 def)
% Number of atoms : 3262 (1124 equ; 0 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 10525 ( 249 ~; 25 |; 182 &;8779 @)
% ( 0 <=>;1290 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 787 ( 787 >; 0 *; 0 +; 0 <<)
% Number of symbols : 188 ( 187 usr; 23 con; 0-5 aty)
% Number of variables : 3388 ( 237 ^;3087 !; 64 ?;3388 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:50:44.726
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
set_Mo5210732246825857808et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
produc4045820344675478307at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_Mo2574807150581459802et_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
monoto8535755219626829232et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Mo6069479339911551325at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Mo5013373542560054436et_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
monoto5483634261523599098et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
monoto8276428299528460797at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto7244996872745832772et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
set_Mo2626137824023173004mula_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_Itf__a_J,type,
monotone_mformula_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (187)
thf(sy_c_Assumptions__and__Approximations_OL0,type,
assumptions_and_L0: nat ).
thf(sy_c_Assumptions__and__Approximations_OL0_H,type,
assumptions_and_L02: nat ).
thf(sy_c_Assumptions__and__Approximations_OM0,type,
assumptions_and_M0: nat ).
thf(sy_c_Assumptions__and__Approximations_OM0_H,type,
assumptions_and_M02: nat ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_OL,type,
assump1710595444109740301irst_L: nat > nat > nat ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_Om,type,
assump1710595444109740334irst_m: nat > nat ).
thf(sy_c_Assumptions__and__Approximations_Osecond__assumptions,type,
assump2881078719466019805ptions: nat > nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Othird__assumptions,type,
assump2119784843035796504ptions: nat > nat > nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_OGraphs,type,
clique5786534781347292306Graphs: set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
clique134924887794942129at_nat: set_nat_nat > set_nat_nat > set_set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Nat__Onat,type,
clique6722202388162463298od_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Nat__Onat_J,type,
clique8906516429304539640et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
clique1181040904276305582et_nat: set_set_set_nat > set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC,type,
clique3210737319928189260st_ACC: nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC__cf,type,
clique951075384711337423ACC_cf: nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OC,type,
clique5033774636164728462irst_C: nat > ( nat > nat ) > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OCLIQUE,type,
clique363107459185959606CLIQUE: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_ONEG,type,
clique3210737375870294875st_NEG: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060F_062,type,
clique2971579238625216137irst_F: nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060G_062l,type,
clique7840962075309931874st_G_l: nat > nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060K_062,type,
clique3326749438856946062irst_K: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060P_062L_092_060G_062l,type,
clique2294137941332549862_L_G_l: nat > nat > nat > set_set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oaccepts,type,
clique3686358387679108662ccepts: set_set_set_nat > set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodot,type,
clique5469973757772500719t_odot: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodotl,type,
clique7966186356931407165_odotl: nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oplucking__step,type,
clique4095374090462327202g_step: nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov,type,
clique5033774636164728513irst_v: set_set_nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov__gs,type,
clique8462013130872731469t_v_gs: set_set_set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
clique5528702923696243640at_nat: nat > nat > nat > set_nat_nat > ( ( nat > nat ) > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001t__Set__Oset_It__Nat__Onat_J,type,
clique522982669833463679et_nat: nat > nat > nat > set_set_nat > ( set_nat > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
clique2455256169097332789et_nat: nat > nat > nat > set_set_set_nat > ( set_set_nat > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
clique3407333501437444587et_nat: nat > nat > nat > set_set_set_set_nat > ( set_set_set_nat > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001tf__a,type,
clique8563529963003110213ions_a: nat > nat > nat > set_a > ( a > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OACC__cf__mf_001tf__a,type,
clique8961599393750669800f_mf_a: nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OACC__mf_001tf__a,type,
clique4708818501384062891C_mf_a: nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OAPR_001tf__a,type,
clique3873310923663319714_APR_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OAPR__rel_001tf__a,type,
clique5870032674357670943_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OSET_001tf__a,type,
clique6509092761774629891_SET_a: ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OSET__rel_001tf__a,type,
clique834332680210058238_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
clique6859621968737270801at_nat: set_nat_nat > set_Mo6069479339911551325at_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001t__Set__Oset_It__Nat__Onat_J,type,
clique9181349226887787864et_nat: set_set_nat > set_Mo5013373542560054436et_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
clique7740924183492588046et_nat: set_set_set_nat > set_Mo2574807150581459802et_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
clique2555064243683067844et_nat: set_set_set_set_nat > set_Mo5210732246825857808et_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001tf__a,type,
clique5987991184601036204th_A_a: set_a > set_Mo2626137824023173004mula_a ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060theta_062_092_060_094sub_062g_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
clique2614152022935308233at_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_set_nat > ( nat > nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060theta_062_092_060_094sub_062g_001t__Set__Oset_It__Nat__Onat_J,type,
clique8265756686838912272et_nat: set_set_nat > ( set_nat > set_nat ) > set_set_nat > set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060theta_062_092_060_094sub_062g_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
clique3281771716360483270et_nat: set_set_set_nat > ( set_set_nat > set_nat ) > set_set_nat > set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060theta_062_092_060_094sub_062g_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
clique1705999508207691644et_nat: set_set_set_set_nat > ( set_set_set_nat > set_nat ) > set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060theta_062_092_060_094sub_062g_001tf__a,type,
clique3148831351753978868ta_g_a: set_a > ( a > set_nat ) > set_set_nat > a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__neg_001tf__a,type,
clique6623365555141101007_neg_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__neg__rel_001tf__a,type,
clique6353239774569474354_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__pos_001tf__a,type,
clique8538548958085942603_pos_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oapprox__pos__rel_001tf__a,type,
clique4465983624924118198_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Odeviate__neg_001tf__a,type,
clique2019076642914533763_neg_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Odeviate__pos_001tf__a,type,
clique3934260045859375359_pos_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oeval__g_001tf__a,type,
clique5859573001277246426al_g_a: set_a > ( a > set_nat ) > ( a > $o ) > set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_Oeval__gs_001tf__a,type,
clique835570645587132141l_gs_a: set_a > ( a > set_nat ) > ( a > $o ) > set_set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Onumbers,type,
clique3652268606331196573umbers: nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU,type,
clique2699557479641037314nd_PLU: nat > nat > nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main,type,
clique429652266423863867U_main: nat > nat > nat > set_set_set_nat > produc4045820344675478307at_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main__rel,type,
clique8954521387433384062in_rel: nat > nat > nat > set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cap,type,
clique1591571987438064265eg_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cup,type,
clique1591571987439376245eg_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cap,type,
clique3314026705535538693os_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cup,type,
clique3314026705536850673os_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcap,type,
clique2586627118206219037_sqcap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcup,type,
clique2586627118207531017_sqcup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite3586981331298542604at_nat: set_set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
finite5926941155766903689et_nat: set_set_set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bij_betw_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bij_be2321430536510320189et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001tf__a,type,
bij_betw_nat_nat_a: ( ( nat > nat ) > a ) > set_nat_nat > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bij_betw_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
bij_betw_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bij_be6938610931847138308et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
bij_betw_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bij_be3438014552859920132et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bij_be5767359585022399418et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Nat__Onat_J_001tf__a,type,
bij_betw_set_nat_a: ( set_nat > a ) > set_set_nat > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
bij_be6199415091885040644at_nat: ( set_set_nat > nat ) > set_set_set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
bij_be4885122793727115194et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001tf__a,type,
bij_be3032674665972365258_nat_a: ( set_set_nat > a ) > set_set_set_nat > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001tf__a,type,
bij_be458158114365198228_nat_a: ( set_set_set_nat > a ) > set_set_set_set_nat > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bij_betw_a_nat_nat: ( a > nat > nat ) > set_a > set_nat_nat > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
bij_betw_a_set_nat: ( a > set_nat ) > set_a > set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bij_be2639851105560558660et_nat: ( a > set_set_nat ) > set_a > set_set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bij_be3030433078811146746et_nat: ( a > set_set_set_nat ) > set_a > set_set_set_set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001tf__a,type,
bij_betw_a_a: ( a > a ) > set_a > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_a_set_nat: ( a > set_nat ) > set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
minus_3113942175840221057et_nat: set_set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_HOL_Oundefined_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
undefi6751788150640612746et_nat: set_set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_inf_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
inf_in5711780100303410308et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
inf_in2396666505901392698et_nat: set_set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_sup_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
sup_su4213647025997063966et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
sup_su3906748206781935060et_nat: set_set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Monotone__Formula_Oeval_001tf__a,type,
monotone_eval_a: ( a > $o ) > monotone_mformula_a > $o ).
thf(sy_c_Monotone__Formula_Omformula_OConj_001tf__a,type,
monotone_Conj_a: monotone_mformula_a > monotone_mformula_a > monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Omformula_ODisj_001tf__a,type,
monotone_Disj_a: monotone_mformula_a > monotone_mformula_a > monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Omformula_OFALSE_001tf__a,type,
monotone_FALSE_a: monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Omformula_OTRUE_001tf__a,type,
monotone_TRUE_a: monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Omformula_OVar_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
monotone_Var_nat_nat: ( nat > nat ) > monoto8276428299528460797at_nat ).
thf(sy_c_Monotone__Formula_Omformula_OVar_001t__Set__Oset_It__Nat__Onat_J,type,
monotone_Var_set_nat: set_nat > monoto7244996872745832772et_nat ).
thf(sy_c_Monotone__Formula_Omformula_OVar_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto3251651810667535926et_nat: set_set_nat > monoto5483634261523599098et_nat ).
thf(sy_c_Monotone__Formula_Omformula_OVar_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
monoto7822445266502226924et_nat: set_set_set_nat > monoto8535755219626829232et_nat ).
thf(sy_c_Monotone__Formula_Omformula_OVar_001tf__a,type,
monotone_Var_a: a > monotone_mformula_a ).
thf(sy_c_Monotone__Formula_Otf__mformula_001tf__a,type,
monoto4877036962378694605mula_a: set_Mo2626137824023173004mula_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bot_nat_nat_o: ( nat > nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
bot_bo6227097192321305471_nat_o: set_set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7376149671870096959at_nat: set_set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
bot_bo193956671110832956et_nat: set_set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Nat__Onat,type,
produc2803780273060847707at_nat: set_set_set_nat > nat > produc4045820344675478307at_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_9186907679027735170et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > set_set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
insert_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Monotone____Formula__Omformula_Itf__a_J,type,
accp_M6162913489380515981mula_a: ( monotone_mformula_a > monotone_mformula_a > $o ) > monotone_mformula_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
accp_set_set_set_nat: ( set_set_set_nat > set_set_set_nat > $o ) > set_set_set_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member435044527007775910at_nat: monoto8276428299528460797at_nat > set_Mo6069479339911551325at_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J,type,
member7623223977307079021et_nat: monoto7244996872745832772et_nat > set_Mo5013373542560054436et_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member4844836972813196067et_nat: monoto5483634261523599098et_nat > set_Mo2574807150581459802et_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
member4689220760989666777et_nat: monoto8535755219626829232et_nat > set_Mo5210732246825857808et_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_Itf__a_J,type,
member535913909593306477mula_a: monotone_mformula_a > set_Mo2626137824023173004mula_a > $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__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_G,type,
g: set_set_nat ).
thf(sy_v__092_060V_062,type,
v: set_a ).
thf(sy_v__092_060phi_062,type,
phi: monotone_mformula_a ).
thf(sy_v__092_060pi_062,type,
pi: a > set_nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1274)
thf(fact_0__092_060theta_062_092_060_094sub_062g__def,axiom,
! [G: set_set_nat,X: a] :
( ( clique3148831351753978868ta_g_a @ v @ pi @ G @ X )
= ( ( member_a @ X @ v )
& ( member_set_nat @ ( pi @ X ) @ G ) ) ) ).
% \<theta>\<^sub>g_def
thf(fact_1_phi_I1_J,axiom,
member535913909593306477mula_a @ phi @ monoto4877036962378694605mula_a ).
% phi(1)
thf(fact_2_phi_I2_J,axiom,
member535913909593306477mula_a @ phi @ ( clique5987991184601036204th_A_a @ v ) ).
% phi(2)
thf(fact_3_eval__g__def,axiom,
! [Theta: a > $o,G: set_set_nat] :
( ( clique5859573001277246426al_g_a @ v @ pi @ Theta @ G )
= ( ! [X2: a] :
( ( member_a @ X2 @ v )
=> ( ( member_set_nat @ ( pi @ X2 ) @ G )
=> ( Theta @ X2 ) ) ) ) ) ).
% eval_g_def
thf(fact_4_forth__assumptions_OACC__mf_Ocong,axiom,
clique4708818501384062891C_mf_a = clique4708818501384062891C_mf_a ).
% forth_assumptions.ACC_mf.cong
thf(fact_5_forth__assumptions_O_092_060theta_062_092_060_094sub_062g_Ocong,axiom,
clique3148831351753978868ta_g_a = clique3148831351753978868ta_g_a ).
% forth_assumptions.\<theta>\<^sub>g.cong
thf(fact_6_inj__on___092_060pi_062,axiom,
inj_on_a_set_nat @ pi @ v ).
% inj_on_\<pi>
thf(fact_7_eval__gs__def,axiom,
! [Theta: a > $o,X3: set_set_set_nat] :
( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ X3 )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X3 )
& ( clique5859573001277246426al_g_a @ v @ pi @ Theta @ X2 ) ) ) ) ).
% eval_gs_def
thf(fact_8_ACC__mf__def,axiom,
! [Phi: monotone_mformula_a] :
( ( clique4708818501384062891C_mf_a @ k @ pi @ Phi )
= ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) ) ).
% ACC_mf_def
thf(fact_9_eval__gs__union,axiom,
! [Theta: a > $o,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) )
= ( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ X3 )
| ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ Y ) ) ) ).
% eval_gs_union
thf(fact_10_G,axiom,
member_set_set_nat @ g @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% G
thf(fact_11_eval__set,axiom,
! [Phi: monotone_mformula_a,Theta: a > $o] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
=> ( ( monotone_eval_a @ Theta @ Phi )
= ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) ) ) ) ).
% eval_set
thf(fact_12_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_13__092_060pi_062m2,axiom,
! [X: a] :
( ( member_a @ X @ v )
=> ( member_set_nat @ ( pi @ X ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ).
% \<pi>m2
thf(fact_14_forth__assumptions_O_092_060theta_062_092_060_094sub_062g__def,axiom,
! [L: nat,P: nat,K: nat,V: set_set_set_set_nat,Pi: set_set_set_nat > set_nat,G: set_set_nat,X: set_set_set_nat] :
( ( clique3407333501437444587et_nat @ L @ P @ K @ V @ Pi )
=> ( ( clique1705999508207691644et_nat @ V @ Pi @ G @ X )
= ( ( member2946998982187404937et_nat @ X @ V )
& ( member_set_nat @ ( Pi @ X ) @ G ) ) ) ) ).
% forth_assumptions.\<theta>\<^sub>g_def
thf(fact_15_forth__assumptions_O_092_060theta_062_092_060_094sub_062g__def,axiom,
! [L: nat,P: nat,K: nat,V: set_set_set_nat,Pi: set_set_nat > set_nat,G: set_set_nat,X: set_set_nat] :
( ( clique2455256169097332789et_nat @ L @ P @ K @ V @ Pi )
=> ( ( clique3281771716360483270et_nat @ V @ Pi @ G @ X )
= ( ( member_set_set_nat @ X @ V )
& ( member_set_nat @ ( Pi @ X ) @ G ) ) ) ) ).
% forth_assumptions.\<theta>\<^sub>g_def
thf(fact_16_forth__assumptions_O_092_060theta_062_092_060_094sub_062g__def,axiom,
! [L: nat,P: nat,K: nat,V: set_set_nat,Pi: set_nat > set_nat,G: set_set_nat,X: set_nat] :
( ( clique522982669833463679et_nat @ L @ P @ K @ V @ Pi )
=> ( ( clique8265756686838912272et_nat @ V @ Pi @ G @ X )
= ( ( member_set_nat @ X @ V )
& ( member_set_nat @ ( Pi @ X ) @ G ) ) ) ) ).
% forth_assumptions.\<theta>\<^sub>g_def
thf(fact_17_forth__assumptions_O_092_060theta_062_092_060_094sub_062g__def,axiom,
! [L: nat,P: nat,K: nat,V: set_nat_nat,Pi: ( nat > nat ) > set_nat,G: set_set_nat,X: nat > nat] :
( ( clique5528702923696243640at_nat @ L @ P @ K @ V @ Pi )
=> ( ( clique2614152022935308233at_nat @ V @ Pi @ G @ X )
= ( ( member_nat_nat @ X @ V )
& ( member_set_nat @ ( Pi @ X ) @ G ) ) ) ) ).
