TPTP Problem File: SLH0192^1.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_01653_063281__16389620_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1542 ( 777 unt; 264 typ; 0 def)
% Number of atoms : 3173 (1415 equ; 0 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 10423 ( 270 ~; 20 |; 203 &;8813 @)
% ( 0 <=>;1117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 1209 (1209 >; 0 *; 0 +; 0 <<)
% Number of symbols : 247 ( 246 usr; 24 con; 0-5 aty)
% Number of variables : 3319 ( 168 ^;3090 !; 61 ?;3319 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:50:44.143
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
set_se7970953024979822686et_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_M_Eo_J_J,type,
set_se8611727395572922045_nat_o: $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__Set__Oset_I_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
set_set_set_nat_o: $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__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
set_nat_nat_o: $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__Set__Oset_It__Nat__Onat_J_M_Eo_J_J,type,
set_set_nat_o: $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__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_M_Eo_J_J,type,
set_a_o: $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 (246)
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,type,
assump5453534214990993103ptions: nat > nat > nat > $o ).
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_Binomial_Obinomial,type,
binomial: nat > 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_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_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_001tf__a,type,
clique5987991184601036204th_A_a: set_a > set_Mo2626137824023173004mula_a ).
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__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_Complete__Lattices_OSup__class_OSup_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
comple8312177224774716605_nat_o: set_nat_nat_o > ( nat > nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
comple3806919086088850358_nat_o: set_set_nat_o > set_nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
comple2068028038703680896_nat_o: set_set_set_nat_o > set_set_nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_M_Eo_J,type,
comple8839946721722594890_nat_o: set_se8611727395572922045_nat_o > set_set_set_nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_Itf__a_M_Eo_J,type,
complete_Sup_Sup_a_o: set_a_o > a > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
comple5448282615319421384at_nat: set_set_nat_nat > set_nat_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
comple548664676211718543et_nat: set_set_set_nat > set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
comple6569609367425551173et_nat: set_set_set_set_nat > set_set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
comple5789376832584316411et_nat: set_se7970953024979822686et_nat > set_set_set_set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Finite__Set_OFpow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite_Fpow_nat_nat: set_nat_nat > set_set_nat_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Nat__Onat_J,type,
finite_Fpow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite7717622420921165910et_nat: set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
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_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1149291290879098388et_nat: set_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__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_Fun_Oinj__on_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on4164537515518332398et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_nat_set_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on8105003582846801791et_nat: ( nat > set_set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
inj_on4976969725211833178et_nat: ( set_nat_nat > set_set_set_nat ) > set_set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on2776966659131765557et_nat: ( set_nat > set_set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001tf__a,type,
inj_on_set_nat_a: ( set_nat > a ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on3419524245016971886at_nat: ( set_set_nat > nat > nat ) > set_set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on1894729867836481333et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001tf__a,type,
inj_on_set_set_nat_a: ( set_set_nat > a ) > set_set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on4386985374303630753et_nat: ( set_set_set_nat > set_set_nat ) > set_set_set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on4942881440122211049et_nat: ( set_a > set_set_nat ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on_a_nat_nat: ( a > nat > nat ) > 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_Fun_Oinj__on_001tf__a_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inj_on_a_set_set_nat: ( a > set_set_nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Fun_Othe__inv__into_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
the_in6552357484096881712et_nat: set_nat_nat > ( ( nat > nat ) > set_set_nat ) > set_set_nat > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
the_in5057678521256355851et_nat: set_nat > ( nat > set_nat ) > set_nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
the_in7111554173081589953et_nat: set_nat > ( nat > set_set_nat ) > set_set_nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001tf__a,type,
the_inv_into_nat_a: set_nat > ( nat > a ) > a > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
the_in3610957794094371777et_nat: set_set_nat > ( set_nat > set_nat ) > set_nat > set_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
the_in4457553895174990967et_nat: set_set_nat > ( set_nat > set_set_nat ) > set_set_nat > set_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_It__Nat__Onat_J_001tf__a,type,
the_in1235728082690079619_nat_a: set_set_nat > ( set_nat > a ) > a > set_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_in5807344213595521200at_nat: set_set_set_nat > ( set_set_nat > nat > nat ) > ( nat > nat ) > set_set_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
the_in3575317103879706743et_nat: set_set_set_nat > ( set_set_nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001tf__a,type,
the_in2240265568404529229_nat_a: set_set_set_nat > ( set_set_nat > a ) > a > set_set_nat ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_in1748411866455846218at_nat: set_a > ( a > nat > nat ) > ( nat > nat ) > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
the_in5098273967424113681et_nat: set_a > ( a > set_nat ) > set_nat > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
the_in1847442007992722631et_nat: set_a > ( a > set_set_nat ) > set_set_nat > a ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001tf__a,type,
the_inv_into_a_a: set_a > ( a > a ) > a > a ).
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__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_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_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inf_inf_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > 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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in710756014367367485at_nat: set_set_nat_nat > set_set_nat_nat > set_set_nat_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
sup_sup_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su2553808219797728471at_nat: set_set_nat_nat > set_set_nat_nat > set_set_nat_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_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_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_001_062_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_M_Eo_J,type,
bot_bo5536612546450143305_nat_o: set_set_set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
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_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_Obind_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
bind_nat_nat_nat: set_nat_nat > ( ( nat > nat ) > set_nat ) > set_nat ).
thf(sy_c_Set_Obind_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bind_nat_nat_set_nat: set_nat_nat > ( ( nat > nat ) > set_set_nat ) > set_set_nat ).
thf(sy_c_Set_Obind_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
bind_nat_nat_nat2: set_nat > ( nat > set_nat_nat ) > set_nat_nat ).
thf(sy_c_Set_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: set_nat > ( nat > set_nat ) > set_nat ).
thf(sy_c_Set_Obind_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
bind_nat_set_nat: set_nat > ( nat > set_set_nat ) > set_set_nat ).
thf(sy_c_Set_Obind_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bind_nat_set_set_nat: set_nat > ( nat > set_set_set_nat ) > set_set_set_nat ).
thf(sy_c_Set_Obind_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
bind_set_nat_nat: set_set_nat > ( set_nat > set_nat ) > set_nat ).
thf(sy_c_Set_Obind_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bind_set_nat_set_nat: set_set_nat > ( set_nat > set_set_nat ) > set_set_nat ).
thf(sy_c_Set_Obind_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bind_s4332039312244270231et_nat: set_set_nat > ( set_nat > set_set_set_nat ) > set_set_set_nat ).
thf(sy_c_Set_Obind_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
bind_set_set_nat_nat: set_set_set_nat > ( set_set_nat > set_nat ) > set_nat ).
thf(sy_c_Set_Obind_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
bind_s3449802520948986007et_nat: set_set_set_nat > ( set_set_nat > set_set_nat ) > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7977807581451749376at_nat: ( ( ( nat > nat ) > $o ) > set_nat_nat ) > set_nat_nat_o > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_3205354838064109189at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7432509271690132940et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > set_set_nat ).
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_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001tf__a,type,
image_nat_nat_a: ( ( nat > nat ) > a ) > set_nat_nat > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_4687162037615663680et_nat: ( ( set_nat > $o ) > set_set_nat ) > set_set_nat_o > set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_3164711303094801856et_nat: ( ( set_set_nat > $o ) > set_set_set_nat ) > set_set_set_nat_o > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_M_Eo_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
image_3156746147002188096et_nat: ( ( set_set_set_nat > $o ) > set_set_set_set_nat ) > set_se8611727395572922045_nat_o > set_se7970953024979822686et_nat ).
thf(sy_c_Set_Oimage_001_062_Itf__a_M_Eo_J_001t__Set__Oset_Itf__a_J,type,
image_a_o_set_a: ( ( a > $o ) > set_a ) > set_a_o > set_set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_7301343469591561292at_nat: ( nat > set_nat_nat ) > set_nat > set_set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_5738044413236618185et_nat: ( nat > set_set_set_nat ) > set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_8569768528772619084at_nat: ( set_nat > nat > nat ) > set_set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_4583741654806091647et_nat: ( set_nat > set_set_set_nat ) > set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001tf__a,type,
image_set_nat_a: ( set_nat > a ) > set_set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_8441894408526374658at_nat: ( set_set_nat > nat > nat ) > set_set_set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
image_1454916318497077779at_nat: ( set_set_nat > nat ) > set_set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001tf__a,type,
image_set_set_nat_a: ( set_set_nat > a ) > set_set_set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7149431738526707583et_nat: ( set_set_set_nat > set_nat ) > set_set_set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2225960715480453173et_nat: ( set_set_set_nat > set_set_nat ) > set_set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001tf__a,type,
image_3422112407882505029_nat_a: ( set_set_set_nat > a ) > set_set_set_set_nat > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_a_nat_nat: ( a > nat > nat ) > set_a > set_nat_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
image_a_set_nat: ( a > set_nat ) > set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_a_set_set_nat: ( a > set_set_nat ) > set_a > set_set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
insert_set_nat_nat: set_nat_nat > set_set_nat_nat > set_set_nat_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_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
insert3687027775829606434et_nat: set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
is_empty_nat_nat: set_nat_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__Nat__Onat_J,type,
is_empty_set_nat: set_set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
is_empty_set_set_nat: set_set_set_nat > $o ).
thf(sy_c_Set_Othe__elem_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
the_elem_nat_nat: set_nat_nat > nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
the_elem_set_set_nat: set_set_set_nat > 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_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_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
member3774042032884853055et_nat: set_set_set_set_nat > set_se7970953024979822686et_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_D____,type,
d: set_set_nat ).
thf(sy_v_E____,type,
e: set_set_nat ).
thf(sy_v_X,type,
x: set_set_set_nat ).
thf(sy_v_Y,type,
y: set_set_set_nat ).
thf(sy_v__092_060V_062,type,
v: set_a ).
thf(sy_v__092_060pi_062,type,
pi: a > set_nat ).
thf(sy_v__092_060theta_062,type,
theta: a > $o ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1277)
thf(fact_0_DE_I2_J,axiom,
member_set_set_nat @ e @ y ).
% DE(2)
thf(fact_1_DE_I1_J,axiom,
member_set_set_nat @ d @ x ).
% DE(1)
thf(fact_2_eval_I2_J,axiom,
clique5859573001277246426al_g_a @ v @ pi @ theta @ e ).
% eval(2)
thf(fact_3_eval_I1_J,axiom,
clique5859573001277246426al_g_a @ v @ pi @ theta @ d ).
% eval(1)
thf(fact_4_eval__g__def,axiom,
! [Theta: a > $o,G: set_set_nat] :
( ( clique5859573001277246426al_g_a @ v @ pi @ Theta @ G )
= ( ! [X: a] :
( ( member_a @ X @ v )
=> ( ( member_set_nat @ ( pi @ X ) @ G )
=> ( Theta @ X ) ) ) ) ) ).
% eval_g_def
thf(fact_5__092_060open_062eval__gs_A_092_060theta_062_AX_A_092_060and_062_Aeval__gs_A_092_060theta_062_AY_092_060close_062,axiom,
( ( clique835570645587132141l_gs_a @ v @ pi @ theta @ x )
& ( clique835570645587132141l_gs_a @ v @ pi @ theta @ y ) ) ).
% \<open>eval_gs \<theta> X \<and> eval_gs \<theta> Y\<close>
thf(fact_6_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_7_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_8_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_9_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_10_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_11_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_12_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_13_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_14_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_15_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_16_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_17_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_18_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_19_sup_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_20_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_21_sup_Oidem,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_22_sup__idem,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_23_sup__idem,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_24_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_25_sup__idem,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_26_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_27_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_28_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_29_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_30_sup__left__idem,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y ) )
= ( sup_sup_set_set_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_31_sup__left__idem,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_32_sup__left__idem,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) )
= ( sup_sup_set_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_33_sup__left__idem,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y ) )
= ( sup_sup_set_nat_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_34_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_35_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_36_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_37_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_38_inj__on___092_060pi_062,axiom,
inj_on_a_set_nat @ pi @ v ).
% inj_on_\<pi>
thf(fact_39_eval__gs__def,axiom,
! [Theta: a > $o,X3: set_set_set_nat] :
( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ X3 )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ X3 )
& ( clique5859573001277246426al_g_a @ v @ pi @ Theta @ X ) ) ) ) ).
% eval_gs_def
thf(fact_40__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062D_AE_O_A_092_060lbrakk_062D_A_092_060in_062_AX_059_AE_A_092_060in_062_AY_059_Aeval__g_A_092_060theta_062_AD_059_Aeval__g_A_092_060theta_062_AE_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [D: set_set_nat] :
( ( member_set_set_nat @ D @ x )
=> ! [E: set_set_nat] :
( ( member_set_set_nat @ E @ y )
=> ( ( clique5859573001277246426al_g_a @ v @ pi @ theta @ D )
=> ~ ( clique5859573001277246426al_g_a @ v @ pi @ theta @ E ) ) ) ) ).
% \<open>\<And>thesis. (\<And>D E. \<lbrakk>D \<in> X; E \<in> Y; eval_g \<theta> D; eval_g \<theta> E\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_41_eval__gs__union,axiom,
! [Theta: a > $o,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) )
= ( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ X3 )
| ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ Y2 ) ) ) ).
% eval_gs_union
thf(fact_42_forth__assumptions_Oeval__gs_Ocong,axiom,
clique835570645587132141l_gs_a = clique835570645587132141l_gs_a ).
% forth_assumptions.eval_gs.cong
thf(fact_43_forth__assumptions_Oeval__g_Ocong,axiom,
clique5859573001277246426al_g_a = clique5859573001277246426al_g_a ).
% forth_assumptions.eval_g.cong
thf(fact_44_sup__left__commute,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_45_sup__left__commute,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_46_sup__left__commute,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_47_sup__left__commute,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z ) )
= ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_48_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_49_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_50_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_51_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_52_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X: set_set_nat,Y3: set_set_nat] : ( sup_sup_set_set_nat @ Y3 @ X ) ) ) ).
% sup_commute
thf(fact_53_sup__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X: set_set_set_nat,Y3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y3 @ X ) ) ) ).
% sup_commute
thf(fact_54_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X ) ) ) ).
% sup_commute
thf(fact_55_sup__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [X: set_nat_nat,Y3: set_nat_nat] : ( sup_sup_set_nat_nat @ Y3 @ X ) ) ) ).
% sup_commute
thf(fact_56_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] : ( sup_sup_set_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_57_sup_Ocommute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_58_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_59_sup_Ocommute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] : ( sup_sup_set_nat_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_60_sup__assoc,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ Z )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_61_sup__assoc,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ Z )
= ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_62_sup__assoc,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_63_sup__assoc,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ Z )
= ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_64_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_65_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_66_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_67_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_68_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X: set_set_nat,Y3: set_set_nat] : ( sup_sup_set_set_nat @ Y3 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_69_inf__sup__aci_I5_J,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X: set_set_set_nat,Y3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y3 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_70_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_71_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat_nat
= ( ^ [X: set_nat_nat,Y3: set_nat_nat] : ( sup_sup_set_nat_nat @ Y3 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_72_inf__sup__aci_I6_J,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ Z )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_73_inf__sup__aci_I6_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ Z )
= ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_74_inf__sup__aci_I6_J,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_75_inf__sup__aci_I6_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ Z )
= ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_76_inf__sup__aci_I7_J,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_77_inf__sup__aci_I7_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_78_inf__sup__aci_I7_J,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_79_inf__sup__aci_I7_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z ) )
= ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_80_inf__sup__aci_I8_J,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y ) )
= ( sup_sup_set_set_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_81_inf__sup__aci_I8_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_82_inf__sup__aci_I8_J,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) )
= ( sup_sup_set_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_83_inf__sup__aci_I8_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y ) )
= ( sup_sup_set_nat_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] : ( sup_sup_set_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_93_Un__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_94_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_95_Un__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( sup_sup_set_nat_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_96_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_97_Un__absorb,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_98_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_99_Un__absorb,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_100_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_101_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_102_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_103_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_104_ball__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( P @ X ) )
& ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_105_ball__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( P @ X ) )
& ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_106_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( P @ X ) )
& ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_107_ball__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A )
=> ( P @ X ) )
& ! [X: nat > nat] :
( ( member_nat_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_108_bex__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ? [X: set_nat] :
( ( member_set_nat @ X @ ( sup_sup_set_set_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( P @ X ) )
| ? [X: set_nat] :
( ( member_set_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_109_bex__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ ( sup_su4213647025997063966et_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
& ( P @ X ) )
| ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_110_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) )
| ? [X: nat] :
( ( member_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_111_bex__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ? [X: nat > nat] :
( ( member_nat_nat @ X @ ( sup_sup_set_nat_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ X ) )
| ? [X: nat > nat] :
( ( member_nat_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_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_120_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_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_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_129_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_130_mem__Collect__eq,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A2 @ ( collect_set_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_131_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_132_mem__Collect__eq,axiom,
! [A2: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_133_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( member2946998982187404937et_nat @ A2 @ ( collec7201453139178570183et_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X: set_set_nat] : ( member_set_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X: nat > nat] : ( member_nat_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A: set_set_set_set_nat] :
( ( collec7201453139178570183et_nat
@ ^ [X: set_set_set_nat] : ( member2946998982187404937et_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_140_Collect__cong,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X4: set_set_set_nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec7201453139178570183et_nat @ P )
= ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_141_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_142_v__gs__union,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ).