% forth_assumptions.\<theta>\<^sub>g_def
thf(fact_18_forth__assumptions_O_092_060theta_062_092_060_094sub_062g__def,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,G: set_set_nat,X: a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique3148831351753978868ta_g_a @ V @ Pi @ G @ X )
= ( ( member_a @ X @ V )
& ( member_set_nat @ ( Pi @ X ) @ G ) ) ) ) ).
% forth_assumptions.\<theta>\<^sub>g_def
thf(fact_19_forth__assumptions__axioms,axiom,
clique8563529963003110213ions_a @ l @ p @ k @ v @ pi ).
% forth_assumptions_axioms
thf(fact_20_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_21_ACC__cf__mf__def,axiom,
! [Phi: monotone_mformula_a] :
( ( clique8961599393750669800f_mf_a @ k @ pi @ Phi )
= ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) ) ).
% ACC_cf_mf_def
thf(fact_22_ACC__union,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X3 ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ).
% ACC_union
thf(fact_23_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% empty_\<G>
thf(fact_24_finite__ACC,axiom,
! [X3: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X3 ) ) ).
% finite_ACC
thf(fact_25_forth__assumptions_O_092_060pi_062m2,axiom,
! [L: nat,P: nat,K: nat,V: set_set_set_set_nat,Pi: set_set_set_nat > set_nat,X: set_set_set_nat] :
( ( clique3407333501437444587et_nat @ L @ P @ K @ V @ Pi )
=> ( ( member2946998982187404937et_nat @ X @ V )
=> ( member_set_nat @ ( Pi @ X ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ).
% forth_assumptions.\<pi>m2
thf(fact_26_forth__assumptions_O_092_060pi_062m2,axiom,
! [L: nat,P: nat,K: nat,V: set_set_set_nat,Pi: set_set_nat > set_nat,X: set_set_nat] :
( ( clique2455256169097332789et_nat @ L @ P @ K @ V @ Pi )
=> ( ( member_set_set_nat @ X @ V )
=> ( member_set_nat @ ( Pi @ X ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ).
% forth_assumptions.\<pi>m2
thf(fact_27_forth__assumptions_O_092_060pi_062m2,axiom,
! [L: nat,P: nat,K: nat,V: set_set_nat,Pi: set_nat > set_nat,X: set_nat] :
( ( clique522982669833463679et_nat @ L @ P @ K @ V @ Pi )
=> ( ( member_set_nat @ X @ V )
=> ( member_set_nat @ ( Pi @ X ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ).
% forth_assumptions.\<pi>m2
thf(fact_28_forth__assumptions_O_092_060pi_062m2,axiom,
! [L: nat,P: nat,K: nat,V: set_nat_nat,Pi: ( nat > nat ) > set_nat,X: nat > nat] :
( ( clique5528702923696243640at_nat @ L @ P @ K @ V @ Pi )
=> ( ( member_nat_nat @ X @ V )
=> ( member_set_nat @ ( Pi @ X ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ).
% forth_assumptions.\<pi>m2
thf(fact_29_forth__assumptions_O_092_060pi_062m2,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,X: a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( member_a @ X @ V )
=> ( member_set_nat @ ( Pi @ X ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ).
% forth_assumptions.\<pi>m2
thf(fact_30_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_31_forth__assumptions_OSET_Ocong,axiom,
clique6509092761774629891_SET_a = clique6509092761774629891_SET_a ).
% forth_assumptions.SET.cong
thf(fact_32_forth__assumptions_Oeval__set,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Theta: a > $o] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ V ) )
=> ( ( monotone_eval_a @ Theta @ Phi )
= ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ ( clique6509092761774629891_SET_a @ Pi @ Phi ) ) ) ) ) ) ).
% forth_assumptions.eval_set
thf(fact_33_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_34_forth__assumptions_O_092_060A_062_Ocong,axiom,
clique5987991184601036204th_A_a = clique5987991184601036204th_A_a ).
% forth_assumptions.\<A>.cong
thf(fact_35_forth__assumptions_Oeval__g__def,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o,G: set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique5859573001277246426al_g_a @ V @ Pi @ Theta @ G )
= ( ! [X2: a] :
( ( member_a @ X2 @ V )
=> ( ( member_set_nat @ ( Pi @ X2 ) @ G )
=> ( Theta @ X2 ) ) ) ) ) ) ).
% forth_assumptions.eval_g_def
thf(fact_36_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_37_forth__assumptions_Oeval__gs__def,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o,X3: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ X3 )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X3 )
& ( clique5859573001277246426al_g_a @ V @ Pi @ Theta @ X2 ) ) ) ) ) ).
% forth_assumptions.eval_gs_def
thf(fact_38_forth__assumptions_Oinj__on___092_060pi_062,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( inj_on_a_set_nat @ Pi @ V ) ) ).
% forth_assumptions.inj_on_\<pi>
thf(fact_39_forth__assumptions_OACC__cf__mf__def,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique8961599393750669800f_mf_a @ K @ Pi @ Phi )
= ( clique951075384711337423ACC_cf @ K @ ( clique6509092761774629891_SET_a @ Pi @ Phi ) ) ) ) ).
% forth_assumptions.ACC_cf_mf_def
thf(fact_40_forth__assumptions_Oeval__gs__union,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) )
= ( ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ X3 )
| ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ Y ) ) ) ) ).
% forth_assumptions.eval_gs_union
thf(fact_41_sameprod__finite,axiom,
! [X3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ X3 )
=> ( finite3586981331298542604at_nat @ ( clique134924887794942129at_nat @ X3 @ X3 ) ) ) ).
% sameprod_finite
thf(fact_42_sameprod__finite,axiom,
! [X3: set_set_nat] :
( ( finite1152437895449049373et_nat @ X3 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X3 @ X3 ) ) ) ).
% sameprod_finite
thf(fact_43_sameprod__finite,axiom,
! [X3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X3 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X3 @ X3 ) ) ) ).
% sameprod_finite
thf(fact_44_sameprod__finite,axiom,
! [X3: set_nat] :
( ( finite_finite_nat @ X3 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X3 @ X3 ) ) ) ).
% sameprod_finite
thf(fact_45_forth__assumptions_OACC__mf__def,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique4708818501384062891C_mf_a @ K @ Pi @ Phi )
= ( clique3210737319928189260st_ACC @ K @ ( clique6509092761774629891_SET_a @ Pi @ Phi ) ) ) ) ).
% forth_assumptions.ACC_mf_def
thf(fact_46_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_47_forth__assumptions_Oeval__gs_Ocong,axiom,
clique835570645587132141l_gs_a = clique835570645587132141l_gs_a ).
% forth_assumptions.eval_gs.cong
thf(fact_48_forth__assumptions_OACC__cf__mf_Ocong,axiom,
clique8961599393750669800f_mf_a = clique8961599393750669800f_mf_a ).
% forth_assumptions.ACC_cf_mf.cong
thf(fact_49_forth__assumptions_Oeval__g_Ocong,axiom,
clique5859573001277246426al_g_a = clique5859573001277246426al_g_a ).
% forth_assumptions.eval_g.cong
thf(fact_50_deviate__finite_I6_J,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k @ A @ B ) ) ).
% deviate_finite(6)
thf(fact_51_deviate__finite_I4_J,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique1591571987439376245eg_cup @ l @ p @ k @ A @ B ) ) ).
% deviate_finite(4)
thf(fact_52_mem__Collect__eq,axiom,
! [A2: set_set_set_nat,P2: set_set_set_nat > $o] :
( ( member2946998982187404937et_nat @ A2 @ ( collec7201453139178570183et_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A2: set_set_nat,P2: set_set_nat > $o] :
( ( member_set_set_nat @ A2 @ ( collect_set_set_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A2: nat > nat,P2: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A: set_set_set_set_nat] :
( ( collec7201453139178570183et_nat
@ ^ [X2: set_set_set_nat] : ( member2946998982187404937et_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X2: nat > nat] : ( member_nat_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_62_finite__approx__neg,axiom,
! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ Phi ) ) ).
% finite_approx_neg
thf(fact_63_deviate__finite_I2_J,axiom,
! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) ).
% deviate_finite(2)
thf(fact_64_SET___092_060G_062,axiom,
! [Phi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
=> ( ord_le9131159989063066194et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% SET_\<G>
thf(fact_65_third__assumptions__axioms,axiom,
assump2119784843035796504ptions @ l @ p @ k ).
% third_assumptions_axioms
thf(fact_66_inj__on__empty,axiom,
! [F: a > set_nat] : ( inj_on_a_set_nat @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_67_finite__Un,axiom,
! [F2: set_set_set_nat,G: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F2 @ G ) )
= ( ( finite6739761609112101331et_nat @ F2 )
& ( finite6739761609112101331et_nat @ G ) ) ) ).
% finite_Un
thf(fact_68_finite__Un,axiom,
! [F2: set_nat_nat,G: set_nat_nat] :
( ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F2 @ G ) )
= ( ( finite2115694454571419734at_nat @ F2 )
& ( finite2115694454571419734at_nat @ G ) ) ) ).
% finite_Un
thf(fact_69_finite__Un,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F2 @ G ) )
= ( ( finite1152437895449049373et_nat @ F2 )
& ( finite1152437895449049373et_nat @ G ) ) ) ).
% finite_Un
thf(fact_70_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_71_Un__empty,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_72_Un__empty,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
= ( ( A = bot_bo7198184520161983622et_nat )
& ( B = bot_bo7198184520161983622et_nat ) ) ) ).
% Un_empty
thf(fact_73_Un__empty,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ( A = bot_bot_set_nat_nat )
& ( B = bot_bot_set_nat_nat ) ) ) ).
% Un_empty
thf(fact_74_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_75_sup__bot__left,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_76_sup__bot__left,axiom,
! [X: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_77_sup__bot__left,axiom,
! [X: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_78_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_79_sup__bot__right,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ bot_bot_set_set_nat )
= X ) ).
% sup_bot_right
thf(fact_80_sup__bot__right,axiom,
! [X: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ bot_bo7198184520161983622et_nat )
= X ) ).
% sup_bot_right
thf(fact_81_sup__bot__right,axiom,
! [X: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ bot_bot_set_nat_nat )
= X ) ).
% sup_bot_right
thf(fact_82_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_83_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_84_ACC__cf__union,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X3 ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_union
thf(fact_85_subset__antisym,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_86_subset__antisym,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_87_subset__antisym,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_88_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_89_subsetI,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ! [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
=> ( member2946998982187404937et_nat @ X4 @ B ) )
=> ( ord_le572741076514265352et_nat @ A @ B ) ) ).
% subsetI
thf(fact_90_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_91_subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( member_set_set_nat @ X4 @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% subsetI
thf(fact_92_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ X4 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_93_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( member_nat_nat @ X4 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_94_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_95_empty__Collect__eq,axiom,
! [P2: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P2 ) )
= ( ! [X2: set_nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_96_empty__Collect__eq,axiom,
! [P2: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P2 ) )
= ( ! [X2: set_set_nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_97_empty__Collect__eq,axiom,
! [P2: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P2 ) )
= ( ! [X2: nat > nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_98_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_99_Collect__empty__eq,axiom,
! [P2: set_nat > $o] :
( ( ( collect_set_nat @ P2 )
= bot_bot_set_set_nat )
= ( ! [X2: set_nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_100_Collect__empty__eq,axiom,
! [P2: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P2 )
= bot_bo7198184520161983622et_nat )
= ( ! [X2: set_set_nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_101_Collect__empty__eq,axiom,
! [P2: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P2 )
= bot_bot_set_nat_nat )
= ( ! [X2: nat > nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_102_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_103_all__not__in__conv,axiom,
! [A: set_set_set_set_nat] :
( ( ! [X2: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ X2 @ A ) )
= ( A = bot_bo193956671110832956et_nat ) ) ).
% all_not_in_conv
thf(fact_104_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_105_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X2: set_nat] :
~ ( member_set_nat @ X2 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_106_all__not__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ! [X2: set_set_nat] :
~ ( member_set_set_nat @ X2 @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_107_all__not__in__conv,axiom,
! [A: set_nat_nat] :
( ( ! [X2: nat > nat] :
~ ( member_nat_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_108_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_109_empty__iff,axiom,
! [C: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ C @ bot_bo193956671110832956et_nat ) ).
% empty_iff
thf(fact_110_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_111_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_112_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_113_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_114_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_115_sup_Oright__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ B2 )
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_116_sup_Oright__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_117_sup_Oright__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_118_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_119_sup__left__idem,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_120_sup__left__idem,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y2 ) )
= ( sup_sup_set_nat_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_121_sup__left__idem,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y2 ) )
= ( sup_sup_set_set_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_122_sup__left__idem,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_123_sup_Oleft__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_124_sup_Oleft__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_125_sup_Oleft__idem,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_126_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_127_sup__idem,axiom,
! [X: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_128_sup__idem,axiom,
! [X: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_129_sup__idem,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_130_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_131_sup_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_132_sup_Oidem,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_133_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_134_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_135_Un__iff,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( sup_su3906748206781935060et_nat @ A @ B ) )
= ( ( member2946998982187404937et_nat @ C @ A )
| ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_136_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_137_Un__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
| ( member_set_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_138_Un__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
| ( member_nat_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_139_Un__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
| ( member_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_140_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_141_UnCI,axiom,
! [C: set_set_set_nat,B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( ~ ( member2946998982187404937et_nat @ C @ B )
=> ( member2946998982187404937et_nat @ C @ A ) )
=> ( member2946998982187404937et_nat @ C @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_142_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_143_UnCI,axiom,
! [C: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( ~ ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ A ) )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_144_UnCI,axiom,
! [C: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( ~ ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ A ) )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_145_UnCI,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ A ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_146_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_147_finite__members___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ).
% finite_members_\<G>
thf(fact_148_finite__numbers,axiom,
! [N: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ).
% finite_numbers
thf(fact_149_empty__subsetI,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% empty_subsetI
thf(fact_150_empty__subsetI,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% empty_subsetI
thf(fact_151_empty__subsetI,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% empty_subsetI
thf(fact_152_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_153_subset__empty,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_154_subset__empty,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_155_subset__empty,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_156_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_157_sup_Obounded__iff,axiom,
! [B2: set_set_set_nat,C: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C ) @ A2 )
= ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
& ( ord_le9131159989063066194et_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_158_sup_Obounded__iff,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
& ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_159_sup_Obounded__iff,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 )
= ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
& ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_160_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_161_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_162_le__sup__iff,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) @ Z )
= ( ( ord_le9131159989063066194et_nat @ X @ Z )
& ( ord_le9131159989063066194et_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_163_le__sup__iff,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X @ Y2 ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X @ Z )
& ( ord_le6893508408891458716et_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_164_le__sup__iff,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X @ Y2 ) @ Z )
= ( ( ord_le9059583361652607317at_nat @ X @ Z )
& ( ord_le9059583361652607317at_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_165_le__sup__iff,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y2 ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_166_le__sup__iff,axiom,
! [X: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z )
= ( ( ord_less_eq_set_nat @ X @ Z )
& ( ord_less_eq_set_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_167_sup__bot_Oright__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_168_sup__bot_Oright__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_169_sup__bot_Oright__neutral,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ bot_bot_set_nat_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_170_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_171_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_172_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_173_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B2 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_174_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_175_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_176_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_177_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_178_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_179_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_180_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_181_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= bot_bot_set_nat_nat )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B2 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_182_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_183_sup__eq__bot__iff,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X @ Y2 )
= bot_bot_set_set_nat )
= ( ( X = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_184_sup__eq__bot__iff,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ X @ Y2 )
= bot_bo7198184520161983622et_nat )
= ( ( X = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_185_sup__eq__bot__iff,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ X @ Y2 )
= bot_bot_set_nat_nat )
= ( ( X = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_186_sup__eq__bot__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y2 )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_187_bot__eq__sup__iff,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X @ Y2 ) )
= ( ( X = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_188_bot__eq__sup__iff,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( ( X = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_189_bot__eq__sup__iff,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ X @ Y2 ) )
= ( ( X = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_190_bot__eq__sup__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y2 ) )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_191_Un__subset__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C2 )
= ( ( ord_le9131159989063066194et_nat @ A @ C2 )
& ( ord_le9131159989063066194et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_192_Un__subset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( ( ord_le6893508408891458716et_nat @ A @ C2 )
& ( ord_le6893508408891458716et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_193_Un__subset__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 )
= ( ( ord_le9059583361652607317at_nat @ A @ C2 )
& ( ord_le9059583361652607317at_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_194_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_195_finite__numbers2,axiom,
! [N: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ).
% finite_numbers2
thf(fact_196_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_197_union___092_060G_062,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% union_\<G>
thf(fact_198_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% finite_\<G>
thf(fact_199_deviate__finite_I3_J,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( finite6739761609112101331et_nat @ ( clique3314026705536850673os_cup @ l @ p @ k @ A @ B ) ) ).
% deviate_finite(3)
thf(fact_200_NEG___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% NEG_\<G>
thf(fact_201_forth__assumptions_Odeviate__neg_Ocong,axiom,
clique2019076642914533763_neg_a = clique2019076642914533763_neg_a ).
% forth_assumptions.deviate_neg.cong
thf(fact_202_Collect__mono__iff,axiom,
! [P2: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P2 ) @ ( collect_set_set_nat @ Q ) )
= ( ! [X2: set_set_nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_203_Collect__mono__iff,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) )
= ( ! [X2: set_nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_204_Collect__mono__iff,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) )
= ( ! [X2: nat > nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_205_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_206_set__eq__subset,axiom,
( ( ^ [Y3: set_set_set_nat,Z2: set_set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_207_set__eq__subset,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_208_set__eq__subset,axiom,
( ( ^ [Y3: set_nat_nat,Z2: set_nat_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_209_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_210_subset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ord_le9131159989063066194et_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_211_subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_212_subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_213_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_214_Collect__mono,axiom,
! [P2: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X4: set_set_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P2 ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_215_Collect__mono,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_216_Collect__mono,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_217_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_218_subset__refl,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ A ) ).
% subset_refl
thf(fact_219_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_220_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_221_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_222_subset__iff,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A3: set_set_set_set_nat,B3: set_set_set_set_nat] :
! [T: set_set_set_nat] :
( ( member2946998982187404937et_nat @ T @ A3 )
=> ( member2946998982187404937et_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_223_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_224_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [T: set_set_nat] :
( ( member_set_set_nat @ T @ A3 )
=> ( member_set_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_225_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_226_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [T: nat > nat] :
( ( member_nat_nat @ T @ A3 )
=> ( member_nat_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_227_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_228_equalityD2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ( ord_le9131159989063066194et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_229_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_230_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_231_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_232_equalityD1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_233_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_234_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_235_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_236_subset__eq,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A3: set_set_set_set_nat,B3: set_set_set_set_nat] :
! [X2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A3 )
=> ( member2946998982187404937et_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_237_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_238_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
=> ( member_set_set_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_239_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( member_set_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_240_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
=> ( member_nat_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_241_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_242_equalityE,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ~ ( ord_le9131159989063066194et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_243_equalityE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ~ ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_244_equalityE,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ~ ( ord_le9059583361652607317at_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_245_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_246_subsetD,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,C: set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( member2946998982187404937et_nat @ C @ A )
=> ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_247_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_248_subsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_249_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_250_subsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_251_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_252_in__mono,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,X: set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( member2946998982187404937et_nat @ X @ A )
=> ( member2946998982187404937et_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_253_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_254_in__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ X @ A )
=> ( member_set_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_255_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_256_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ X @ A )
=> ( member_nat_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_257_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_258_second__assumptions_Odeviate__neg__cup_Ocong,axiom,
clique1591571987439376245eg_cup = clique1591571987439376245eg_cup ).
% second_assumptions.deviate_neg_cup.cong
thf(fact_259_second__assumptions_Odeviate__neg__cap_Ocong,axiom,
clique1591571987438064265eg_cap = clique1591571987438064265eg_cap ).
% second_assumptions.deviate_neg_cap.cong
thf(fact_260_forth__assumptions_Oapprox__neg_Ocong,axiom,
clique6623365555141101007_neg_a = clique6623365555141101007_neg_a ).
% forth_assumptions.approx_neg.cong
thf(fact_261_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_262_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_263_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_264_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_265_finite__has__minimal2,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( member_nat_nat @ A2 @ A )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
& ( ord_less_eq_nat_nat @ X4 @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_266_finite__has__minimal2,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( member2946998982187404937et_nat @ A2 @ A )
=> ? [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
& ( ord_le9131159989063066194et_nat @ X4 @ A2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_267_finite__has__minimal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
& ( ord_le6893508408891458716et_nat @ X4 @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_268_finite__has__minimal2,axiom,
! [A: set_set_nat_nat,A2: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( member_set_nat_nat @ A2 @ A )
=> ? [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ A )
& ( ord_le9059583361652607317at_nat @ X4 @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_269_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ X4 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_270_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ X4 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_271_finite__has__maximal2,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( member_nat_nat @ A2 @ A )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
& ( ord_less_eq_nat_nat @ A2 @ X4 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_272_finite__has__maximal2,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( member2946998982187404937et_nat @ A2 @ A )
=> ? [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
& ( ord_le9131159989063066194et_nat @ A2 @ X4 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_273_finite__has__maximal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
& ( ord_le6893508408891458716et_nat @ A2 @ X4 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_274_finite__has__maximal2,axiom,
! [A: set_set_nat_nat,A2: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( member_set_nat_nat @ A2 @ A )
=> ? [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ A )
& ( ord_le9059583361652607317at_nat @ A2 @ X4 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_275_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_eq_nat @ A2 @ X4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_276_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ( ord_less_eq_set_nat @ A2 @ X4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_277_rev__finite__subset,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( finite6739761609112101331et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_278_rev__finite__subset,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_279_rev__finite__subset,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( finite2115694454571419734at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_280_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_281_infinite__super,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S @ T2 )
=> ( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_282_infinite__super,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_283_infinite__super,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S @ T2 )
=> ( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_284_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_285_finite__subset,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( finite6739761609112101331et_nat @ B )
=> ( finite6739761609112101331et_nat @ A ) ) ) ).
% finite_subset
thf(fact_286_finite__subset,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% finite_subset
thf(fact_287_finite__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( finite2115694454571419734at_nat @ A ) ) ) ).
% finite_subset
thf(fact_288_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_289_sup_OcoboundedI2,axiom,
! [C: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ B2 )
=> ( ord_le9131159989063066194et_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_290_sup_OcoboundedI2,axiom,
! [C: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ B2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_291_sup_OcoboundedI2,axiom,
! [C: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ B2 )
=> ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_292_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_293_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_294_sup_OcoboundedI1,axiom,
! [C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ A2 )
=> ( ord_le9131159989063066194et_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_295_sup_OcoboundedI1,axiom,
! [C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_296_sup_OcoboundedI1,axiom,
! [C: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_297_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_298_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_299_sup_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_300_sup_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_301_sup_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_302_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_303_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_304_sup_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_305_sup_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_306_sup_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_307_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_308_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_309_sup_Ocobounded2,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_310_sup_Ocobounded2,axiom,
! [B2: set_set_nat,A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_311_sup_Ocobounded2,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_312_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_313_sup_Ocobounded2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_314_sup_Ocobounded1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_315_sup_Ocobounded1,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_316_sup_Ocobounded1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_317_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_318_sup_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_319_sup_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( A4
= ( sup_su4213647025997063966et_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_320_sup_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( A4
= ( sup_sup_set_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_321_sup_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( A4
= ( sup_sup_set_nat_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_322_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_323_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_324_sup_OboundedI,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_325_sup_OboundedI,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_326_sup_OboundedI,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_327_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_328_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_329_sup_OboundedE,axiom,
! [B2: set_set_set_nat,C: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ~ ( ord_le9131159989063066194et_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_330_sup_OboundedE,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_331_sup_OboundedE,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_332_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_333_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_334_sup__absorb2,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y2 )
=> ( ( sup_su4213647025997063966et_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_335_sup__absorb2,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y2 )
=> ( ( sup_sup_set_set_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_336_sup__absorb2,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( sup_sup_set_nat_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_337_sup__absorb2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( sup_sup_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_338_sup__absorb2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_339_sup__absorb1,axiom,
! [Y2: set_set_set_nat,X: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X )
=> ( ( sup_su4213647025997063966et_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_340_sup__absorb1,axiom,
! [Y2: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X )
=> ( ( sup_sup_set_set_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_341_sup__absorb1,axiom,
! [Y2: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X )
=> ( ( sup_sup_set_nat_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_342_sup__absorb1,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( sup_sup_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_343_sup__absorb1,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_344_sup_Oabsorb2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_345_sup_Oabsorb2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_346_sup_Oabsorb2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_347_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_348_sup_Oabsorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_349_sup_Oabsorb1,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_350_sup_Oabsorb1,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_351_sup_Oabsorb1,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_352_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_353_sup_Oabsorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_354_sup__unique,axiom,
! [F: set_set_set_nat > set_set_set_nat > set_set_set_nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ! [X4: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y4 @ X4 )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ X4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_su4213647025997063966et_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_355_sup__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X: set_set_nat,Y2: set_set_nat] :
( ! [X4: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_nat,Y4: set_set_nat,Z3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y4 @ X4 )
=> ( ( ord_le6893508408891458716et_nat @ Z3 @ X4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_set_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_356_sup__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X: set_nat_nat,Y2: set_nat_nat] :
( ! [X4: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat,Z3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y4 @ X4 )
=> ( ( ord_le9059583361652607317at_nat @ Z3 @ X4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_nat_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_357_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y2: nat] :
( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_358_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y2: set_nat] :
( ! [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X4 )
=> ( ord_less_eq_set_nat @ ( F @ Y4 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_359_sup_OorderI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_360_sup_OorderI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_361_sup_OorderI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2
= ( sup_sup_set_nat_nat @ A2 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_362_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_363_sup_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_364_sup_OorderE,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( A2
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_365_sup_OorderE,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_366_sup_OorderE,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_367_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_368_sup_OorderE,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_369_le__iff__sup,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_370_le__iff__sup,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X2: set_set_nat,Y5: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_371_le__iff__sup,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_372_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y5: nat] :
( ( sup_sup_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_373_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_374_sup__least,axiom,
! [Y2: set_set_set_nat,X: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X )
=> ( ( ord_le9131159989063066194et_nat @ Z @ X )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_375_sup__least,axiom,
! [Y2: set_set_nat,X: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X )
=> ( ( ord_le6893508408891458716et_nat @ Z @ X )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_376_sup__least,axiom,
! [Y2: set_nat_nat,X: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X )
=> ( ( ord_le9059583361652607317at_nat @ Z @ X )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_377_sup__least,axiom,
! [Y2: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_378_sup__least,axiom,
! [Y2: set_nat,X: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ Z @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_379_sup__mono,axiom,
! [A2: set_set_set_nat,C: set_set_set_nat,B2: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ ( sup_su4213647025997063966et_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_380_sup__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_381_sup__mono,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ ( sup_sup_set_nat_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_382_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_383_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_384_sup_Omono,axiom,
! [C: set_set_set_nat,A2: set_set_set_nat,D: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ D @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ C @ D ) @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_385_sup_Omono,axiom,
! [C: set_set_nat,A2: set_set_nat,D: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ D @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ C @ D ) @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_386_sup_Omono,axiom,
! [C: set_nat_nat,A2: set_nat_nat,D: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ D @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ C @ D ) @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_387_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_388_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_389_le__supI2,axiom,
! [X: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ B2 )
=> ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_390_le__supI2,axiom,
! [X: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ B2 )
=> ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_391_le__supI2,axiom,
! [X: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ B2 )
=> ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_392_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_393_le__supI2,axiom,
! [X: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_394_le__supI1,axiom,
! [X: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ A2 )
=> ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_395_le__supI1,axiom,
! [X: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ A2 )
=> ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_396_le__supI1,axiom,
! [X: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ A2 )
=> ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_397_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_398_le__supI1,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_399_sup__ge2,axiom,
! [Y2: set_set_set_nat,X: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_400_sup__ge2,axiom,
! [Y2: set_set_nat,X: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y2 @ ( sup_sup_set_set_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_401_sup__ge2,axiom,
! [Y2: set_nat_nat,X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y2 @ ( sup_sup_set_nat_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_402_sup__ge2,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_403_sup__ge2,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_404_sup__ge1,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_405_sup__ge1,axiom,
! [X: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_406_sup__ge1,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_407_sup__ge1,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_408_sup__ge1,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_409_le__supI,axiom,
! [A2: set_set_set_nat,X: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ X )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ X )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_410_le__supI,axiom,
! [A2: set_set_nat,X: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ X )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_411_le__supI,axiom,
! [A2: set_nat_nat,X: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ X )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_412_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_413_le__supI,axiom,
! [A2: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_414_le__supE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ X )
=> ~ ( ord_le9131159989063066194et_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_415_le__supE,axiom,
! [A2: set_set_nat,B2: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ X )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_416_le__supE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ X )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_417_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_418_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X )
=> ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_419_inf__sup__ord_I3_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_420_inf__sup__ord_I3_J,axiom,
! [X: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ ( sup_sup_set_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_421_inf__sup__ord_I3_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_422_inf__sup__ord_I3_J,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_423_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_424_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_set_nat,X: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_425_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_nat,X: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y2 @ ( sup_sup_set_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_426_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat_nat,X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y2 @ ( sup_sup_set_nat_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_427_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_428_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_429_subset__Un__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_430_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_431_subset__Un__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_432_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_433_subset__UnE,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ~ ! [A5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ A )
=> ! [B5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B5 @ B )
=> ( C2
!= ( sup_su4213647025997063966et_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_434_subset__UnE,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ~ ! [A5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ A )
=> ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ B )
=> ( C2
!= ( sup_sup_set_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_435_subset__UnE,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ~ ! [A5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ A )
=> ! [B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ B )
=> ( C2
!= ( sup_sup_set_nat_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_436_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A )
=> ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_437_Un__absorb2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( sup_su4213647025997063966et_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_438_Un__absorb2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( sup_sup_set_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_439_Un__absorb2,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( sup_sup_set_nat_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_440_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_441_Un__absorb1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( sup_su4213647025997063966et_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_442_Un__absorb1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( sup_sup_set_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_443_Un__absorb1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( sup_sup_set_nat_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_444_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_445_Un__upper2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_446_Un__upper2,axiom,
! [B: set_set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_447_Un__upper2,axiom,
! [B: set_nat_nat,A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B @ ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_448_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_449_Un__upper1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_450_Un__upper1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_451_Un__upper1,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_452_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_453_Un__least,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_454_Un__least,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_455_Un__least,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_456_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_457_Un__mono,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_458_Un__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_459_Un__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_460_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_461_subset__inj__on,axiom,
! [F: a > set_nat,B: set_a,A: set_a] :
( ( inj_on_a_set_nat @ F @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( inj_on_a_set_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_462_inj__on__subset,axiom,
! [F: a > set_nat,A: set_a,B: set_a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( inj_on_a_set_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_463_sameprod__mono,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y )
=> ( ord_le572741076514265352et_nat @ ( clique1181040904276305582et_nat @ X3 @ X3 ) @ ( clique1181040904276305582et_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_464_sameprod__mono,axiom,
! [X3: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X3 @ X3 ) @ ( clique8906516429304539640et_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_465_sameprod__mono,axiom,
! [X3: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y )
=> ( ord_le4954213926817602059at_nat @ ( clique134924887794942129at_nat @ X3 @ X3 ) @ ( clique134924887794942129at_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_466_sameprod__mono,axiom,
! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X3 @ X3 ) @ ( clique6722202388162463298od_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_467_finite__has__minimal,axiom,
! [A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( A != bot_bot_set_nat_nat )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_468_finite__has__minimal,axiom,
! [A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( A != bot_bo193956671110832956et_nat )
=> ? [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_469_finite__has__minimal,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_470_finite__has__minimal,axiom,
! [A: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( A != bot_bo7376149671870096959at_nat )
=> ? [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_471_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_472_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_473_finite__has__maximal,axiom,
! [A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( A != bot_bot_set_nat_nat )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_474_finite__has__maximal,axiom,
! [A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( A != bot_bo193956671110832956et_nat )
=> ? [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_475_finite__has__maximal,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_476_finite__has__maximal,axiom,
! [A: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( A != bot_bo7376149671870096959at_nat )
=> ? [X4: set_nat_nat] :
( ( member_set_nat_nat @ X4 @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_477_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_478_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_479_forth__assumptions_Oaxioms_I1_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( assump2119784843035796504ptions @ L @ P @ K ) ) ).