% v_gs_union
thf(fact_143_inj__onD,axiom,
! [F: a > set_nat,A: set_a,X2: a,Y: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_a @ X2 @ A )
=> ( ( member_a @ Y @ A )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_144_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_145_inj__on__def,axiom,
( inj_on_a_set_nat
= ( ^ [F2: a > set_nat,A4: set_a] :
! [X: a] :
( ( member_a @ X @ A4 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ A4 )
=> ( ( ( F2 @ X )
= ( F2 @ Y3 ) )
=> ( X = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_146_inj__on__cong,axiom,
! [A: set_a,F: a > set_nat,G2: a > set_nat] :
( ! [A5: a] :
( ( member_a @ A5 @ A )
=> ( ( F @ A5 )
= ( G2 @ A5 ) ) )
=> ( ( inj_on_a_set_nat @ F @ A )
= ( inj_on_a_set_nat @ G2 @ A ) ) ) ).
% inj_on_cong
thf(fact_147_inj__on__eq__iff,axiom,
! [F: a > set_nat,A: set_a,X2: a,Y: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( member_a @ X2 @ A )
=> ( ( member_a @ Y @ A )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_148_inj__on__contraD,axiom,
! [F: a > set_nat,A: set_a,X2: a,Y: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( X2 != Y )
=> ( ( member_a @ X2 @ A )
=> ( ( member_a @ Y @ A )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_149_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_150_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_nat,K: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( B
= ( sup_sup_set_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_set_nat @ A2 @ B )
= ( sup_sup_set_set_nat @ K @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_151_boolean__algebra__cancel_Osup2,axiom,
! [B: set_set_set_nat,K: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( B
= ( sup_su4213647025997063966et_nat @ K @ B2 ) )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B )
= ( sup_su4213647025997063966et_nat @ K @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_152_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_153_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat_nat,K: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( B
= ( sup_sup_set_nat_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat_nat @ A2 @ B )
= ( sup_sup_set_nat_nat @ K @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_154_boolean__algebra__cancel_Osup1,axiom,
! [A: set_set_nat,K: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( A
= ( sup_sup_set_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_set_nat @ A @ B2 )
= ( sup_sup_set_set_nat @ K @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_155_boolean__algebra__cancel_Osup1,axiom,
! [A: set_set_set_nat,K: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A
= ( sup_su4213647025997063966et_nat @ K @ A2 ) )
=> ( ( sup_su4213647025997063966et_nat @ A @ B2 )
= ( sup_su4213647025997063966et_nat @ K @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_156_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_157_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat_nat,K: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( A
= ( sup_sup_set_nat_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat_nat @ A @ B2 )
= ( sup_sup_set_nat_nat @ K @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_158_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_159_first__assumptions_Ov__union,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_union
thf(fact_160_first__assumptions_Ov__gs__union,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ).
% first_assumptions.v_gs_union
thf(fact_161_forth__assumptions_Oeval__gs__def,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o,X3: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ X3 )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ X3 )
& ( clique5859573001277246426al_g_a @ V @ Pi @ Theta @ X ) ) ) ) ) ).
% forth_assumptions.eval_gs_def
thf(fact_162_forth__assumptions_Oeval__gs__union,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) )
= ( ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ X3 )
| ( clique835570645587132141l_gs_a @ V @ Pi @ Theta @ Y2 ) ) ) ) ).
% forth_assumptions.eval_gs_union
thf(fact_163_inj__on__Un__image__eq__iff,axiom,
! [F: a > set_nat,A: set_a,B: set_a] :
( ( inj_on_a_set_nat @ F @ ( sup_sup_set_a @ A @ B ) )
=> ( ( ( image_a_set_nat @ F @ A )
= ( image_a_set_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_164_inj__on__Un__image__eq__iff,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( ( ( image_5842784325960735177et_nat @ F @ A )
= ( image_5842784325960735177et_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_165_inj__on__Un__image__eq__iff,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,B: set_nat_nat] :
( ( inj_on4164537515518332398et_nat @ F @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( ( ( image_9186907679027735170et_nat @ F @ A )
= ( image_9186907679027735170et_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_166_forth__assumptions_Oinj__on___092_060pi_062,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( inj_on_a_set_nat @ Pi @ V ) ) ).
% forth_assumptions.inj_on_\<pi>
thf(fact_167_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_168_Union__Un__distrib,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( comple6569609367425551173et_nat @ ( sup_su3906748206781935060et_nat @ A @ B ) )
= ( sup_su4213647025997063966et_nat @ ( comple6569609367425551173et_nat @ A ) @ ( comple6569609367425551173et_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_169_Union__Un__distrib,axiom,
! [A: set_set_nat_nat,B: set_set_nat_nat] :
( ( comple5448282615319421384at_nat @ ( sup_su2553808219797728471at_nat @ A @ B ) )
= ( sup_sup_set_nat_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_170_Union__Un__distrib,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_171_Union__Un__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( comple548664676211718543et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_172_inj__on__empty,axiom,
! [F: a > set_nat] : ( inj_on_a_set_nat @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_173_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_174_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_175_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_176_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_177_image__eqI,axiom,
! [B2: a,F: a > a,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_178_image__eqI,axiom,
! [B2: a,F: set_nat > a,X2: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_a @ B2 @ ( image_set_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_179_image__eqI,axiom,
! [B2: set_nat,F: a > set_nat,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_set_nat @ B2 @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_180_image__eqI,axiom,
! [B2: a,F: set_set_nat > a,X2: set_set_nat,A: set_set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_set_set_nat @ X2 @ A )
=> ( member_a @ B2 @ ( image_set_set_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_181_image__eqI,axiom,
! [B2: set_nat,F: set_nat > set_nat,X2: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_182_image__eqI,axiom,
! [B2: a,F: ( nat > nat ) > a,X2: nat > nat,A: set_nat_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_a @ B2 @ ( image_nat_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_183_image__eqI,axiom,
! [B2: set_set_nat,F: a > set_set_nat,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_set_set_nat @ B2 @ ( image_a_set_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_184_image__eqI,axiom,
! [B2: nat > nat,F: a > nat > nat,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_nat_nat @ B2 @ ( image_a_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_185_image__eqI,axiom,
! [B2: a,F: set_set_set_nat > a,X2: set_set_set_nat,A: set_set_set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member2946998982187404937et_nat @ X2 @ A )
=> ( member_a @ B2 @ ( image_3422112407882505029_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_186_image__eqI,axiom,
! [B2: set_nat,F: set_set_nat > set_nat,X2: set_set_nat,A: set_set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_set_set_nat @ X2 @ A )
=> ( member_set_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_187_empty__Collect__eq,axiom,
! [P: set_set_set_nat > $o] :
( ( bot_bo193956671110832956et_nat
= ( collec7201453139178570183et_nat @ P ) )
= ( ! [X: set_set_set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_188_empty__Collect__eq,axiom,
! [P: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P ) )
= ( ! [X: set_set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_189_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_190_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_191_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X: nat > nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_192_Collect__empty__eq,axiom,
! [P: set_set_set_nat > $o] :
( ( ( collec7201453139178570183et_nat @ P )
= bot_bo193956671110832956et_nat )
= ( ! [X: set_set_set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_193_Collect__empty__eq,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( ! [X: set_set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_194_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_195_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_196_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X: nat > nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_197_all__not__in__conv,axiom,
! [A: set_set_set_set_nat] :
( ( ! [X: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ X @ A ) )
= ( A = bot_bo193956671110832956et_nat ) ) ).