% forth_assumptions.axioms(1)
thf(fact_480_ex__in__conv,axiom,
! [A: set_set_set_set_nat] :
( ( ? [X2: set_set_set_nat] : ( member2946998982187404937et_nat @ X2 @ A ) )
= ( A != bot_bo193956671110832956et_nat ) ) ).
% ex_in_conv
thf(fact_481_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_482_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X2: set_nat] : ( member_set_nat @ X2 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_483_ex__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ? [X2: set_set_nat] : ( member_set_set_nat @ X2 @ A ) )
= ( A != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_484_ex__in__conv,axiom,
! [A: set_nat_nat] :
( ( ? [X2: nat > nat] : ( member_nat_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_485_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_486_equals0I,axiom,
! [A: set_set_set_set_nat] :
( ! [Y4: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ Y4 @ A )
=> ( A = bot_bo193956671110832956et_nat ) ) ).
% equals0I
thf(fact_487_equals0I,axiom,
! [A: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_488_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y4: set_nat] :
~ ( member_set_nat @ Y4 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_489_equals0I,axiom,
! [A: set_set_set_nat] :
( ! [Y4: set_set_nat] :
~ ( member_set_set_nat @ Y4 @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_490_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y4: nat > nat] :
~ ( member_nat_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_491_equals0I,axiom,
! [A: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_492_equals0D,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( A = bot_bo193956671110832956et_nat )
=> ~ ( member2946998982187404937et_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_493_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_494_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_495_equals0D,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( A = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_496_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_497_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_498_emptyE,axiom,
! [A2: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ A2 @ bot_bo193956671110832956et_nat ) ).
% emptyE
thf(fact_499_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_500_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_501_emptyE,axiom,
! [A2: set_set_nat] :
~ ( member_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_502_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_503_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_504_sup__left__commute,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_505_sup__left__commute,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_506_sup__left__commute,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_507_sup__left__commute,axiom,
! [X: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_508_sup_Oleft__commute,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ B2 @ ( sup_su4213647025997063966et_nat @ A2 @ C ) )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_509_sup_Oleft__commute,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ B2 @ ( sup_sup_set_nat_nat @ A2 @ C ) )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_510_sup_Oleft__commute,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ C ) )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_511_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_512_sup__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_513_sup__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] : ( sup_sup_set_nat_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_514_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X2: set_set_nat,Y5: set_set_nat] : ( sup_sup_set_set_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_515_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] : ( sup_sup_set_nat @ Y5 @ X2 ) ) ) ).
% sup_commute
thf(fact_516_sup_Ocommute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_517_sup_Ocommute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( sup_sup_set_nat_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_518_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] : ( sup_sup_set_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_519_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_520_sup__assoc,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) @ Z )
= ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_521_sup__assoc,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y2 ) @ Z )
= ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_522_sup__assoc,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_523_sup__assoc,axiom,
! [X: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_524_sup_Oassoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ C )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_525_sup_Oassoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_526_sup_Oassoc,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_527_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_528_inf__sup__aci_I5_J,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_529_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] : ( sup_sup_set_nat_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_530_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X2: set_set_nat,Y5: set_set_nat] : ( sup_sup_set_set_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_531_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y5: set_nat] : ( sup_sup_set_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_532_inf__sup__aci_I6_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) @ Z )
= ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_533_inf__sup__aci_I6_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X @ Y2 ) @ Z )
= ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_534_inf__sup__aci_I6_J,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_535_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_536_inf__sup__aci_I7_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_537_inf__sup__aci_I7_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_538_inf__sup__aci_I7_J,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_539_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_540_inf__sup__aci_I8_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_541_inf__sup__aci_I8_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y2 ) )
= ( sup_sup_set_nat_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_542_inf__sup__aci_I8_J,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y2 ) )
= ( sup_sup_set_set_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_543_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_544_Un__left__commute,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) )
= ( sup_su4213647025997063966et_nat @ B @ ( sup_su4213647025997063966et_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_545_Un__left__commute,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) )
= ( sup_sup_set_nat_nat @ B @ ( sup_sup_set_nat_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_546_Un__left__commute,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_547_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_548_Un__left__absorb,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_549_Un__left__absorb,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_550_Un__left__absorb,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_551_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_552_Un__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_553_Un__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] : ( sup_sup_set_nat_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_554_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] : ( sup_sup_set_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_555_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_556_Un__absorb,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_557_Un__absorb,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_558_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_559_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_560_Un__assoc,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C2 )
= ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_561_Un__assoc,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_562_Un__assoc,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_563_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_564_ball__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P2: set_set_nat > $o] :
( ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( P2 @ X2 ) )
& ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_565_ball__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P2: ( nat > nat ) > $o] :
( ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( P2 @ X2 ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_566_ball__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P2: set_nat > $o] :
( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( P2 @ X2 ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_567_ball__Un,axiom,
! [A: set_nat,B: set_nat,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( P2 @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_568_bex__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P2: set_set_nat > $o] :
( ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
& ( P2 @ X2 ) )
| ? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_569_bex__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P2: ( nat > nat ) > $o] :
( ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ A @ B ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A )
& ( P2 @ X2 ) )
| ? [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_570_bex__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P2: set_nat > $o] :
( ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( sup_sup_set_set_nat @ A @ B ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( P2 @ X2 ) )
| ? [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_571_bex__Un,axiom,
! [A: set_nat,B: set_nat,P2: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A @ B ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P2 @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_572_UnI2,axiom,
! [C: set_set_set_nat,B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ B )
=> ( member2946998982187404937et_nat @ C @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_573_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_574_UnI2,axiom,
! [C: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_575_UnI2,axiom,
! [C: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_576_UnI2,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_577_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_578_UnI1,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ A )
=> ( member2946998982187404937et_nat @ C @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_579_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_580_UnI1,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_581_UnI1,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_582_UnI1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_583_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_584_UnE,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( sup_su3906748206781935060et_nat @ A @ B ) )
=> ( ~ ( member2946998982187404937et_nat @ C @ A )
=> ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% UnE
thf(fact_585_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_586_UnE,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( ~ ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_587_UnE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( ~ ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% UnE
thf(fact_588_UnE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_589_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_590_inj__on__inverseI,axiom,
! [A: set_a,G2: set_nat > a,F: a > set_nat] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ( G2 @ ( F @ X4 ) )
= X4 ) )
=> ( inj_on_a_set_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_591_inj__on__contraD,axiom,
! [F: a > set_nat,A: set_a,X: a,Y2: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( X != Y2 )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( F @ X )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_592_inj__on__eq__iff,axiom,
! [F: a > set_nat,A: set_a,X: a,Y2: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
= ( X = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_593_inj__on__cong,axiom,
! [A: set_a,F: a > set_nat,G2: a > set_nat] :
( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( ( F @ A6 )
= ( G2 @ A6 ) ) )
=> ( ( inj_on_a_set_nat @ F @ A )
= ( inj_on_a_set_nat @ G2 @ A ) ) ) ).
% inj_on_cong
thf(fact_594_inj__on__def,axiom,
( inj_on_a_set_nat
= ( ^ [F3: a > set_nat,A3: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ A3 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y5 ) )
=> ( X2 = Y5 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_595_inj__onI,axiom,
! [A: set_a,F: a > set_nat] :
( ! [X4: a,Y4: a] :
( ( member_a @ X4 @ A )
=> ( ( member_a @ Y4 @ A )
=> ( ( ( F @ X4 )
= ( F @ Y4 ) )
=> ( X4 = Y4 ) ) ) )
=> ( inj_on_a_set_nat @ F @ A ) ) ).
% inj_onI
thf(fact_596_inj__onD,axiom,
! [F: a > set_nat,A: set_a,X: a,Y2: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y2 ) )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y2 @ A )
=> ( X = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_597_forth__assumptions_Odeviate__finite_I4_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( finite2115694454571419734at_nat @ ( clique1591571987439376245eg_cup @ L @ P @ K @ A @ B ) ) ) ).
% forth_assumptions.deviate_finite(4)
thf(fact_598_forth__assumptions_Odeviate__finite_I6_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( finite2115694454571419734at_nat @ ( clique1591571987438064265eg_cap @ L @ P @ K @ A @ B ) ) ) ).
% forth_assumptions.deviate_finite(6)
thf(fact_599_forth__assumptions_Odeviate__finite_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( finite2115694454571419734at_nat @ ( clique2019076642914533763_neg_a @ L @ P @ K @ Pi @ Phi ) ) ) ).
% forth_assumptions.deviate_finite(2)
thf(fact_600_forth__assumptions_Ofinite__approx__neg,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( finite2115694454571419734at_nat @ ( clique6623365555141101007_neg_a @ L @ P @ K @ Pi @ Phi ) ) ) ).
% forth_assumptions.finite_approx_neg
thf(fact_601_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_602_infinite__imp__nonempty,axiom,
! [S: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S )
=> ( S != bot_bo7198184520161983622et_nat ) ) ).
% infinite_imp_nonempty
thf(fact_603_infinite__imp__nonempty,axiom,
! [S: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ S )
=> ( S != bot_bot_set_nat_nat ) ) ).
% infinite_imp_nonempty
thf(fact_604_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_605_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_606_finite_OemptyI,axiom,
finite6739761609112101331et_nat @ bot_bo7198184520161983622et_nat ).
% finite.emptyI
thf(fact_607_finite_OemptyI,axiom,
finite2115694454571419734at_nat @ bot_bot_set_nat_nat ).
% finite.emptyI
thf(fact_608_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_609_Un__empty__right,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% Un_empty_right
thf(fact_610_Un__empty__right,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ bot_bo7198184520161983622et_nat )
= A ) ).
% Un_empty_right
thf(fact_611_Un__empty__right,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ bot_bot_set_nat_nat )
= A ) ).
% Un_empty_right
thf(fact_612_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_613_Un__empty__left,axiom,
! [B: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_614_Un__empty__left,axiom,
! [B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_615_Un__empty__left,axiom,
! [B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_616_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_617_infinite__Un,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( ~ ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) ) )
= ( ~ ( finite6739761609112101331et_nat @ S )
| ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_618_infinite__Un,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( ~ ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) ) )
= ( ~ ( finite2115694454571419734at_nat @ S )
| ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_619_infinite__Un,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) )
= ( ~ ( finite1152437895449049373et_nat @ S )
| ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_620_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_621_Un__infinite,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_622_Un__infinite,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_623_Un__infinite,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_624_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_625_finite__UnI,axiom,
! [F2: set_set_set_nat,G: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( ( finite6739761609112101331et_nat @ G )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_626_finite__UnI,axiom,
! [F2: set_nat_nat,G: set_nat_nat] :
( ( finite2115694454571419734at_nat @ F2 )
=> ( ( finite2115694454571419734at_nat @ G )
=> ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_627_finite__UnI,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( finite1152437895449049373et_nat @ G )
=> ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_628_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_629_forth__assumptions_OSET___092_060G_062,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ V ) )
=> ( ord_le9131159989063066194et_nat @ ( clique6509092761774629891_SET_a @ Pi @ Phi ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% forth_assumptions.SET_\<G>
thf(fact_630_deviate__finite_I5_J,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( finite6739761609112101331et_nat @ ( clique3314026705535538693os_cap @ l @ p @ k @ A @ B ) ) ).
% deviate_finite(5)
thf(fact_631_eval__gs__odot,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Theta: a > $o] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ ( clique5469973757772500719t_odot @ X3 @ Y ) )
= ( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ X3 )
& ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ Y ) ) ) ) ) ).
% eval_gs_odot
thf(fact_632_finite__approx__pos,axiom,
! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ Phi ) ) ).
% finite_approx_pos
thf(fact_633_odot___092_060G_062,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X3 @ Y ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% odot_\<G>
thf(fact_634_deviate__finite_I1_J,axiom,
! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) ).
% deviate_finite(1)
thf(fact_635__092_060K_062___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% \<K>_\<G>
thf(fact_636_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_637_approx__neg_Osimps_I4_J,axiom,
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bot_set_nat_nat ) ).
% approx_neg.simps(4)
thf(fact_638_approx__neg_Osimps_I3_J,axiom,
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ monotone_TRUE_a )
= bot_bot_set_nat_nat ) ).
% approx_neg.simps(3)
thf(fact_639_finite__v__gs,axiom,
! [X3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) ) ).
% finite_v_gs
thf(fact_640_bij__betw___092_060pi_062,axiom,
bij_betw_a_set_nat @ pi @ v @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% bij_betw_\<pi>
thf(fact_641_v__gs__mono,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X3 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_mono
thf(fact_642_v__gs__union,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_union
thf(fact_643_ACC__cf__mono,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X3 ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_mono
thf(fact_644_ACC__cf___092_060F_062,axiom,
! [X3: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X3 ) @ ( clique2971579238625216137irst_F @ k ) ) ).
% ACC_cf_\<F>
thf(fact_645__092_060A_062__simps_I1_J,axiom,
member535913909593306477mula_a @ monotone_FALSE_a @ ( clique5987991184601036204th_A_a @ v ) ).
% \<A>_simps(1)
thf(fact_646_SET_Osimps_I1_J,axiom,
( ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% SET.simps(1)
thf(fact_647_L0,axiom,
ord_less_eq_nat @ assumptions_and_L0 @ l ).
% L0
thf(fact_648_L0_H,axiom,
ord_less_eq_nat @ assumptions_and_L02 @ l ).
% L0'
thf(fact_649_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_650_M0,axiom,
ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ k ) ).
% M0
thf(fact_651_approx__pos_Osimps_I3_J,axiom,
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% approx_pos.simps(3)
thf(fact_652_approx__pos_Osimps_I2_J,axiom,
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ monotone_TRUE_a )
= bot_bo7198184520161983622et_nat ) ).
% approx_pos.simps(2)
thf(fact_653_M0_H,axiom,
ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ k ) ).
% M0'
thf(fact_654_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_655_ACC__SET_I2_J,axiom,
( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a ) )
= bot_bo7198184520161983622et_nat ) ).
% ACC_SET(2)
thf(fact_656_ACC__cf__SET_I2_J,axiom,
( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a ) )
= bot_bot_set_nat_nat ) ).
% ACC_cf_SET(2)
thf(fact_657_no__deviation_I1_J,axiom,
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% no_deviation(1)
thf(fact_658_no__deviation_I2_J,axiom,
( ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bot_set_nat_nat ) ).
% no_deviation(2)
thf(fact_659_joinl__join,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X3 @ Y ) @ ( clique5469973757772500719t_odot @ X3 @ Y ) ) ).
% joinl_join
thf(fact_660_bij__betw__iff__bijections,axiom,
( bij_betw_a_a
= ( ^ [F3: a > a,A3: set_a,B3: set_a] :
? [G3: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member_a @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( member_a @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_661_bij__betw__iff__bijections,axiom,
( bij_betw_set_nat_a
= ( ^ [F3: set_nat > a,A3: set_set_nat,B3: set_a] :
? [G3: a > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ( member_a @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( member_set_nat @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_662_bij__betw__iff__bijections,axiom,
( bij_betw_a_set_nat
= ( ^ [F3: a > set_nat,A3: set_a,B3: set_set_nat] :
? [G3: set_nat > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member_set_nat @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B3 )
=> ( ( member_a @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_663_bij__betw__iff__bijections,axiom,
( bij_be2639851105560558660et_nat
= ( ^ [F3: a > set_set_nat,A3: set_a,B3: set_set_set_nat] :
? [G3: set_set_nat > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member_set_set_nat @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B3 )
=> ( ( member_a @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_664_bij__betw__iff__bijections,axiom,
( bij_be3438014552859920132et_nat
= ( ^ [F3: set_nat > set_nat,A3: set_set_nat,B3: set_set_nat] :
? [G3: set_nat > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ( member_set_nat @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B3 )
=> ( ( member_set_nat @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_665_bij__betw__iff__bijections,axiom,
( bij_betw_a_nat_nat
= ( ^ [F3: a > nat > nat,A3: set_a,B3: set_nat_nat] :
? [G3: ( nat > nat ) > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member_nat_nat @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ B3 )
=> ( ( member_a @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_666_bij__betw__iff__bijections,axiom,
( bij_be3032674665972365258_nat_a
= ( ^ [F3: set_set_nat > a,A3: set_set_set_nat,B3: set_a] :
? [G3: a > set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A3 )
=> ( ( member_a @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( member_set_set_nat @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_667_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat_a
= ( ^ [F3: ( nat > nat ) > a,A3: set_nat_nat,B3: set_a] :
? [G3: a > nat > nat] :
( ! [X2: nat > nat] :
( ( member_nat_nat @ X2 @ A3 )
=> ( ( member_a @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( ( member_nat_nat @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_668_bij__betw__iff__bijections,axiom,
( bij_be3030433078811146746et_nat
= ( ^ [F3: a > set_set_set_nat,A3: set_a,B3: set_set_set_set_nat] :
? [G3: set_set_set_nat > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member2946998982187404937et_nat @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ B3 )
=> ( ( member_a @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_669_bij__betw__iff__bijections,axiom,
( bij_be5767359585022399418et_nat
= ( ^ [F3: set_nat > set_set_nat,A3: set_set_nat,B3: set_set_set_nat] :
? [G3: set_set_nat > set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A3 )
=> ( ( member_set_set_nat @ ( F3 @ X2 ) @ B3 )
& ( ( G3 @ ( F3 @ X2 ) )
= X2 ) ) )
& ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ B3 )
=> ( ( member_set_nat @ ( G3 @ X2 ) @ A3 )
& ( ( F3 @ ( G3 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_670_bij__betw__apply,axiom,
! [F: a > a,A: set_a,B: set_a,A2: a] :
( ( bij_betw_a_a @ F @ A @ B )
=> ( ( member_a @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_671_bij__betw__apply,axiom,
! [F: set_nat > a,A: set_set_nat,B: set_a,A2: set_nat] :
( ( bij_betw_set_nat_a @ F @ A @ B )
=> ( ( member_set_nat @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_672_bij__betw__apply,axiom,
! [F: a > set_nat,A: set_a,B: set_set_nat,A2: a] :
( ( bij_betw_a_set_nat @ F @ A @ B )
=> ( ( member_a @ A2 @ A )
=> ( member_set_nat @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_673_bij__betw__apply,axiom,
! [F: set_set_nat > a,A: set_set_set_nat,B: set_a,A2: set_set_nat] :
( ( bij_be3032674665972365258_nat_a @ F @ A @ B )
=> ( ( member_set_set_nat @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_674_bij__betw__apply,axiom,
! [F: set_nat > set_nat,A: set_set_nat,B: set_set_nat,A2: set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A @ B )
=> ( ( member_set_nat @ A2 @ A )
=> ( member_set_nat @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_675_bij__betw__apply,axiom,
! [F: ( nat > nat ) > a,A: set_nat_nat,B: set_a,A2: nat > nat] :
( ( bij_betw_nat_nat_a @ F @ A @ B )
=> ( ( member_nat_nat @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_676_bij__betw__apply,axiom,
! [F: a > set_set_nat,A: set_a,B: set_set_set_nat,A2: a] :
( ( bij_be2639851105560558660et_nat @ F @ A @ B )
=> ( ( member_a @ A2 @ A )
=> ( member_set_set_nat @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_677_bij__betw__apply,axiom,
! [F: a > nat > nat,A: set_a,B: set_nat_nat,A2: a] :
( ( bij_betw_a_nat_nat @ F @ A @ B )
=> ( ( member_a @ A2 @ A )
=> ( member_nat_nat @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_678_bij__betw__apply,axiom,
! [F: set_set_set_nat > a,A: set_set_set_set_nat,B: set_a,A2: set_set_set_nat] :
( ( bij_be458158114365198228_nat_a @ F @ A @ B )
=> ( ( member2946998982187404937et_nat @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_679_bij__betw__apply,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,B: set_set_nat,A2: set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A @ B )
=> ( ( member_set_set_nat @ A2 @ A )
=> ( member_set_nat @ ( F @ A2 ) @ B ) ) ) ).
% bij_betw_apply
thf(fact_680_bij__betw__cong,axiom,
! [A: set_a,F: a > set_nat,G2: a > set_nat,A7: set_set_nat] :
( ! [A6: a] :
( ( member_a @ A6 @ A )
=> ( ( F @ A6 )
= ( G2 @ A6 ) ) )
=> ( ( bij_betw_a_set_nat @ F @ A @ A7 )
= ( bij_betw_a_set_nat @ G2 @ A @ A7 ) ) ) ).
% bij_betw_cong
thf(fact_681_bij__betw__ball,axiom,
! [F: a > set_nat,A: set_a,B: set_set_nat,Phi2: set_nat > $o] :
( ( bij_betw_a_set_nat @ F @ A @ B )
=> ( ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( Phi2 @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( Phi2 @ ( F @ X2 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_682_bij__betw__inv,axiom,
! [F: set_nat > a,A: set_set_nat,B: set_a] :
( ( bij_betw_set_nat_a @ F @ A @ B )
=> ? [G4: a > set_nat] : ( bij_betw_a_set_nat @ G4 @ B @ A ) ) ).
% bij_betw_inv
thf(fact_683_bij__betw__inv,axiom,
! [F: a > set_nat,A: set_a,B: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ A @ B )
=> ? [G4: set_nat > a] : ( bij_betw_set_nat_a @ G4 @ B @ A ) ) ).
% bij_betw_inv
thf(fact_684_bij__betwE,axiom,
! [F: a > set_nat,A: set_a,B: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ A @ B )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( member_set_nat @ ( F @ X5 ) @ B ) ) ) ).
% bij_betwE
thf(fact_685_forth__assumptions_Oapprox__pos_Ocong,axiom,
clique8538548958085942603_pos_a = clique8538548958085942603_pos_a ).
% forth_assumptions.approx_pos.cong
thf(fact_686_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_687_second__assumptions_Odeviate__pos__cap_Ocong,axiom,
clique3314026705535538693os_cap = clique3314026705535538693os_cap ).
% second_assumptions.deviate_pos_cap.cong
thf(fact_688_forth__assumptions_Odeviate__pos_Ocong,axiom,
clique3934260045859375359_pos_a = clique3934260045859375359_pos_a ).
% forth_assumptions.deviate_pos.cong
thf(fact_689_second__assumptions_Odeviate__pos__cup_Ocong,axiom,
clique3314026705536850673os_cup = clique3314026705536850673os_cup ).
% second_assumptions.deviate_pos_cup.cong
thf(fact_690_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_691_forth__assumptions_Ono__deviation_I1_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique3934260045859375359_pos_a @ L @ P @ K @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.no_deviation(1)
thf(fact_692_forth__assumptions_Oapprox__pos_Osimps_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P @ K @ Pi @ monotone_TRUE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(2)
thf(fact_693_forth__assumptions_Oapprox__pos_Osimps_I3_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P @ K @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(3)
thf(fact_694_bij__betw__empty1,axiom,
! [F: nat > nat,A: set_nat] :
( ( bij_betw_nat_nat @ F @ bot_bot_set_nat @ A )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_695_bij__betw__empty1,axiom,
! [F: a > set_nat,A: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ bot_bot_set_a @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% bij_betw_empty1
thf(fact_696_bij__betw__empty1,axiom,
! [F: set_nat > nat,A: set_nat] :
( ( bij_betw_set_nat_nat @ F @ bot_bot_set_set_nat @ A )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_697_bij__betw__empty1,axiom,
! [F: nat > set_nat,A: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ bot_bot_set_nat @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% bij_betw_empty1
thf(fact_698_bij__betw__empty1,axiom,
! [F: set_nat > set_nat,A: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ bot_bot_set_set_nat @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% bij_betw_empty1
thf(fact_699_bij__betw__empty1,axiom,
! [F: set_set_nat > nat,A: set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ bot_bo7198184520161983622et_nat @ A )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_700_bij__betw__empty1,axiom,
! [F: ( nat > nat ) > nat,A: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ bot_bot_set_nat_nat @ A )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty1
thf(fact_701_bij__betw__empty1,axiom,
! [F: nat > set_set_nat,A: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ bot_bot_set_nat @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty1
thf(fact_702_bij__betw__empty1,axiom,
! [F: nat > nat > nat,A: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ bot_bot_set_nat @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% bij_betw_empty1
thf(fact_703_bij__betw__empty1,axiom,
! [F: set_nat > set_set_nat,A: set_set_set_nat] :
( ( bij_be5767359585022399418et_nat @ F @ bot_bot_set_set_nat @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty1
thf(fact_704_bij__betw__empty2,axiom,
! [F: nat > nat,A: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_705_bij__betw__empty2,axiom,
! [F: a > set_nat,A: set_a] :
( ( bij_betw_a_set_nat @ F @ A @ bot_bot_set_set_nat )
=> ( A = bot_bot_set_a ) ) ).
% bij_betw_empty2
thf(fact_706_bij__betw__empty2,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( bij_betw_nat_set_nat @ F @ A @ bot_bot_set_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_707_bij__betw__empty2,axiom,
! [F: set_nat > nat,A: set_set_nat] :
( ( bij_betw_set_nat_nat @ F @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_set_nat ) ) ).
% bij_betw_empty2
thf(fact_708_bij__betw__empty2,axiom,
! [F: set_nat > set_nat,A: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A @ bot_bot_set_set_nat )
=> ( A = bot_bot_set_set_nat ) ) ).
% bij_betw_empty2
thf(fact_709_bij__betw__empty2,axiom,
! [F: nat > set_set_nat,A: set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A @ bot_bo7198184520161983622et_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_710_bij__betw__empty2,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A @ bot_bot_set_nat_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bij_betw_empty2
thf(fact_711_bij__betw__empty2,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A @ bot_bot_set_nat )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty2
thf(fact_712_bij__betw__empty2,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( bij_betw_nat_nat_nat @ F @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat_nat ) ) ).
% bij_betw_empty2
thf(fact_713_bij__betw__empty2,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( bij_be4885122793727115194et_nat @ F @ A @ bot_bot_set_set_nat )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% bij_betw_empty2
thf(fact_714_bij__betw__finite,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B )
=> ( ( finite_finite_nat @ A )
= ( finite_finite_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_715_bij__betw__finite,axiom,
! [F: a > set_nat,A: set_a,B: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ A @ B )
=> ( ( finite_finite_a @ A )
= ( finite1152437895449049373et_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_716_bij__betw__finite,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_nat] :
( ( bij_betw_set_nat_nat @ F @ A @ B )
=> ( ( finite1152437895449049373et_nat @ A )
= ( finite_finite_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_717_bij__betw__finite,axiom,
! [F: nat > set_nat,A: set_nat,B: set_set_nat] :
( ( bij_betw_nat_set_nat @ F @ A @ B )
=> ( ( finite_finite_nat @ A )
= ( finite1152437895449049373et_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_718_bij__betw__finite,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,B: set_nat] :
( ( bij_betw_nat_nat_nat @ F @ A @ B )
=> ( ( finite2115694454571419734at_nat @ A )
= ( finite_finite_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_719_bij__betw__finite,axiom,
! [F: set_nat > set_nat,A: set_set_nat,B: set_set_nat] :
( ( bij_be3438014552859920132et_nat @ F @ A @ B )
=> ( ( finite1152437895449049373et_nat @ A )
= ( finite1152437895449049373et_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_720_bij__betw__finite,axiom,
! [F: nat > nat > nat,A: set_nat,B: set_nat_nat] :
( ( bij_betw_nat_nat_nat2 @ F @ A @ B )
=> ( ( finite_finite_nat @ A )
= ( finite2115694454571419734at_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_721_bij__betw__finite,axiom,
! [F: nat > set_set_nat,A: set_nat,B: set_set_set_nat] :
( ( bij_be6938610931847138308et_nat @ F @ A @ B )
=> ( ( finite_finite_nat @ A )
= ( finite6739761609112101331et_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_722_bij__betw__finite,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat,B: set_nat] :
( ( bij_be6199415091885040644at_nat @ F @ A @ B )
=> ( ( finite6739761609112101331et_nat @ A )
= ( finite_finite_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_723_bij__betw__finite,axiom,
! [F: ( nat > nat ) > set_nat,A: set_nat_nat,B: set_set_nat] :
( ( bij_be2321430536510320189et_nat @ F @ A @ B )
=> ( ( finite2115694454571419734at_nat @ A )
= ( finite1152437895449049373et_nat @ B ) ) ) ).
% bij_betw_finite
thf(fact_724_bij__betw__imp__inj__on,axiom,
! [F: a > set_nat,A: set_a,B: set_set_nat] :
( ( bij_betw_a_set_nat @ F @ A @ B )
=> ( inj_on_a_set_nat @ F @ A ) ) ).
% bij_betw_imp_inj_on
thf(fact_725_numbers2__mono,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X ) @ ( clique3652268606331196573umbers @ X ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y2 ) @ ( clique3652268606331196573umbers @ Y2 ) ) ) ) ).
% numbers2_mono
thf(fact_726_forth__assumptions_Odeviate__finite_I5_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( finite6739761609112101331et_nat @ ( clique3314026705535538693os_cap @ L @ P @ K @ A @ B ) ) ) ).
% forth_assumptions.deviate_finite(5)
thf(fact_727_forth__assumptions_Odeviate__finite_I1_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( finite6739761609112101331et_nat @ ( clique3934260045859375359_pos_a @ L @ P @ K @ Pi @ Phi ) ) ) ).
% forth_assumptions.deviate_finite(1)
thf(fact_728_forth__assumptions_Ofinite__approx__pos,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( finite6739761609112101331et_nat @ ( clique8538548958085942603_pos_a @ L @ P @ K @ Pi @ Phi ) ) ) ).
% forth_assumptions.finite_approx_pos
thf(fact_729_forth__assumptions_OSET_Osimps_I1_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique6509092761774629891_SET_a @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.SET.simps(1)
thf(fact_730_forth__assumptions_Oeval__simps_I4_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( monotone_eval_a @ Theta @ monotone_TRUE_a ) ) ).
% forth_assumptions.eval_simps(4)
thf(fact_731_forth__assumptions_Oeval__simps_I3_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ~ ( monotone_eval_a @ Theta @ monotone_FALSE_a ) ) ).
% forth_assumptions.eval_simps(3)
thf(fact_732_forth__assumptions_O_092_060A_062__simps_I1_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( member535913909593306477mula_a @ monotone_FALSE_a @ ( clique5987991184601036204th_A_a @ V ) ) ) ).
% forth_assumptions.\<A>_simps(1)
thf(fact_733_forth__assumptions_Odeviate__finite_I3_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( finite6739761609112101331et_nat @ ( clique3314026705536850673os_cup @ L @ P @ K @ A @ B ) ) ) ).
% forth_assumptions.deviate_finite(3)
thf(fact_734_forth__assumptions_Ono__deviation_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique2019076642914533763_neg_a @ L @ P @ K @ Pi @ monotone_FALSE_a )
= bot_bot_set_nat_nat ) ) ).
% forth_assumptions.no_deviation(2)
thf(fact_735_forth__assumptions_Oapprox__neg_Osimps_I3_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique6623365555141101007_neg_a @ L @ P @ K @ Pi @ monotone_TRUE_a )
= bot_bot_set_nat_nat ) ) ).
% forth_assumptions.approx_neg.simps(3)
thf(fact_736_forth__assumptions_Oapprox__neg_Osimps_I4_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique6623365555141101007_neg_a @ L @ P @ K @ Pi @ monotone_FALSE_a )
= bot_bot_set_nat_nat ) ) ).
% forth_assumptions.approx_neg.simps(4)
thf(fact_737_forth__assumptions_OACC__cf__SET_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique951075384711337423ACC_cf @ K @ ( clique6509092761774629891_SET_a @ Pi @ monotone_FALSE_a ) )
= bot_bot_set_nat_nat ) ) ).
% forth_assumptions.ACC_cf_SET(2)
thf(fact_738_forth__assumptions_OACC__SET_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique6509092761774629891_SET_a @ Pi @ monotone_FALSE_a ) )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.ACC_SET(2)
thf(fact_739_forth__assumptions_Obij__betw___092_060pi_062,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( bij_betw_a_set_nat @ Pi @ V @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% forth_assumptions.bij_betw_\<pi>
thf(fact_740_forth__assumptions_Oeval__gs__odot,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,X3: set_set_set_nat,Y: set_set_set_nat,Theta: a > $o] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ ( clique5469973757772500719t_odot @ X3 @ Y ) )
= ( ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ X3 )
& ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ Y ) ) ) ) ) ) ).
% forth_assumptions.eval_gs_odot
thf(fact_741_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X6: set_set_set_nat,G5: set_set_nat] :
? [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ X6 )
& ( ord_le6893508408891458716et_nat @ X2 @ G5 ) ) ) ) ).
% accepts_def
thf(fact_742_finite__v__gs__Gl,axiom,
! [X3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) ) ).