% all_not_in_conv
thf(fact_198_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_199_all__not__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ! [X: set_set_nat] :
~ ( member_set_set_nat @ X @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_200_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X: set_nat] :
~ ( member_set_nat @ X @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_201_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_202_all__not__in__conv,axiom,
! [A: set_nat_nat] :
( ( ! [X: nat > nat] :
~ ( member_nat_nat @ X @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_203_empty__iff,axiom,
! [C: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ C @ bot_bo193956671110832956et_nat ) ).
% empty_iff
thf(fact_204_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_205_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_206_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_207_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_208_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_209_UnionI,axiom,
! [X3: set_set_set_set_nat,C2: set_se7970953024979822686et_nat,A: set_set_set_nat] :
( ( member3774042032884853055et_nat @ X3 @ C2 )
=> ( ( member2946998982187404937et_nat @ A @ X3 )
=> ( member2946998982187404937et_nat @ A @ ( comple5789376832584316411et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_210_UnionI,axiom,
! [X3: set_nat_nat,C2: set_set_nat_nat,A: nat > nat] :
( ( member_set_nat_nat @ X3 @ C2 )
=> ( ( member_nat_nat @ A @ X3 )
=> ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_211_UnionI,axiom,
! [X3: set_a,C2: set_set_a,A: a] :
( ( member_set_a @ X3 @ C2 )
=> ( ( member_a @ A @ X3 )
=> ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_212_UnionI,axiom,
! [X3: set_set_set_nat,C2: set_set_set_set_nat,A: set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ C2 )
=> ( ( member_set_set_nat @ A @ X3 )
=> ( member_set_set_nat @ A @ ( comple6569609367425551173et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_213_UnionI,axiom,
! [X3: set_set_nat,C2: set_set_set_nat,A: set_nat] :
( ( member_set_set_nat @ X3 @ C2 )
=> ( ( member_set_nat @ A @ X3 )
=> ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_214_UnionI,axiom,
! [X3: set_nat,C2: set_set_nat,A: nat] :
( ( member_set_nat @ X3 @ C2 )
=> ( ( member_nat @ A @ X3 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) ) ) ) ).
% UnionI
thf(fact_215_Union__iff,axiom,
! [A: set_set_set_nat,C2: set_se7970953024979822686et_nat] :
( ( member2946998982187404937et_nat @ A @ ( comple5789376832584316411et_nat @ C2 ) )
= ( ? [X: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ X @ C2 )
& ( member2946998982187404937et_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_216_Union__iff,axiom,
! [A: set_set_nat,C2: set_set_set_set_nat] :
( ( member_set_set_nat @ A @ ( comple6569609367425551173et_nat @ C2 ) )
= ( ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ C2 )
& ( member_set_set_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_217_Union__iff,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ C2 )
& ( member_set_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_218_Union__iff,axiom,
! [A: nat > nat,C2: set_set_nat_nat] :
( ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C2 ) )
= ( ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ C2 )
& ( member_nat_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_219_Union__iff,axiom,
! [A: a,C2: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) )
= ( ? [X: set_a] :
( ( member_set_a @ X @ C2 )
& ( member_a @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_220_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_221_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_222_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_223_image__is__empty,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( ( image_nat_set_nat @ F @ A )
= bot_bot_set_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_224_image__is__empty,axiom,
! [F: set_nat > nat,A: set_set_nat] :
( ( ( image_set_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_225_image__is__empty,axiom,
! [F: nat > set_set_nat,A: set_nat] :
( ( ( image_2194112158459175443et_nat @ F @ A )
= bot_bo7198184520161983622et_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_226_image__is__empty,axiom,
! [F: set_nat > set_nat,A: set_set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ A )
= bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_227_image__is__empty,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat] :
( ( ( image_1454916318497077779at_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_228_image__is__empty,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( ( image_nat_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_229_image__is__empty,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( ( image_nat_nat_nat2 @ F @ A )
= bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_230_image__is__empty,axiom,
! [F: set_nat > set_set_nat,A: set_set_nat] :
( ( ( image_6725021117256019401et_nat @ F @ A )
= bot_bo7198184520161983622et_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_231_image__is__empty,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( ( image_5842784325960735177et_nat @ F @ A )
= bot_bot_set_set_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_232_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_233_empty__is__image,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_234_empty__is__image,axiom,
! [F: set_nat > nat,A: set_set_nat] :
( ( bot_bot_set_nat
= ( image_set_nat_nat @ F @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_235_empty__is__image,axiom,
! [F: nat > set_set_nat,A: set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( image_2194112158459175443et_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_236_empty__is__image,axiom,
! [F: set_nat > set_nat,A: set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_7916887816326733075et_nat @ F @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_237_empty__is__image,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat] :
( ( bot_bot_set_nat
= ( image_1454916318497077779at_nat @ F @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_238_empty__is__image,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_239_empty__is__image,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( bot_bot_set_nat_nat
= ( image_nat_nat_nat2 @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_240_empty__is__image,axiom,
! [F: set_nat > set_set_nat,A: set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( image_6725021117256019401et_nat @ F @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_241_empty__is__image,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_5842784325960735177et_nat @ F @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_242_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_243_image__empty,axiom,
! [F: set_nat > nat] :
( ( image_set_nat_nat @ F @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_244_image__empty,axiom,
! [F: nat > set_nat] :
( ( image_nat_set_nat @ F @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_245_image__empty,axiom,
! [F: set_set_nat > nat] :
( ( image_1454916318497077779at_nat @ F @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_246_image__empty,axiom,
! [F: set_nat > set_nat] :
( ( image_7916887816326733075et_nat @ F @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_247_image__empty,axiom,
! [F: nat > set_set_nat] :
( ( image_2194112158459175443et_nat @ F @ bot_bot_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_248_image__empty,axiom,
! [F: nat > nat > nat] :
( ( image_nat_nat_nat2 @ F @ bot_bot_set_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_249_image__empty,axiom,
! [F: ( nat > nat ) > nat] :
( ( image_nat_nat_nat @ F @ bot_bot_set_nat_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_250_image__empty,axiom,
! [F: set_set_nat > set_nat] :
( ( image_5842784325960735177et_nat @ F @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_251_image__empty,axiom,
! [F: set_nat > set_set_nat] :
( ( image_6725021117256019401et_nat @ F @ bot_bot_set_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_252_Sup__bot__conv_I2_J,axiom,
! [A: set_set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( comple6569609367425551173et_nat @ A ) )
= ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
=> ( X = bot_bo7198184520161983622et_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_253_Sup__bot__conv_I2_J,axiom,
! [A: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( comple548664676211718543et_nat @ A ) )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( X = bot_bot_set_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_254_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_255_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( comple5448282615319421384at_nat @ A ) )
= ( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A )
=> ( X = bot_bot_set_nat_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_256_Sup__bot__conv_I1_J,axiom,
! [A: set_set_set_set_nat] :
( ( ( comple6569609367425551173et_nat @ A )
= bot_bo7198184520161983622et_nat )
= ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
=> ( X = bot_bo7198184520161983622et_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_257_Sup__bot__conv_I1_J,axiom,
! [A: set_set_set_nat] :
( ( ( comple548664676211718543et_nat @ A )
= bot_bot_set_set_nat )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( X = bot_bot_set_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_258_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_259_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat_nat] :
( ( ( comple5448282615319421384at_nat @ A )
= bot_bot_set_nat_nat )
= ( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A )
=> ( X = bot_bot_set_nat_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_260_sup__bot_Oright__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_261_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_262_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_263_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_264_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_265_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_266_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_267_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_268_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_269_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_270_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_271_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_272_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_273_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_274_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_275_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_276_sup__eq__bot__iff,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ X2 @ Y )
= bot_bo7198184520161983622et_nat )
= ( ( X2 = bot_bo7198184520161983622et_nat )
& ( Y = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_277_sup__eq__bot__iff,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X2 @ Y )
= bot_bot_set_set_nat )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_278_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_279_sup__eq__bot__iff,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ X2 @ Y )
= bot_bot_set_nat_nat )
= ( ( X2 = bot_bot_set_nat_nat )
& ( Y = bot_bot_set_nat_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_280_bot__eq__sup__iff,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( ( X2 = bot_bo7198184520161983622et_nat )
& ( Y = bot_bo7198184520161983622et_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_281_bot__eq__sup__iff,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_282_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_283_bot__eq__sup__iff,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_nat_nat )
& ( Y = bot_bot_set_nat_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_284_sup__bot__right,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= X2 ) ).