% finite_v_gs_Gl
thf(fact_743_deviate__pos__cup,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( clique3314026705536850673os_cup @ l @ p @ k @ X3 @ Y )
= bot_bo7198184520161983622et_nat ) ) ) ).
% deviate_pos_cup
thf(fact_744_v___092_060G_062__2,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% v_\<G>_2
thf(fact_745_finite__vG,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ).
% finite_vG
thf(fact_746_approx__neg_Osimps_I5_J,axiom,
! [V2: a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Var_a @ V2 ) )
= bot_bot_set_nat_nat ) ).
% approx_neg.simps(5)
thf(fact_747_approx__pos_Osimps_I5_J,axiom,
! [V2: monotone_mformula_a,Va: monotone_mformula_a] :
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ V2 @ Va ) )
= bot_bo7198184520161983622et_nat ) ).
% approx_pos.simps(5)
thf(fact_748_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_749_v__mono,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_mono
thf(fact_750_v__union,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_union
thf(fact_751_dual__order_Orefl,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_752_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_753_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_754_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_755_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_756_order__refl,axiom,
! [X: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X @ X ) ).
% order_refl
thf(fact_757_order__refl,axiom,
! [X: set_set_nat] : ( ord_le6893508408891458716et_nat @ X @ X ) ).
% order_refl
thf(fact_758_order__refl,axiom,
! [X: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X @ X ) ).
% order_refl
thf(fact_759_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_760_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_761_SET_Osimps_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi ) @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ).
% SET.simps(3)
thf(fact_762_v___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% v_\<G>
thf(fact_763_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_764_approx__pos_Osimps_I4_J,axiom,
! [V2: a] :
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Var_a @ V2 ) )
= bot_bo7198184520161983622et_nat ) ).
% approx_pos.simps(4)
thf(fact_765_acceptsI,axiom,
! [D2: set_set_nat,G: set_set_nat,X3: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D2 @ G )
=> ( ( member_set_set_nat @ D2 @ X3 )
=> ( clique3686358387679108662ccepts @ X3 @ G ) ) ) ).
% acceptsI
thf(fact_766__092_060A_062__simps_I2_J,axiom,
! [X: a] :
( ( member535913909593306477mula_a @ ( monotone_Var_a @ X ) @ ( clique5987991184601036204th_A_a @ v ) )
= ( member_a @ X @ v ) ) ).
% \<A>_simps(2)
thf(fact_767__092_060A_062__simps_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ ( monotone_Disj_a @ Phi @ Psi ) @ ( clique5987991184601036204th_A_a @ v ) )
= ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
& ( member535913909593306477mula_a @ Psi @ ( clique5987991184601036204th_A_a @ v ) ) ) ) ).
% \<A>_simps(4)
thf(fact_768_empty___092_060P_062L_092_060G_062l,axiom,
member2946998982187404937et_nat @ bot_bo7198184520161983622et_nat @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ).
% empty_\<P>L\<G>l
thf(fact_769_ACC__I,axiom,
! [G: set_set_nat,X3: set_set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X3 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ k @ X3 ) ) ) ) ).
% ACC_I
thf(fact_770_ACC__SET_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).
% ACC_SET(3)
thf(fact_771_ACC__cf__SET_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).
% ACC_cf_SET(3)
thf(fact_772_no__deviation_I3_J,axiom,
! [X: a] :
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X ) )
= bot_bo7198184520161983622et_nat ) ).
% no_deviation(3)
thf(fact_773_no__deviation_I4_J,axiom,
! [X: a] :
( ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X ) )
= bot_bot_set_nat_nat ) ).
% no_deviation(4)
thf(fact_774_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_775_first__assumptions_O_092_060P_062L_092_060G_062l_Ocong,axiom,
clique2294137941332549862_L_G_l = clique2294137941332549862_L_G_l ).
% first_assumptions.\<P>L\<G>l.cong
thf(fact_776_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_777_forth__assumptions_Oeval__simps_I5_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o,X: a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( monotone_eval_a @ Theta @ ( monotone_Var_a @ X ) )
= ( Theta @ X ) ) ) ).
% forth_assumptions.eval_simps(5)
thf(fact_778_forth__assumptions_Oeval__simps_I6_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( monotone_eval_a @ Theta @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( ( monotone_eval_a @ Theta @ Phi )
| ( monotone_eval_a @ Theta @ Psi ) ) ) ) ).
% forth_assumptions.eval_simps(6)
thf(fact_779_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_set_set_set_nat,Pi: set_set_set_nat > set_nat,X: set_set_set_nat] :
( ( clique3407333501437444587et_nat @ L @ P @ K @ V @ Pi )
=> ( ( member4689220760989666777et_nat @ ( monoto7822445266502226924et_nat @ X ) @ ( clique2555064243683067844et_nat @ V ) )
= ( member2946998982187404937et_nat @ X @ V ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_780_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_set_set_nat,Pi: set_set_nat > set_nat,X: set_set_nat] :
( ( clique2455256169097332789et_nat @ L @ P @ K @ V @ Pi )
=> ( ( member4844836972813196067et_nat @ ( monoto3251651810667535926et_nat @ X ) @ ( clique7740924183492588046et_nat @ V ) )
= ( member_set_set_nat @ X @ V ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_781_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_set_nat,Pi: set_nat > set_nat,X: set_nat] :
( ( clique522982669833463679et_nat @ L @ P @ K @ V @ Pi )
=> ( ( member7623223977307079021et_nat @ ( monotone_Var_set_nat @ X ) @ ( clique9181349226887787864et_nat @ V ) )
= ( member_set_nat @ X @ V ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_782_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_nat_nat,Pi: ( nat > nat ) > set_nat,X: nat > nat] :
( ( clique5528702923696243640at_nat @ L @ P @ K @ V @ Pi )
=> ( ( member435044527007775910at_nat @ ( monotone_Var_nat_nat @ X ) @ ( clique6859621968737270801at_nat @ V ) )
= ( member_nat_nat @ X @ V ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_783_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,X: a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( member535913909593306477mula_a @ ( monotone_Var_a @ X ) @ ( clique5987991184601036204th_A_a @ V ) )
= ( member_a @ X @ V ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_784_forth__assumptions_O_092_060A_062__simps_I4_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( member535913909593306477mula_a @ ( monotone_Disj_a @ Phi @ Psi ) @ ( clique5987991184601036204th_A_a @ V ) )
= ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ V ) )
& ( member535913909593306477mula_a @ Psi @ ( clique5987991184601036204th_A_a @ V ) ) ) ) ) ).
% forth_assumptions.\<A>_simps(4)
thf(fact_785_forth__assumptions_Oempty___092_060P_062L_092_060G_062l,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( member2946998982187404937et_nat @ bot_bo7198184520161983622et_nat @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) ) ) ).
% forth_assumptions.empty_\<P>L\<G>l
thf(fact_786_order__antisym__conv,axiom,
! [Y2: set_set_set_nat,X: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X )
=> ( ( ord_le9131159989063066194et_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_787_order__antisym__conv,axiom,
! [Y2: set_set_nat,X: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X )
=> ( ( ord_le6893508408891458716et_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_788_order__antisym__conv,axiom,
! [Y2: set_nat_nat,X: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X )
=> ( ( ord_le9059583361652607317at_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_789_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_790_order__antisym__conv,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_791_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_792_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_793_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_794_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_795_ord__le__eq__subst,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_796_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_797_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_798_ord__le__eq__subst,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_799_ord__le__eq__subst,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_800_ord__le__eq__subst,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_801_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_802_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_803_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_804_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_805_ord__eq__le__subst,axiom,
! [A2: nat,F: set_set_nat > nat,B2: set_set_nat,C: set_set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_806_ord__eq__le__subst,axiom,
! [A2: set_set_nat,F: nat > set_set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_807_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_808_ord__eq__le__subst,axiom,
! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_809_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_set_nat > set_nat,B2: set_set_nat,C: set_set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_810_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat_nat > nat,B2: set_nat_nat,C: set_nat_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_811_ord__eq__le__subst,axiom,
! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_812_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_813_order__eq__refl,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( X = Y2 )
=> ( ord_le9131159989063066194et_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_814_order__eq__refl,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( X = Y2 )
=> ( ord_le6893508408891458716et_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_815_order__eq__refl,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( X = Y2 )
=> ( ord_le9059583361652607317at_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_816_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_817_order__eq__refl,axiom,
! [X: set_nat,Y2: set_nat] :
( ( X = Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_818_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_819_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_820_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_821_order__subst2,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_822_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_823_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_824_order__subst2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_825_order__subst2,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_826_order__subst2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,F: set_nat_nat > nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_827_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_828_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_829_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_830_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_831_order__subst1,axiom,
! [A2: set_set_nat,F: nat > set_set_nat,B2: nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_832_order__subst1,axiom,
! [A2: nat,F: set_set_nat > nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_833_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_834_order__subst1,axiom,
! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_835_order__subst1,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat,B2: set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_836_order__subst1,axiom,
! [A2: set_nat_nat,F: nat > set_nat_nat,B2: nat,C: nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_837_order__subst1,axiom,
! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_838_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_set_nat,Z2: set_set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A4 @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_839_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_840_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat_nat,Z2: set_nat_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_841_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_842_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_843_antisym,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_844_antisym,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_845_antisym,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_846_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_847_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_848_dual__order_Otrans,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C @ B2 )
=> ( ord_le9131159989063066194et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_849_dual__order_Otrans,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ B2 )
=> ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_850_dual__order_Otrans,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ B2 )
=> ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_851_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_852_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_853_dual__order_Oantisym,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_854_dual__order_Oantisym,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_855_dual__order_Oantisym,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_856_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_857_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_858_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_set_nat,Z2: set_set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B4 @ A4 )
& ( ord_le9131159989063066194et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_859_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
& ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_860_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat_nat,Z2: set_nat_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A4 )
& ( ord_le9059583361652607317at_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_861_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_862_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_863_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P2 @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P2 @ B6 @ A6 )
=> ( P2 @ A6 @ B6 ) )
=> ( P2 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_864_order__trans,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ Z )
=> ( ord_le9131159989063066194et_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_865_order__trans,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ Z )
=> ( ord_le6893508408891458716et_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_866_order__trans,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z )
=> ( ord_le9059583361652607317at_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_867_order__trans,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_868_order__trans,axiom,
! [X: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z )
=> ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_869_order_Otrans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_870_order_Otrans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_871_order_Otrans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_872_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_873_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_874_order__antisym,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_875_order__antisym,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_876_order__antisym,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_877_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_878_order__antisym,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_879_ord__le__eq__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_880_ord__le__eq__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_881_ord__le__eq__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_882_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_883_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_884_ord__eq__le__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( A2 = B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_885_ord__eq__le__trans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( A2 = B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_886_ord__eq__le__trans,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( A2 = B2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_887_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_888_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_889_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_set_nat,Z2: set_set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y5 )
& ( ord_le9131159989063066194et_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_890_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [X2: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y5 )
& ( ord_le6893508408891458716et_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_891_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat_nat,Z2: set_nat_nat] : ( Y3 = Z2 ) )
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y5 )
& ( ord_le9059583361652607317at_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_892_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_893_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [X2: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_894_le__cases3,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_895_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_896_forth__assumptions_OSET_Osimps_I3_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique6509092761774629891_SET_a @ Pi @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ Pi @ Phi ) @ ( clique6509092761774629891_SET_a @ Pi @ Psi ) ) ) ) ).
% forth_assumptions.SET.simps(3)
thf(fact_897_forth__assumptions_Ono__deviation_I3_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,X: a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique3934260045859375359_pos_a @ L @ P @ K @ Pi @ ( monotone_Var_a @ X ) )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.no_deviation(3)
thf(fact_898_forth__assumptions_Ono__deviation_I4_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,X: a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique2019076642914533763_neg_a @ L @ P @ K @ Pi @ ( monotone_Var_a @ X ) )
= bot_bot_set_nat_nat ) ) ).
% forth_assumptions.no_deviation(4)
thf(fact_899_forth__assumptions_Oapprox__pos_Osimps_I4_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,V2: a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P @ K @ Pi @ ( monotone_Var_a @ V2 ) )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(4)
thf(fact_900_forth__assumptions_Oapprox__pos_Osimps_I5_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,V2: monotone_mformula_a,Va: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P @ K @ Pi @ ( monotone_Disj_a @ V2 @ Va ) )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(5)
thf(fact_901_forth__assumptions_Oapprox__neg_Osimps_I5_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,V2: a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique6623365555141101007_neg_a @ L @ P @ K @ Pi @ ( monotone_Var_a @ V2 ) )
= bot_bot_set_nat_nat ) ) ).
% forth_assumptions.approx_neg.simps(5)
thf(fact_902_forth__assumptions_OACC__SET_I3_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique6509092761774629891_SET_a @ Pi @ ( monotone_Disj_a @ Phi @ Psi ) ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K @ ( clique6509092761774629891_SET_a @ Pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique6509092761774629891_SET_a @ Pi @ Psi ) ) ) ) ) ).
% forth_assumptions.ACC_SET(3)
thf(fact_903_forth__assumptions_OACC__cf__SET_I3_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique951075384711337423ACC_cf @ K @ ( clique6509092761774629891_SET_a @ Pi @ ( monotone_Disj_a @ Phi @ Psi ) ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique6509092761774629891_SET_a @ Pi @ Phi ) ) @ ( clique951075384711337423ACC_cf @ K @ ( clique6509092761774629891_SET_a @ Pi @ Psi ) ) ) ) ) ).
% forth_assumptions.ACC_cf_SET(3)
thf(fact_904_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_905_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_906_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_907_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_908_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_909_bot_Oextremum__unique,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_unique
thf(fact_910_bot_Oextremum__unique,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_911_bot_Oextremum__unique,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_912_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_913_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_914_bot_Oextremum,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ).
% bot.extremum
thf(fact_915_bot_Oextremum,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% bot.extremum
thf(fact_916_bot_Oextremum,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% bot.extremum
thf(fact_917_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_918_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_919_PLU__joinl,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X3 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% PLU_joinl
thf(fact_920_PLU__union,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% PLU_union
thf(fact_921_sqcap,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X3 @ Y ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% sqcap
thf(fact_922_sqcup,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% sqcup
thf(fact_923_deviate__subset__Disj_I2_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9059583361652607317at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) @ ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% deviate_subset_Disj(2)
thf(fact_924_deviate__subset__Disj_I1_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9131159989063066194et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi @ Psi ) ) @ ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3314026705536850673os_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% deviate_subset_Disj(1)
thf(fact_925_APR,axiom,
! [Phi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
=> ( member2946998982187404937et_nat @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% APR
thf(fact_926_sqcup__def,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ) ).
% sqcup_def
thf(fact_927_sqcap__def,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118206219037_sqcap @ l @ p @ k @ X3 @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X3 @ Y ) ) ) ).
% sqcap_def
thf(fact_928_APR_Osimps_I1_J,axiom,
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% APR.simps(1)
thf(fact_929_APR_Osimps_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( clique2586627118207531017_sqcup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% APR.simps(3)
thf(fact_930_approx__neg_Osimps_I2_J,axiom,
! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi2 @ Psi2 ) )
= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) ).
% approx_neg.simps(2)
thf(fact_931_second__assumptions_OPLU_Ocong,axiom,
clique2699557479641037314nd_PLU = clique2699557479641037314nd_PLU ).
% second_assumptions.PLU.cong
thf(fact_932_forth__assumptions_OAPR_Ocong,axiom,
clique3873310923663319714_APR_a = clique3873310923663319714_APR_a ).
% forth_assumptions.APR.cong
thf(fact_933_second__assumptions_Osqcap_Ocong,axiom,
clique2586627118206219037_sqcap = clique2586627118206219037_sqcap ).
% second_assumptions.sqcap.cong
thf(fact_934_second__assumptions_Osqcup_Ocong,axiom,
clique2586627118207531017_sqcup = clique2586627118207531017_sqcup ).
% second_assumptions.sqcup.cong
thf(fact_935_forth__assumptions_OAPR_Osimps_I3_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( clique2586627118207531017_sqcup @ L @ P @ K @ ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ Phi ) @ ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ Psi ) ) ) ) ).
% forth_assumptions.APR.simps(3)
thf(fact_936_forth__assumptions_OAPR_Osimps_I1_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.APR.simps(1)
thf(fact_937_forth__assumptions_OAPR,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ V ) )
=> ( member2946998982187404937et_nat @ ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ Phi ) @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) ) ) ) ) ).