% sup_bot_right
thf(fact_285_sup__bot__right,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ bot_bot_set_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_286_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_287_sup__bot__right,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= X2 ) ).
% sup_bot_right
thf(fact_288_sup__bot__left,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_289_sup__bot__left,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_290_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_291_sup__bot__left,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_292_UN__ball__bex__simps_I4_J,axiom,
! [B: set_set_nat > set_nat,A: set_set_set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ B @ A ) ) )
& ( P @ X ) ) )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X ) )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_293_UN__ball__bex__simps_I4_J,axiom,
! [B: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_nat > $o] :
( ( ? [X: set_nat] :
( ( member_set_nat @ X @ ( comple548664676211718543et_nat @ ( image_9186907679027735170et_nat @ B @ A ) ) )
& ( P @ X ) ) )
= ( ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ? [Y3: set_nat] :
( ( member_set_nat @ Y3 @ ( B @ X ) )
& ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_294_UN__ball__bex__simps_I2_J,axiom,
! [B: set_set_nat > set_nat,A: set_set_set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ B @ A ) ) )
=> ( P @ X ) ) )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X ) )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_295_UN__ball__bex__simps_I2_J,axiom,
! [B: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_nat > $o] :
( ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( comple548664676211718543et_nat @ ( image_9186907679027735170et_nat @ B @ A ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A )
=> ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ ( B @ X ) )
=> ( P @ Y3 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_296_bex__UN,axiom,
! [B: set_set_nat > set_nat,A: set_set_set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ B @ A ) ) )
& ( P @ X ) ) )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
& ? [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X ) )
& ( P @ Y3 ) ) ) ) ) ).
% bex_UN
thf(fact_297_bex__UN,axiom,
! [B: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_nat > $o] :
( ( ? [X: set_nat] :
( ( member_set_nat @ X @ ( comple548664676211718543et_nat @ ( image_9186907679027735170et_nat @ B @ A ) ) )
& ( P @ X ) ) )
= ( ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ? [Y3: set_nat] :
( ( member_set_nat @ Y3 @ ( B @ X ) )
& ( P @ Y3 ) ) ) ) ) ).
% bex_UN
thf(fact_298_ball__UN,axiom,
! [B: set_set_nat > set_nat,A: set_set_set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ B @ A ) ) )
=> ( P @ X ) ) )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( B @ X ) )
=> ( P @ Y3 ) ) ) ) ) ).
% ball_UN
thf(fact_299_ball__UN,axiom,
! [B: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_nat > $o] :
( ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( comple548664676211718543et_nat @ ( image_9186907679027735170et_nat @ B @ A ) ) )
=> ( P @ X ) ) )
= ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A )
=> ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ ( B @ X ) )
=> ( P @ Y3 ) ) ) ) ) ).
% ball_UN
thf(fact_300_Sup__empty,axiom,
( ( comple6569609367425551173et_nat @ bot_bo193956671110832956et_nat )
= bot_bo7198184520161983622et_nat ) ).
% Sup_empty
thf(fact_301_Sup__empty,axiom,
( ( comple5448282615319421384at_nat @ bot_bo7376149671870096959at_nat )
= bot_bot_set_nat_nat ) ).
% Sup_empty
thf(fact_302_Sup__empty,axiom,
( ( comple548664676211718543et_nat @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% Sup_empty
thf(fact_303_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_304_forth__assumptions_OSET_Osimps_I3_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P2 @ 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_305_inj__on__image,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ ( comple6569609367425551173et_nat @ A ) )
=> ( inj_on4386985374303630753et_nat @ ( image_5842784325960735177et_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_306_inj__on__image,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_set_nat_nat] :
( ( inj_on4164537515518332398et_nat @ F @ ( comple5448282615319421384at_nat @ A ) )
=> ( inj_on4976969725211833178et_nat @ ( image_9186907679027735170et_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_307_inj__on__image,axiom,
! [F: a > set_nat,A: set_set_a] :
( ( inj_on_a_set_nat @ F @ ( comple2307003609928055243_set_a @ A ) )
=> ( inj_on4942881440122211049et_nat @ ( image_a_set_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_308_first__assumptions_Ov__gs__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ) ).
% first_assumptions.v_gs_empty
thf(fact_309_first__assumptions_Ov__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_310_Inf_OINF__cong,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat > set_nat,D2: set_set_nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ B )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_5842784325960735177et_nat @ C2 @ A ) )
= ( Inf @ ( image_5842784325960735177et_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_311_Inf_OINF__cong,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: ( nat > nat ) > set_set_nat,D2: ( nat > nat ) > set_set_nat,Inf: set_set_set_nat > set_set_nat] :
( ( A = B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_9186907679027735170et_nat @ C2 @ A ) )
= ( Inf @ ( image_9186907679027735170et_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_312_Sup_OSUP__cong,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat > set_nat,D2: set_set_nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ B )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_5842784325960735177et_nat @ C2 @ A ) )
= ( Sup @ ( image_5842784325960735177et_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_313_Sup_OSUP__cong,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: ( nat > nat ) > set_set_nat,D2: ( nat > nat ) > set_set_nat,Sup: set_set_set_nat > set_set_nat] :
( ( A = B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_9186907679027735170et_nat @ C2 @ A ) )
= ( Sup @ ( image_9186907679027735170et_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_314_SUP__cong,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat > set_nat,D2: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ B )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ C2 @ A ) )
= ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_315_SUP__cong,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: ( nat > nat ) > set_set_nat,D2: ( nat > nat ) > set_set_nat] :
( ( A = B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ B )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple548664676211718543et_nat @ ( image_9186907679027735170et_nat @ C2 @ A ) )
= ( comple548664676211718543et_nat @ ( image_9186907679027735170et_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_316_SUP__eq__const,axiom,
! [I: set_set_set_nat,F: set_set_nat > set_nat,X2: set_nat] :
( ( I != bot_bo7198184520161983622et_nat )
=> ( ! [I2: set_set_nat] :
( ( member_set_set_nat @ I2 @ I )
=> ( ( F @ I2 )
= X2 ) )
=> ( ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ F @ I ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_317_SUP__eq__const,axiom,
! [I: set_nat_nat,F: ( nat > nat ) > set_set_nat,X2: set_set_nat] :
( ( I != bot_bot_set_nat_nat )
=> ( ! [I2: nat > nat] :
( ( member_nat_nat @ I2 @ I )
=> ( ( F @ I2 )
= X2 ) )
=> ( ( comple548664676211718543et_nat @ ( image_9186907679027735170et_nat @ F @ I ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_318_forth__assumptions_OSET_Ocong,axiom,
clique6509092761774629891_SET_a = clique6509092761774629891_SET_a ).
% forth_assumptions.SET.cong
thf(fact_319_UnionE,axiom,
! [A: nat,C2: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C2 ) )
=> ~ ! [X5: set_nat] :
( ( member_nat @ A @ X5 )
=> ~ ( member_set_nat @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_320_UnionE,axiom,
! [A: set_set_set_nat,C2: set_se7970953024979822686et_nat] :
( ( member2946998982187404937et_nat @ A @ ( comple5789376832584316411et_nat @ C2 ) )
=> ~ ! [X5: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ A @ X5 )
=> ~ ( member3774042032884853055et_nat @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_321_UnionE,axiom,
! [A: set_set_nat,C2: set_set_set_set_nat] :
( ( member_set_set_nat @ A @ ( comple6569609367425551173et_nat @ C2 ) )
=> ~ ! [X5: set_set_set_nat] :
( ( member_set_set_nat @ A @ X5 )
=> ~ ( member2946998982187404937et_nat @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_322_UnionE,axiom,
! [A: set_nat,C2: set_set_set_nat] :
( ( member_set_nat @ A @ ( comple548664676211718543et_nat @ C2 ) )
=> ~ ! [X5: set_set_nat] :
( ( member_set_nat @ A @ X5 )
=> ~ ( member_set_set_nat @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_323_UnionE,axiom,
! [A: nat > nat,C2: set_set_nat_nat] :
( ( member_nat_nat @ A @ ( comple5448282615319421384at_nat @ C2 ) )
=> ~ ! [X5: set_nat_nat] :
( ( member_nat_nat @ A @ X5 )
=> ~ ( member_set_nat_nat @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_324_UnionE,axiom,
! [A: a,C2: set_set_a] :
( ( member_a @ A @ ( comple2307003609928055243_set_a @ C2 ) )
=> ~ ! [X5: set_a] :
( ( member_a @ A @ X5 )
=> ~ ( member_set_a @ X5 @ C2 ) ) ) ).
% UnionE
thf(fact_325_Union__empty,axiom,
( ( comple6569609367425551173et_nat @ bot_bo193956671110832956et_nat )
= bot_bo7198184520161983622et_nat ) ).
% Union_empty
thf(fact_326_Union__empty,axiom,
( ( comple5448282615319421384at_nat @ bot_bo7376149671870096959at_nat )
= bot_bot_set_nat_nat ) ).
% Union_empty
thf(fact_327_Union__empty,axiom,
( ( comple548664676211718543et_nat @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% Union_empty
thf(fact_328_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_329_Union__empty__conv,axiom,
! [A: set_set_set_set_nat] :
( ( ( comple6569609367425551173et_nat @ A )
= bot_bo7198184520161983622et_nat )
= ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
=> ( X = bot_bo7198184520161983622et_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_330_Union__empty__conv,axiom,
! [A: set_set_set_nat] :
( ( ( comple548664676211718543et_nat @ A )
= bot_bot_set_set_nat )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( X = bot_bot_set_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_331_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_332_Union__empty__conv,axiom,
! [A: set_set_nat_nat] :
( ( ( comple5448282615319421384at_nat @ A )
= bot_bot_set_nat_nat )
= ( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A )
=> ( X = bot_bot_set_nat_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_333_empty__Union__conv,axiom,
! [A: set_set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( comple6569609367425551173et_nat @ A ) )
= ( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
=> ( X = bot_bo7198184520161983622et_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_334_empty__Union__conv,axiom,
! [A: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( comple548664676211718543et_nat @ A ) )
= ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( X = bot_bot_set_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_335_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_336_empty__Union__conv,axiom,
! [A: set_set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( comple5448282615319421384at_nat @ A ) )
= ( ! [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A )
=> ( X = bot_bot_set_nat_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_337_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_338_rev__image__eqI,axiom,
! [X2: set_nat,A: set_set_nat,B2: a,F: set_nat > a] :
( ( member_set_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_set_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_339_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: set_nat,F: a > set_nat] :
( ( member_a @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_nat @ B2 @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_340_rev__image__eqI,axiom,
! [X2: set_set_nat,A: set_set_set_nat,B2: a,F: set_set_nat > a] :
( ( member_set_set_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_set_set_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_341_rev__image__eqI,axiom,
! [X2: set_nat,A: set_set_nat,B2: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_342_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B2: a,F: ( nat > nat ) > a] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_nat_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_343_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: set_set_nat,F: a > set_set_nat] :
( ( member_a @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_set_nat @ B2 @ ( image_a_set_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_344_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: nat > nat,F: a > nat > nat] :
( ( member_a @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat_nat @ B2 @ ( image_a_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_345_rev__image__eqI,axiom,
! [X2: set_set_set_nat,A: set_set_set_set_nat,B2: a,F: set_set_set_nat > a] :
( ( member2946998982187404937et_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a @ B2 @ ( image_3422112407882505029_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_346_rev__image__eqI,axiom,
! [X2: set_set_nat,A: set_set_set_nat,B2: set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_347_ball__imageD,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,P: set_nat > $o] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_5842784325960735177et_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_348_ball__imageD,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_set_nat > $o] :
( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ ( image_9186907679027735170et_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X6: nat > nat] :
( ( member_nat_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_349_image__cong,axiom,
! [M: set_set_set_nat,N: set_set_set_nat,F: set_set_nat > set_nat,G2: set_set_nat > set_nat] :
( ( M = N )
=> ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_5842784325960735177et_nat @ F @ M )
= ( image_5842784325960735177et_nat @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_350_image__cong,axiom,
! [M: set_nat_nat,N: set_nat_nat,F: ( nat > nat ) > set_set_nat,G2: ( nat > nat ) > set_set_nat] :
( ( M = N )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_9186907679027735170et_nat @ F @ M )
= ( image_9186907679027735170et_nat @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_351_ex__in__conv,axiom,
! [A: set_set_set_set_nat] :
( ( ? [X: set_set_set_nat] : ( member2946998982187404937et_nat @ X @ A ) )
= ( A != bot_bo193956671110832956et_nat ) ) ).
% ex_in_conv
thf(fact_352_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_353_ex__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ? [X: set_set_nat] : ( member_set_set_nat @ X @ A ) )
= ( A != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_354_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X: set_nat] : ( member_set_nat @ X @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_355_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_356_ex__in__conv,axiom,
! [A: set_nat_nat] :
( ( ? [X: nat > nat] : ( member_nat_nat @ X @ A ) )
= ( A != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_357_bex__imageD,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_5842784325960735177et_nat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_358_bex__imageD,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_set_nat > $o] :
( ? [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ ( image_9186907679027735170et_nat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_359_image__iff,axiom,
! [Z: set_set_nat,F: ( nat > nat ) > set_set_nat,A: set_nat_nat] :
( ( member_set_set_nat @ Z @ ( image_9186907679027735170et_nat @ F @ A ) )
= ( ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_360_image__iff,axiom,
! [Z: set_nat,F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( member_set_nat @ Z @ ( image_5842784325960735177et_nat @ F @ A ) )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_361_equals0I,axiom,
! [A: set_set_set_set_nat] :
( ! [Y4: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ Y4 @ A )
=> ( A = bot_bo193956671110832956et_nat ) ) ).
% equals0I
thf(fact_362_equals0I,axiom,
! [A: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_363_equals0I,axiom,
! [A: set_set_set_nat] :
( ! [Y4: set_set_nat] :
~ ( member_set_set_nat @ Y4 @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_364_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y4: set_nat] :
~ ( member_set_nat @ Y4 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_365_equals0I,axiom,
! [A: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_366_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y4: nat > nat] :
~ ( member_nat_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_367_equals0D,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( A = bot_bo193956671110832956et_nat )
=> ~ ( member2946998982187404937et_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_368_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_369_equals0D,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( A = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_370_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_371_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_372_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_373_imageI,axiom,
! [X2: a,A: set_a,F: a > a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_374_imageI,axiom,
! [X2: set_nat,A: set_set_nat,F: set_nat > a] :
( ( member_set_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_set_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_375_imageI,axiom,
! [X2: a,A: set_a,F: a > set_nat] :
( ( member_a @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_a_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_376_imageI,axiom,
! [X2: set_set_nat,A: set_set_set_nat,F: set_set_nat > a] :
( ( member_set_set_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_set_set_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_377_imageI,axiom,
! [X2: set_nat,A: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_378_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > a] :
( ( member_nat_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_nat_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_379_imageI,axiom,
! [X2: a,A: set_a,F: a > set_set_nat] :
( ( member_a @ X2 @ A )
=> ( member_set_set_nat @ ( F @ X2 ) @ ( image_a_set_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_380_imageI,axiom,
! [X2: a,A: set_a,F: a > nat > nat] :
( ( member_a @ X2 @ A )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_a_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_381_imageI,axiom,
! [X2: set_set_set_nat,A: set_set_set_set_nat,F: set_set_set_nat > a] :
( ( member2946998982187404937et_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ ( image_3422112407882505029_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_382_imageI,axiom,
! [X2: set_set_nat,A: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_383_emptyE,axiom,
! [A2: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ A2 @ bot_bo193956671110832956et_nat ) ).