% forth_assumptions.APR
thf(fact_938_forth__assumptions_Oapprox__neg_Osimps_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ( clique6623365555141101007_neg_a @ L @ P @ K @ Pi @ ( monotone_Disj_a @ Phi2 @ Psi2 ) )
= ( clique1591571987439376245eg_cup @ L @ P @ K @ ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ Psi2 ) ) ) ) ).
% forth_assumptions.approx_neg.simps(2)
thf(fact_939_forth__assumptions_Odeviate__subset__Disj_I1_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ord_le9131159989063066194et_nat @ ( clique3934260045859375359_pos_a @ L @ P @ K @ Pi @ ( monotone_Disj_a @ Phi @ Psi ) ) @ ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3314026705536850673os_cup @ L @ P @ K @ ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ Phi ) @ ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ Psi ) ) @ ( clique3934260045859375359_pos_a @ L @ P @ K @ Pi @ Phi ) ) @ ( clique3934260045859375359_pos_a @ L @ P @ K @ Pi @ Psi ) ) ) ) ).
% forth_assumptions.deviate_subset_Disj(1)
thf(fact_940_forth__assumptions_Odeviate__subset__Disj_I2_J,axiom,
! [L: nat,P: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P @ K @ V @ Pi )
=> ( ord_le9059583361652607317at_nat @ ( clique2019076642914533763_neg_a @ L @ P @ K @ Pi @ ( monotone_Disj_a @ Phi @ Psi ) ) @ ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( clique1591571987439376245eg_cup @ L @ P @ K @ ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ Phi ) @ ( clique3873310923663319714_APR_a @ L @ P @ K @ Pi @ Psi ) ) @ ( clique2019076642914533763_neg_a @ L @ P @ K @ Pi @ Phi ) ) @ ( clique2019076642914533763_neg_a @ L @ P @ K @ Pi @ Psi ) ) ) ) ).
% forth_assumptions.deviate_subset_Disj(2)
thf(fact_941_approx__neg_Oelims,axiom,
! [X: monotone_mformula_a,Y2: set_nat_nat] :
( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) ) )
=> ( ( ( X = monotone_TRUE_a )
=> ( Y2 != bot_bot_set_nat_nat ) )
=> ( ( ( X = monotone_FALSE_a )
=> ( Y2 != bot_bot_set_nat_nat ) )
=> ~ ( ? [V3: a] :
( X
= ( monotone_Var_a @ V3 ) )
=> ( Y2 != bot_bot_set_nat_nat ) ) ) ) ) ) ) ).
% approx_neg.elims
thf(fact_942_approx__pos_Oelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) ) )
=> ( ( ( X = monotone_TRUE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ( ( X = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ( ? [V3: a] :
( X
= ( monotone_Var_a @ V3 ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ~ ( ? [V3: monotone_mformula_a,Va2: monotone_mformula_a] :
( X
= ( monotone_Disj_a @ V3 @ Va2 ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ).
% approx_pos.elims
thf(fact_943_sqcup__sub,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y ) ) ) ) ) ).
% sqcup_sub
thf(fact_944_deviate__neg__def,axiom,
! [Phi: monotone_mformula_a] :
( ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique8961599393750669800f_mf_a @ k @ pi @ Phi ) ) ) ).
% deviate_neg_def
thf(fact_945_deviate__subset__Conj_I2_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9059583361652607317at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) @ ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% deviate_subset_Conj(2)
thf(fact_946_deviate__subset__Conj_I1_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] : ( ord_le9131159989063066194et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) @ ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% deviate_subset_Conj(1)
thf(fact_947_inf_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_948_inf_Oidem,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_949_inf__idem,axiom,
! [X: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_950_inf__idem,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_951_inf_Oleft__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_952_inf_Oleft__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_953_inf__left__idem,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ X @ Y2 ) ) ).
% inf_left_idem
thf(fact_954_inf__left__idem,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y2 ) )
= ( inf_inf_set_nat_nat @ X @ Y2 ) ) ).
% inf_left_idem
thf(fact_955_inf_Oright__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ B2 )
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_956_inf_Oright__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_957_inf__right__idem,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ Y2 )
= ( inf_in5711780100303410308et_nat @ X @ Y2 ) ) ).
% inf_right_idem
thf(fact_958_inf__right__idem,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ Y2 )
= ( inf_inf_set_nat_nat @ X @ Y2 ) ) ).
% inf_right_idem
thf(fact_959_IntI,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ A )
=> ( ( member2946998982187404937et_nat @ C @ B )
=> ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_960_IntI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_961_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_962_IntI,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A )
=> ( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_963_IntI,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_964_Int__iff,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) )
= ( ( member2946998982187404937et_nat @ C @ A )
& ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_965_Int__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ( member_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_966_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_967_Int__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
& ( member_set_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_968_Int__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
& ( member_nat_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_969_DiffI,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ A )
=> ( ~ ( member2946998982187404937et_nat @ C @ B )
=> ( member2946998982187404937et_nat @ C @ ( minus_3113942175840221057et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_970_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_971_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_972_DiffI,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( ~ ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_973_DiffI,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A )
=> ( ~ ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_974_Diff__iff,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( minus_3113942175840221057et_nat @ A @ B ) )
= ( ( member2946998982187404937et_nat @ C @ A )
& ~ ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_975_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ~ ( member_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_976_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_977_Diff__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
& ~ ( member_nat_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_978_Diff__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
& ~ ( member_set_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_979_Diff__idemp,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ B )
= ( minus_8121590178497047118at_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_980_Diff__idemp,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ B )
= ( minus_2447799839930672331et_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_981_ACC__odot,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X3 @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X3 ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ).
% ACC_odot
thf(fact_982_SET_Osimps_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi ) @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ).
% SET.simps(4)
thf(fact_983_approx__pos_Ocases,axiom,
! [X: monotone_mformula_a] :
( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X
!= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( X != monotone_TRUE_a )
=> ( ( X != monotone_FALSE_a )
=> ( ! [V3: a] :
( X
!= ( monotone_Var_a @ V3 ) )
=> ~ ! [V3: monotone_mformula_a,Va2: monotone_mformula_a] :
( X
!= ( monotone_Disj_a @ V3 @ Va2 ) ) ) ) ) ) ).
% approx_pos.cases
thf(fact_984_approx__neg_Ocases,axiom,
! [X: monotone_mformula_a] :
( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X
!= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X
!= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( X != monotone_TRUE_a )
=> ( ( X != monotone_FALSE_a )
=> ~ ! [V3: a] :
( X
!= ( monotone_Var_a @ V3 ) ) ) ) ) ) ).
% approx_neg.cases
thf(fact_985_SET_Ocases,axiom,
! [X: monotone_mformula_a] :
( ( X != monotone_FALSE_a )
=> ( ! [X4: a] :
( X
!= ( monotone_Var_a @ X4 ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( X
!= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( X
!= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( X = monotone_TRUE_a ) ) ) ) ) ).
% SET.cases
thf(fact_986_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_987_inf_Obounded__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) )
= ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_988_inf_Obounded__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_989_inf_Obounded__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_990_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_991_inf_Obounded__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_992_le__inf__iff,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) )
= ( ( ord_le9131159989063066194et_nat @ X @ Y2 )
& ( ord_le9131159989063066194et_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_993_le__inf__iff,axiom,
! [X: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ ( inf_inf_set_set_nat @ Y2 @ Z ) )
= ( ( ord_le6893508408891458716et_nat @ X @ Y2 )
& ( ord_le6893508408891458716et_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_994_le__inf__iff,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z ) )
= ( ( ord_le9059583361652607317at_nat @ X @ Y2 )
& ( ord_le9059583361652607317at_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_995_le__inf__iff,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_996_le__inf__iff,axiom,
! [X: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_set_nat @ X @ Y2 )
& ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_997_inf__bot__right,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_998_inf__bot__right,axiom,
! [X: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_right
thf(fact_999_inf__bot__right,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% inf_bot_right
thf(fact_1000_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_1001_inf__bot__left,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_1002_inf__bot__left,axiom,
! [X: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_left
thf(fact_1003_inf__bot__left,axiom,
! [X: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X )
= bot_bot_set_nat_nat ) ).
% inf_bot_left
thf(fact_1004_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_1005_Int__subset__iff,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( ord_le9131159989063066194et_nat @ C2 @ A )
& ( ord_le9131159989063066194et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_1006_Int__subset__iff,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( ord_le6893508408891458716et_nat @ C2 @ A )
& ( ord_le6893508408891458716et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_1007_Int__subset__iff,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C2 @ A )
& ( ord_le9059583361652607317at_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_1008_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_1009_finite__Int,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ F2 )
| ( finite1152437895449049373et_nat @ G ) )
=> ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_1010_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_1011_finite__Int,axiom,
! [F2: set_set_set_nat,G: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ F2 )
| ( finite6739761609112101331et_nat @ G ) )
=> ( finite6739761609112101331et_nat @ ( inf_in5711780100303410308et_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_1012_finite__Int,axiom,
! [F2: set_nat_nat,G: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ F2 )
| ( finite2115694454571419734at_nat @ G ) )
=> ( finite2115694454571419734at_nat @ ( inf_inf_set_nat_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_1013_Diff__cancel,axiom,
! [A: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ A )
= bot_bot_set_set_nat ) ).
% Diff_cancel
thf(fact_1014_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1015_Diff__cancel,axiom,
! [A: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ A )
= bot_bot_set_nat_nat ) ).
% Diff_cancel
thf(fact_1016_Diff__cancel,axiom,
! [A: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A @ A )
= bot_bo7198184520161983622et_nat ) ).
% Diff_cancel
thf(fact_1017_empty__Diff,axiom,
! [A: set_set_nat] :
( ( minus_2163939370556025621et_nat @ bot_bot_set_set_nat @ A )
= bot_bot_set_set_nat ) ).
% empty_Diff
thf(fact_1018_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1019_empty__Diff,axiom,
! [A: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ bot_bot_set_nat_nat @ A )
= bot_bot_set_nat_nat ) ).
% empty_Diff
thf(fact_1020_empty__Diff,axiom,
! [A: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ bot_bo7198184520161983622et_nat @ A )
= bot_bo7198184520161983622et_nat ) ).
% empty_Diff
thf(fact_1021_Diff__empty,axiom,
! [A: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% Diff_empty
thf(fact_1022_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_1023_Diff__empty,axiom,
! [A: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ bot_bot_set_nat_nat )
= A ) ).
% Diff_empty
thf(fact_1024_Diff__empty,axiom,
! [A: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A @ bot_bo7198184520161983622et_nat )
= A ) ).
% Diff_empty
thf(fact_1025_sup__inf__absorb,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_1026_sup__inf__absorb,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_1027_sup__inf__absorb,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ ( inf_inf_set_set_nat @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_1028_sup__inf__absorb,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_1029_inf__sup__absorb,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_1030_inf__sup__absorb,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( sup_sup_set_nat_nat @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_1031_inf__sup__absorb,axiom,
! [X: set_set_nat,Y2: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ ( sup_sup_set_set_nat @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_1032_inf__sup__absorb,axiom,
! [X: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_1033_odotl__def,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k @ X3 @ Y )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X3 @ Y ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).
% odotl_def
thf(fact_1034_finite__Diff2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( finite1152437895449049373et_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1035_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1036_finite__Diff2,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) )
= ( finite2115694454571419734at_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1037_finite__Diff2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) )
= ( finite6739761609112101331et_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_1038_finite__Diff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1039_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1040_finite__Diff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1041_finite__Diff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_1042_Un__Int__eq_I1_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1043_Un__Int__eq_I1_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1044_Un__Int__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1045_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1046_Un__Int__eq_I2_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1047_Un__Int__eq_I2_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1048_Un__Int__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1049_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1050_Un__Int__eq_I3_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ S @ ( sup_su4213647025997063966et_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1051_Un__Int__eq_I3_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S @ ( sup_sup_set_nat_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1052_Un__Int__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ S @ ( sup_sup_set_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1053_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1054_Un__Int__eq_I4_J,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ T2 @ ( sup_su4213647025997063966et_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1055_Un__Int__eq_I4_J,axiom,
! [T2: set_nat_nat,S: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T2 @ ( sup_sup_set_nat_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1056_Un__Int__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_set_nat @ T2 @ ( sup_sup_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1057_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1058_Int__Un__eq_I1_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1059_Int__Un__eq_I1_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1060_Int__Un__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1061_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1062_Int__Un__eq_I2_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1063_Int__Un__eq_I2_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1064_Int__Un__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1065_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1066_Int__Un__eq_I3_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ S @ ( inf_in5711780100303410308et_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1067_Int__Un__eq_I3_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S @ ( inf_inf_set_nat_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1068_Int__Un__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ S @ ( inf_inf_set_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1069_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1070_Int__Un__eq_I4_J,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ T2 @ ( inf_in5711780100303410308et_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1071_Int__Un__eq_I4_J,axiom,
! [T2: set_nat_nat,S: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T2 @ ( inf_inf_set_nat_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1072_Int__Un__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_set_nat @ T2 @ ( inf_inf_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1073_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1074_Un__Diff__cancel2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ B @ A ) @ A )
= ( sup_sup_set_set_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1075_Un__Diff__cancel2,axiom,
! [B: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A ) @ A )
= ( sup_sup_set_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1076_Un__Diff__cancel2,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ B @ A ) @ A )
= ( sup_sup_set_nat_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1077_Un__Diff__cancel2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ B @ A ) @ A )
= ( sup_su4213647025997063966et_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_1078_Un__Diff__cancel,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B @ A ) )
= ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1079_Un__Diff__cancel,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1080_Un__Diff__cancel,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B @ A ) )
= ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1081_Un__Diff__cancel,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( minus_2447799839930672331et_nat @ B @ A ) )
= ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_1082_APR_Osimps_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi ) ) ) ).
% APR.simps(4)
thf(fact_1083_deviate__neg__cup__def,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987439376245eg_cup @ l @ p @ k @ X3 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y ) ) @ ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ) ) ).
% deviate_neg_cup_def
thf(fact_1084_deviate__neg__cap__def,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987438064265eg_cap @ l @ p @ k @ X3 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X3 @ Y ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X3 @ Y ) ) ) ) ).
% deviate_neg_cap_def
thf(fact_1085_approx__neg_Osimps_I1_J,axiom,
! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi2 @ Psi2 ) )
= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) ).
% approx_neg.simps(1)
thf(fact_1086_approx__pos_Osimps_I1_J,axiom,
! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi2 @ Psi2 ) )
= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) ).
% approx_pos.simps(1)
thf(fact_1087_Diff__eq__empty__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( minus_2447799839930672331et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
= ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1088_Diff__eq__empty__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1089_Diff__eq__empty__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1090_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_1091_Diff__disjoint,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B @ A ) )
= bot_bot_set_set_nat ) ).
% Diff_disjoint
thf(fact_1092_Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1093_Diff__disjoint,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B @ A ) )
= bot_bot_set_nat_nat ) ).
% Diff_disjoint
thf(fact_1094_Diff__disjoint,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( minus_2447799839930672331et_nat @ B @ A ) )
= bot_bo7198184520161983622et_nat ) ).
% Diff_disjoint
thf(fact_1095__092_060A_062__simps_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ ( monotone_Conj_a @ Phi @ Psi ) @ ( clique5987991184601036204th_A_a @ v ) )
= ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ v ) )
& ( member535913909593306477mula_a @ Psi @ ( clique5987991184601036204th_A_a @ v ) ) ) ) ).
% \<A>_simps(3)
thf(fact_1096_ACC__SET_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).
% ACC_SET(4)
thf(fact_1097_Int__Diff__disjoint,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ B ) )
= bot_bot_set_set_nat ) ).
% Int_Diff_disjoint
thf(fact_1098_Int__Diff__disjoint,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_1099_Int__Diff__disjoint,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ B ) )
= bot_bot_set_nat_nat ) ).
% Int_Diff_disjoint
thf(fact_1100_Int__Diff__disjoint,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ A @ B ) )
= bot_bo7198184520161983622et_nat ) ).