% emptyE
thf(fact_384_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_385_emptyE,axiom,
! [A2: set_set_nat] :
~ ( member_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_386_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_387_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_388_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_389_first__assumptions_Ov__gs__def,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ X3 )
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ X3 ) ) ) ).
% first_assumptions.v_gs_def
thf(fact_390_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_391_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ bot_bot_set_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_392_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_393_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_394_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_395_image__Un,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_set_nat] :
( ( image_set_nat_nat @ F @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_set_nat_nat @ F @ A ) @ ( image_set_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_396_image__Un,axiom,
! [F: nat > set_nat,A: set_nat,B: set_nat] :
( ( image_nat_set_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_397_image__Un,axiom,
! [F: set_nat > set_nat,A: set_set_nat,B: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ ( image_7916887816326733075et_nat @ F @ A ) @ ( image_7916887816326733075et_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_398_image__Un,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( image_1454916318497077779at_nat @ F @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_1454916318497077779at_nat @ F @ A ) @ ( image_1454916318497077779at_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_399_image__Un,axiom,
! [F: nat > set_set_nat,A: set_nat,B: set_nat] :
( ( image_2194112158459175443et_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_su4213647025997063966et_nat @ ( image_2194112158459175443et_nat @ F @ A ) @ ( image_2194112158459175443et_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_400_image__Un,axiom,
! [F: nat > nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat_nat2 @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat_nat @ ( image_nat_nat_nat2 @ F @ A ) @ ( image_nat_nat_nat2 @ F @ B ) ) ) ).
% image_Un
thf(fact_401_image__Un,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,B: set_nat_nat] :
( ( image_nat_nat_nat @ F @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat_nat @ F @ A ) @ ( image_nat_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_402_image__Un,axiom,
! [F: set_nat > set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( image_6725021117256019401et_nat @ F @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_su4213647025997063966et_nat @ ( image_6725021117256019401et_nat @ F @ A ) @ ( image_6725021117256019401et_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_403_image__Un,axiom,
! [F: set_nat > nat > nat,A: set_set_nat,B: set_set_nat] :
( ( image_8569768528772619084at_nat @ F @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat_nat @ ( image_8569768528772619084at_nat @ F @ A ) @ ( image_8569768528772619084at_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_404_inj__on__image__iff,axiom,
! [A: set_a,G2: a > set_nat,F: a > a] :
( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ( G2 @ ( F @ X4 ) )
= ( G2 @ ( F @ Xa ) ) )
= ( ( G2 @ X4 )
= ( G2 @ Xa ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A )
=> ( ( inj_on_a_set_nat @ G2 @ ( image_a_a @ F @ A ) )
= ( inj_on_a_set_nat @ G2 @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_405_Un__empty__right,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ bot_bo7198184520161983622et_nat )
= A ) ).
% Un_empty_right
thf(fact_406_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_407_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_408_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_409_Un__empty__left,axiom,
! [B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_410_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_411_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_412_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_413_Sup__union__distrib,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( comple6569609367425551173et_nat @ ( sup_su3906748206781935060et_nat @ A @ B ) )
= ( sup_su4213647025997063966et_nat @ ( comple6569609367425551173et_nat @ A ) @ ( comple6569609367425551173et_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_414_Sup__union__distrib,axiom,
! [A: set_set_nat_nat,B: set_set_nat_nat] :
( ( comple5448282615319421384at_nat @ ( sup_su2553808219797728471at_nat @ A @ B ) )
= ( sup_sup_set_nat_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_415_Sup__union__distrib,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_416_Sup__union__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( comple548664676211718543et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( sup_sup_set_set_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_417_forth__assumptions_Oeval__g__def,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,Theta: a > $o,G: set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique5859573001277246426al_g_a @ V @ Pi @ Theta @ G )
= ( ! [X: a] :
( ( member_a @ X @ V )
=> ( ( member_set_nat @ ( Pi @ X ) @ G )
=> ( Theta @ X ) ) ) ) ) ) ).
% forth_assumptions.eval_g_def
thf(fact_418_ccSup__empty,axiom,
( ( comple6569609367425551173et_nat @ bot_bo193956671110832956et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ccSup_empty
thf(fact_419_ccSup__empty,axiom,
( ( comple5448282615319421384at_nat @ bot_bo7376149671870096959at_nat )
= bot_bot_set_nat_nat ) ).
% ccSup_empty
thf(fact_420_ccSup__empty,axiom,
( ( comple548664676211718543et_nat @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% ccSup_empty
thf(fact_421_ccSup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% ccSup_empty
thf(fact_422_Union__image__empty,axiom,
! [A: set_nat,F: nat > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_423_Union__image__empty,axiom,
! [A: set_nat,F: set_nat > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ F @ bot_bot_set_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_424_Union__image__empty,axiom,
! [A: set_set_nat,F: nat > set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( comple548664676211718543et_nat @ ( image_2194112158459175443et_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_425_Union__image__empty,axiom,
! [A: set_nat,F: set_set_nat > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ F @ bot_bo7198184520161983622et_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_426_Union__image__empty,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( comple548664676211718543et_nat @ ( image_6725021117256019401et_nat @ F @ bot_bot_set_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_427_Union__image__empty,axiom,
! [A: set_set_set_nat,F: nat > set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( comple6569609367425551173et_nat @ ( image_5738044413236618185et_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_428_Union__image__empty,axiom,
! [A: set_nat_nat,F: nat > set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( comple5448282615319421384at_nat @ ( image_7301343469591561292at_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_429_Union__image__empty,axiom,
! [A: set_nat,F: ( nat > nat ) > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_7432509271690132940et_nat @ F @ bot_bot_set_nat_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_430_Union__image__empty,axiom,
! [A: set_set_nat,F: set_set_nat > set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( comple548664676211718543et_nat @ ( image_7884819252390400639et_nat @ F @ bot_bo7198184520161983622et_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_431_Union__image__empty,axiom,
! [A: set_set_set_nat,F: set_nat > set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( comple6569609367425551173et_nat @ ( image_4583741654806091647et_nat @ F @ bot_bot_set_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_432_mformula_Oinject_I3_J,axiom,
! [X51: monotone_mformula_a,X52: monotone_mformula_a,Y51: monotone_mformula_a,Y52: monotone_mformula_a] :
( ( ( monotone_Disj_a @ X51 @ X52 )
= ( monotone_Disj_a @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% mformula.inject(3)
thf(fact_433_SET_Osimps_I1_J,axiom,
( ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% SET.simps(1)
thf(fact_434_the__elem__image__unique,axiom,
! [A: set_set_set_nat,F: set_set_nat > set_nat,X2: set_set_nat] :
( ( A != bot_bo7198184520161983622et_nat )
=> ( ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ A )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_nat @ ( image_5842784325960735177et_nat @ F @ A ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_435_the__elem__image__unique,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > set_set_nat,X2: nat > nat] :
( ( A != bot_bot_set_nat_nat )
=> ( ! [Y4: nat > nat] :
( ( member_nat_nat @ Y4 @ A )
=> ( ( F @ Y4 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_set_nat @ ( image_9186907679027735170et_nat @ F @ A ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_436_Set_Ois__empty__def,axiom,
( is_empty_set_set_nat
= ( ^ [A4: set_set_set_nat] : ( A4 = bot_bo7198184520161983622et_nat ) ) ) ).
% Set.is_empty_def
thf(fact_437_Set_Ois__empty__def,axiom,
( is_empty_set_nat
= ( ^ [A4: set_set_nat] : ( A4 = bot_bot_set_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_438_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_439_Set_Ois__empty__def,axiom,
( is_empty_nat_nat
= ( ^ [A4: set_nat_nat] : ( A4 = bot_bot_set_nat_nat ) ) ) ).