% Int_Diff_disjoint
thf(fact_1101_Diff__triv,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
=> ( ( minus_2163939370556025621et_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1102_Diff__triv,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1103_Diff__triv,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
=> ( ( minus_8121590178497047118at_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1104_Diff__triv,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
=> ( ( minus_2447799839930672331et_nat @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_1105_Diff__Un,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( inf_inf_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1106_Diff__Un,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1107_Diff__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1108_Diff__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) )
= ( inf_in5711780100303410308et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1109_Diff__Int,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1110_Diff__Int,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1111_Diff__Int,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1112_Diff__Int,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1113_Int__Diff__Un,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1114_Int__Diff__Un,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( minus_minus_set_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1115_Int__Diff__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1116_Int__Diff__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1117_Un__Diff__Int,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( inf_inf_set_set_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1118_Un__Diff__Int,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1119_Un__Diff__Int,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1120_Un__Diff__Int,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1121_inf__sup__aci_I4_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ X @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_1122_inf__sup__aci_I4_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ X @ Y2 ) )
= ( inf_inf_set_nat_nat @ X @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_1123_inf__sup__aci_I3_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) )
= ( inf_in5711780100303410308et_nat @ Y2 @ ( inf_in5711780100303410308et_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_1124_inf__sup__aci_I3_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat_nat @ Y2 @ ( inf_inf_set_nat_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_1125_inf__sup__aci_I2_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ Z )
= ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_1126_inf__sup__aci_I2_J,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ Z )
= ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_1127_inf__sup__aci_I1_J,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1128_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] : ( inf_inf_set_nat_nat @ Y5 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_1129_inf_Oassoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_1130_inf_Oassoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_1131_inf__assoc,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X @ Y2 ) @ Z )
= ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_1132_inf__assoc,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X @ Y2 ) @ Z )
= ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_1133_inf_Ocommute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_1134_inf_Ocommute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( inf_inf_set_nat_nat @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_1135_inf__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X2: set_set_set_nat,Y5: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y5 @ X2 ) ) ) ).
% inf_commute
thf(fact_1136_inf__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [X2: set_nat_nat,Y5: set_nat_nat] : ( inf_inf_set_nat_nat @ Y5 @ X2 ) ) ) ).
% inf_commute
thf(fact_1137_inf_Oleft__commute,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ B2 @ ( inf_in5711780100303410308et_nat @ A2 @ C ) )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_1138_inf_Oleft__commute,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ B2 @ ( inf_inf_set_nat_nat @ A2 @ C ) )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_1139_inf__left__commute,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X @ ( inf_in5711780100303410308et_nat @ Y2 @ Z ) )
= ( inf_in5711780100303410308et_nat @ Y2 @ ( inf_in5711780100303410308et_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_1140_inf__left__commute,axiom,
! [X: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X @ ( inf_inf_set_nat_nat @ Y2 @ Z ) )
= ( inf_inf_set_nat_nat @ Y2 @ ( inf_inf_set_nat_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_1141_IntE,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) )
=> ~ ( ( member2946998982187404937et_nat @ C @ A )
=> ~ ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% IntE
thf(fact_1142_IntE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ~ ( member_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_1143_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_1144_IntE,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ~ ( ( member_set_set_nat @ C @ A )
=> ~ ( member_set_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_1145_IntE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ~ ( ( member_nat_nat @ C @ A )
=> ~ ( member_nat_nat @ C @ B ) ) ) ).
% IntE
thf(fact_1146_DiffE,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( minus_3113942175840221057et_nat @ A @ B ) )
=> ~ ( ( member2946998982187404937et_nat @ C @ A )
=> ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_1147_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_1148_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_1149_DiffE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
=> ~ ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_1150_DiffE,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
=> ~ ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_1151_IntD1,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ( member_set_set_nat @ C @ A ) ) ).
% IntD1
thf(fact_1152_IntD1,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C @ A ) ) ).
% IntD1
thf(fact_1153_deviate__pos__def,axiom,
! [Phi: monotone_mformula_a] :
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique4708818501384062891C_mf_a @ k @ pi @ Phi ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi ) ) ) ) ).
% deviate_pos_def
thf(fact_1154_deviate__pos__cup__def,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705536850673os_cup @ l @ p @ k @ X3 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y ) ) ) ) ).
% deviate_pos_cup_def
thf(fact_1155_deviate__pos__cap__def,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705535538693os_cap @ l @ p @ k @ X3 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X3 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X3 @ Y ) ) ) ) ).
% deviate_pos_cap_def
thf(fact_1156_ACC__cf__odot,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X3 @ Y ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X3 ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_odot
thf(fact_1157_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_1158_ACC__cf__SET_I4_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ ( monotone_Conj_a @ Phi @ Psi ) ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Phi ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique6509092761774629891_SET_a @ pi @ Psi ) ) ) ) ).
% ACC_cf_SET(4)
thf(fact_1159_Lm,axiom,
ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lm
thf(fact_1160_third__assumptions_OM0,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P @ K )
=> ( ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% third_assumptions.M0
thf(fact_1161_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_1162_third__assumptions_OL0_H,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P @ K )
=> ( ord_less_eq_nat @ assumptions_and_L02 @ L ) ) ).
% third_assumptions.L0'
thf(fact_1163_third__assumptions_OL0,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P @ K )
=> ( ord_less_eq_nat @ assumptions_and_L0 @ L ) ) ).
% third_assumptions.L0
thf(fact_1164_third__assumptions_OM0_H,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P @ K )
=> ( ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% third_assumptions.M0'
thf(fact_1165_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_1166_approx__neg_Opelims,axiom,
! [X: monotone_mformula_a,Y2: set_nat_nat] :
( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ X )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Disj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ( ( X = monotone_TRUE_a )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_TRUE_a ) ) )
=> ( ( ( X = monotone_FALSE_a )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_FALSE_a ) ) )
=> ~ ! [V3: a] :
( ( X
= ( monotone_Var_a @ V3 ) )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Var_a @ V3 ) ) ) ) ) ) ) ) ) ) ).
% approx_neg.pelims
thf(fact_1167_second__assumptions__axioms,axiom,
assump2881078719466019805ptions @ l @ p @ k ).
% second_assumptions_axioms
thf(fact_1168_third__assumptions_Oaxioms_I1_J,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2119784843035796504ptions @ L @ P @ K )
=> ( assump2881078719466019805ptions @ L @ P @ K ) ) ).
% third_assumptions.axioms(1)
thf(fact_1169_second__assumptions_Osqcup,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X3 @ Y ) @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) ) ) ) ) ).
% second_assumptions.sqcup
thf(fact_1170_second__assumptions_Osqcap,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118206219037_sqcap @ L @ P @ K @ X3 @ Y ) @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) ) ) ) ) ).
% second_assumptions.sqcap
thf(fact_1171_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1172_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_1173_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1174_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1175_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1176_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1177_second__assumptions_OLm,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ord_less_eq_nat @ ( assump1710595444109740334irst_m @ K ) @ ( assump1710595444109740301irst_L @ L @ P ) ) ) ).
% second_assumptions.Lm
thf(fact_1178_second__assumptions_Ov__sameprod__subset,axiom,
! [L: nat,P: nat,K: nat,Vs: set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ) ).
% second_assumptions.v_sameprod_subset
thf(fact_1179_second__assumptions_Odeviate__pos__cup,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( clique3314026705536850673os_cup @ L @ P @ K @ X3 @ Y )
= bot_bo7198184520161983622et_nat ) ) ) ) ).
% second_assumptions.deviate_pos_cup
thf(fact_1180_second__assumptions_OPLU__union,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) ) ) ) ) ).
% second_assumptions.PLU_union
thf(fact_1181_second__assumptions_Osqcup__def,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique2586627118207531017_sqcup @ L @ P @ K @ X3 @ Y )
= ( clique2699557479641037314nd_PLU @ L @ P @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ) ) ).
% second_assumptions.sqcup_def
thf(fact_1182_second__assumptions_OPLU__joinl,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P @ K @ ( clique7966186356931407165_odotl @ L @ K @ X3 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) ) ) ) ) ).
% second_assumptions.PLU_joinl
thf(fact_1183_second__assumptions_Osqcap__def,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique2586627118206219037_sqcap @ L @ P @ K @ X3 @ Y )
= ( clique2699557479641037314nd_PLU @ L @ P @ K @ ( clique7966186356931407165_odotl @ L @ K @ X3 @ Y ) ) ) ) ).
% second_assumptions.sqcap_def
thf(fact_1184_second__assumptions_Odeviate__neg__cup__def,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique1591571987439376245eg_cup @ L @ P @ K @ X3 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X3 @ Y ) ) @ ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_neg_cup_def
thf(fact_1185_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1186_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1187_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1188_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1189_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1190_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1191_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1192_second__assumptions_Odeviate__neg__cap__def,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique1591571987438064265eg_cap @ L @ P @ K @ X3 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118206219037_sqcap @ L @ P @ K @ X3 @ Y ) ) @ ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X3 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_neg_cap_def
thf(fact_1193_second__assumptions_Osqcup__sub,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X3 @ Y ) ) ) ) ) ) ).
% second_assumptions.sqcup_sub
thf(fact_1194_second__assumptions_Odeviate__pos__cap__def,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique3314026705535538693os_cap @ L @ P @ K @ X3 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X3 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118206219037_sqcap @ L @ P @ K @ X3 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_pos_cap_def
thf(fact_1195_second__assumptions_Odeviate__pos__cup__def,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique3314026705536850673os_cup @ L @ P @ K @ X3 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X3 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_pos_cup_def
thf(fact_1196_approx__pos_Opelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ X )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ( ( X = monotone_TRUE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_TRUE_a ) ) )
=> ( ( ( X = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [V3: a] :
( ( X
= ( monotone_Var_a @ V3 ) )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Var_a @ V3 ) ) ) )
=> ~ ! [V3: monotone_mformula_a,Va2: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ V3 @ Va2 ) )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Disj_a @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% approx_pos.pelims
thf(fact_1197_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_1198_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_1199_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_1200_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_1201_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_1202_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_1203_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1204_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1205_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1206_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1207_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1208_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1209_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1210_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1211_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_1212_plucking__step_I3_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ X3 ) ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ) ) ).
% plucking_step(3)
thf(fact_1213__092_060pi_062__singleton_I1_J,axiom,
! [X: a] :
( ( member_a @ X @ v )
=> ( member_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ).
% \<pi>_singleton(1)
thf(fact_1214_plucking__step_I2_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ Y @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ) ) ).
% plucking_step(2)
thf(fact_1215_plucking__step_I5_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X3 ) )
=> ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ).
% plucking_step(5)
thf(fact_1216_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_1217_second__assumptions_Oplucking__step_I2_J,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ Y @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ) ) ).
% second_assumptions.plucking_step(2)
thf(fact_1218_second__assumptions_Oplucking__step_I5_J,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P @ X3 ) )
=> ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% second_assumptions.plucking_step(5)
thf(fact_1219_second__assumptions_Oplucking__step_I3_J,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ X3 ) ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ) ) ).
% second_assumptions.plucking_step(3)
thf(fact_1220_APR_Osimps_I2_J,axiom,
! [X: a] :
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).
% APR.simps(2)
thf(fact_1221_card__v__gs__join,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat,Z4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z4 @ ( clique5469973757772500719t_odot @ X3 @ Y ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z4 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ) ) ) ).
% card_v_gs_join
thf(fact_1222_SET_Osimps_I2_J,axiom,
! [X: a] :
( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Var_a @ X ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).
% SET.simps(2)
thf(fact_1223_card__numbers,axiom,
! [N: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N ) )
= N ) ).
% card_numbers
thf(fact_1224__092_060pi_062__singleton_I2_J,axiom,
! [X: a] :
( ( member_a @ X @ v )
=> ( member2946998982187404937et_nat @ ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ).
% \<pi>_singleton(2)
thf(fact_1225_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1226_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1227_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_1228_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_1229_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_1230_APR_Oelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ( ( X = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ! [X4: a] :
( ( X
= ( monotone_Var_a @ X4 ) )
=> ( Y2
!= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
=> ( Y2
!= ( clique2586627118207531017_sqcup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( Y2
!= ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) ) )
=> ~ ( ( X = monotone_TRUE_a )
=> ( Y2 != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ).
% APR.elims
thf(fact_1231_SET_Oelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique6509092761774629891_SET_a @ pi @ X )
= Y2 )
=> ( ( ( X = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ! [X4: a] :
( ( X
= ( monotone_Var_a @ X4 ) )
=> ( Y2
!= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
=> ( Y2
!= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi4 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi4 ) ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( Y2
!= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi4 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi4 ) ) ) )
=> ~ ( ( X = monotone_TRUE_a )
=> ( Y2 != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ).
% SET.elims
thf(fact_1232_APR_Opelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ X )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ X )
=> ( ( ( X = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X4: a] :
( ( X
= ( monotone_Var_a @ X4 ) )
=> ( ( Y2
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Var_a @ X4 ) ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
=> ( ( Y2
= ( clique2586627118207531017_sqcup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Disj_a @ Phi4 @ Psi4 ) ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( ( Y2
= ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Conj_a @ Phi4 @ Psi4 ) ) ) )
=> ~ ( ( X = monotone_TRUE_a )
=> ( ( Y2 = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% APR.pelims
thf(fact_1233_SET_Opelims,axiom,
! [X: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique6509092761774629891_SET_a @ pi @ X )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ X )
=> ( ( ( X = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X4: a] :
( ( X
= ( monotone_Var_a @ X4 ) )
=> ( ( Y2
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Var_a @ X4 ) ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
=> ( ( Y2
= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi4 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi4 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Disj_a @ Phi4 @ Psi4 ) ) ) )
=> ( ! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( X
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
=> ( ( Y2
= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi4 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi4 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Conj_a @ Phi4 @ Psi4 ) ) ) )
=> ~ ( ( X = monotone_TRUE_a )
=> ( ( Y2 = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% SET.pelims
thf(fact_1234_PLU__main_Opinduct,axiom,
! [A0: set_set_set_nat,P2: set_set_set_nat > $o] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k ) @ A0 )
=> ( ! [X7: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k ) @ X7 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X7 @ ( clique7840962075309931874st_G_l @ l @ k ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X7 ) ) ) )
=> ( P2 @ ( clique4095374090462327202g_step @ p @ X7 ) ) )
=> ( P2 @ X7 ) ) )
=> ( P2 @ A0 ) ) ) ).
% PLU_main.pinduct
thf(fact_1235_second__assumptions_OPLU__main_Opinduct,axiom,
! [L: nat,P: nat,K: nat,A0: set_set_set_nat,P2: set_set_set_nat > $o] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P @ K ) @ A0 )
=> ( ! [X7: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P @ K ) @ X7 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X7 @ ( clique7840962075309931874st_G_l @ L @ K ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X7 ) ) ) )
=> ( P2 @ ( clique4095374090462327202g_step @ P @ X7 ) ) )
=> ( P2 @ X7 ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% second_assumptions.PLU_main.pinduct
thf(fact_1236_ACC__cf__I,axiom,
! [F2: nat > nat,X3: set_set_set_nat] :
( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ k ) )
=> ( ( clique3686358387679108662ccepts @ X3 @ ( clique5033774636164728462irst_C @ k @ F2 ) )
=> ( member_nat_nat @ F2 @ ( clique951075384711337423ACC_cf @ k @ X3 ) ) ) ) ).
% ACC_cf_I
thf(fact_1237_plucking__step_I1_J,axiom,
! [X3: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X3 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) @ p ) @ one_one_nat ) ) ) ) ) ).
% plucking_step(1)
thf(fact_1238_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1239_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1240_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1241_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1242_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1243_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1244_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1245_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1246_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1247_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1248_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1249_add__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 @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1250_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1251_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1252_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1253_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1254_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1255_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1256_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1257_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1258_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1259_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1260_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1261_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1262_second__assumptions_Oplucking__step_I1_J,axiom,
! [L: nat,P: nat,K: nat,X3: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P @ X3 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) @ P ) @ one_one_nat ) ) ) ) ) ) ).
% second_assumptions.plucking_step(1)
thf(fact_1263_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1264_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1265_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1266_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1267_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1268_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1269_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1270_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1271_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_1272_PLU__main__n,axiom,
! [X3: set_set_set_nat,Z4: set_set_set_nat,N: nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ( clique429652266423863867U_main @ l @ p @ k @ X3 )
= ( produc2803780273060847707at_nat @ Z4 @ N ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ p @ one_one_nat ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) ) ) ) ).
% PLU_main_n
thf(fact_1273_second__assumptions_OPLU__main_Ocong,axiom,
clique429652266423863867U_main = clique429652266423863867U_main ).
% second_assumptions.PLU_main.cong
% Conjectures (1)
thf(conj_0,conjecture,
( ( monotone_eval_a @ ( clique3148831351753978868ta_g_a @ v @ pi @ g ) @ phi )
= ( member_set_set_nat @ g @ ( clique4708818501384062891C_mf_a @ k @ pi @ phi ) ) ) ).
%------------------------------------------------------------------------------