% Set.is_empty_def
thf(fact_440__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_441_forth__assumptions_OACC__SET_I3_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P2 @ 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_442_forth__assumptions_Oapprox__pos_Osimps_I5_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,V2: monotone_mformula_a,Va: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P2 @ K @ Pi @ ( monotone_Disj_a @ V2 @ Va ) )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(5)
thf(fact_443__092_060A_062__simps_I1_J,axiom,
member535913909593306477mula_a @ monotone_FALSE_a @ ( clique5987991184601036204th_A_a @ v ) ).
% \<A>_simps(1)
thf(fact_444_forth__assumptions_O_092_060A_062_Ocong,axiom,
clique5987991184601036204th_A_a = clique5987991184601036204th_A_a ).
% forth_assumptions.\<A>.cong
thf(fact_445_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_446_forth__assumptions_Oapprox__pos_Ocong,axiom,
clique8538548958085942603_pos_a = clique8538548958085942603_pos_a ).
% forth_assumptions.approx_pos.cong
thf(fact_447_mformula_Odistinct_I13_J,axiom,
! [X51: monotone_mformula_a,X52: monotone_mformula_a] :
( monotone_FALSE_a
!= ( monotone_Disj_a @ X51 @ X52 ) ) ).
% mformula.distinct(13)
thf(fact_448_forth__assumptions_O_092_060A_062__simps_I1_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( member535913909593306477mula_a @ monotone_FALSE_a @ ( clique5987991184601036204th_A_a @ V ) ) ) ).
% forth_assumptions.\<A>_simps(1)
thf(fact_449_forth__assumptions_Oapprox__pos_Osimps_I3_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P2 @ K @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(3)
thf(fact_450_bot__set__def,axiom,
( bot_bo193956671110832956et_nat
= ( collec7201453139178570183et_nat @ bot_bo5536612546450143305_nat_o ) ) ).
% bot_set_def
thf(fact_451_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_452_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_453_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_454_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_455_forth__assumptions_OACC__SET_I2_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique6509092761774629891_SET_a @ Pi @ monotone_FALSE_a ) )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.ACC_SET(2)
thf(fact_456_first__assumptions_OACC__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.ACC_empty
thf(fact_457_forth__assumptions_OSET_Osimps_I1_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique6509092761774629891_SET_a @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.SET.simps(1)
thf(fact_458_first__assumptions_OACC__union,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K @ X3 ) @ ( clique3210737319928189260st_ACC @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_union
thf(fact_459_forth__assumptions_O_092_060A_062__simps_I4_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P2 @ 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_460_forth__assumptions_OACC__mf__def,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique4708818501384062891C_mf_a @ K @ Pi @ Phi )
= ( clique3210737319928189260st_ACC @ K @ ( clique6509092761774629891_SET_a @ Pi @ Phi ) ) ) ) ).
% forth_assumptions.ACC_mf_def
thf(fact_461_forth__assumptions_Ono__deviation_I1_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique3934260045859375359_pos_a @ L @ P2 @ K @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.no_deviation(1)
thf(fact_462_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_463_ACC__SET_I2_J,axiom,
( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a ) )
= bot_bo7198184520161983622et_nat ) ).
% ACC_SET(2)
thf(fact_464_first__assumptions_Oempty__CLIQUE,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.empty_CLIQUE
thf(fact_465_forth__assumptions_Oapprox__neg_Osimps_I4_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique6623365555141101007_neg_a @ L @ P2 @ K @ Pi @ monotone_FALSE_a )
= bot_bot_set_nat_nat ) ) ).
% forth_assumptions.approx_neg.simps(4)
thf(fact_466_member__bind,axiom,
! [X2: nat,A: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_nat @ X2 @ ( bind_set_set_nat_nat @ A @ F ) )
= ( member_nat @ X2 @ ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ) ).
% member_bind
thf(fact_467_member__bind,axiom,
! [X2: set_nat,A: set_nat_nat,F: ( nat > nat ) > set_set_nat] :
( ( member_set_nat @ X2 @ ( bind_nat_nat_set_nat @ A @ F ) )
= ( member_set_nat @ X2 @ ( comple548664676211718543et_nat @ ( image_9186907679027735170et_nat @ F @ A ) ) ) ) ).
% member_bind
thf(fact_468_inj__on__image__Fpow,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A )
=> ( inj_on4386985374303630753et_nat @ ( image_5842784325960735177et_nat @ F ) @ ( finite7717622420921165910et_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_469_inj__on__image__Fpow,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat] :
( ( inj_on4164537515518332398et_nat @ F @ A )
=> ( inj_on4976969725211833178et_nat @ ( image_9186907679027735170et_nat @ F ) @ ( finite_Fpow_nat_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_470_inj__on__image__Fpow,axiom,
! [F: a > set_nat,A: set_a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( inj_on4942881440122211049et_nat @ ( image_a_set_nat @ F ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_471_the__inv__into__onto,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A )
=> ( ( image_6725021117256019401et_nat @ ( the_in3575317103879706743et_nat @ A @ F ) @ ( image_5842784325960735177et_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_472_the__inv__into__onto,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat] :
( ( inj_on4164537515518332398et_nat @ F @ A )
=> ( ( image_8441894408526374658at_nat @ ( the_in6552357484096881712et_nat @ A @ F ) @ ( image_9186907679027735170et_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_473_the__inv__into__onto,axiom,
! [F: set_nat > set_set_nat,A: set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ A )
=> ( ( image_5842784325960735177et_nat @ ( the_in4457553895174990967et_nat @ A @ F ) @ ( image_6725021117256019401et_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_474_the__inv__into__onto,axiom,
! [F: set_set_nat > nat > nat,A: set_set_set_nat] :
( ( inj_on3419524245016971886at_nat @ F @ A )
=> ( ( image_9186907679027735170et_nat @ ( the_in5807344213595521200at_nat @ A @ F ) @ ( image_8441894408526374658at_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_475_the__inv__into__onto,axiom,
! [F: a > set_nat,A: set_a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( image_set_nat_a @ ( the_in5098273967424113681et_nat @ A @ F ) @ ( image_a_set_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_476_ACC__union,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X3 ) @ ( clique3210737319928189260st_ACC @ k @ Y2 ) ) ) ).
% ACC_union
thf(fact_477_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_478_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_479_empty__bind,axiom,
! [F: nat > set_nat] :
( ( bind_nat_nat @ bot_bot_set_nat @ F )
= bot_bot_set_nat ) ).
% empty_bind
thf(fact_480_empty__bind,axiom,
! [F: set_nat > set_nat] :
( ( bind_set_nat_nat @ bot_bot_set_set_nat @ F )
= bot_bot_set_nat ) ).
% empty_bind
thf(fact_481_empty__bind,axiom,
! [F: nat > set_set_nat] :
( ( bind_nat_set_nat @ bot_bot_set_nat @ F )
= bot_bot_set_set_nat ) ).
% empty_bind
thf(fact_482_empty__bind,axiom,
! [F: set_set_nat > set_nat] :
( ( bind_set_set_nat_nat @ bot_bo7198184520161983622et_nat @ F )
= bot_bot_set_nat ) ).
% empty_bind
thf(fact_483_empty__bind,axiom,
! [F: set_nat > set_set_nat] :
( ( bind_set_nat_set_nat @ bot_bot_set_set_nat @ F )
= bot_bot_set_set_nat ) ).
% empty_bind
thf(fact_484_empty__bind,axiom,
! [F: nat > set_set_set_nat] :
( ( bind_nat_set_set_nat @ bot_bot_set_nat @ F )
= bot_bo7198184520161983622et_nat ) ).
% empty_bind
thf(fact_485_empty__bind,axiom,
! [F: nat > set_nat_nat] :
( ( bind_nat_nat_nat2 @ bot_bot_set_nat @ F )
= bot_bot_set_nat_nat ) ).
% empty_bind
thf(fact_486_empty__bind,axiom,
! [F: ( nat > nat ) > set_nat] :
( ( bind_nat_nat_nat @ bot_bot_set_nat_nat @ F )
= bot_bot_set_nat ) ).
% empty_bind
thf(fact_487_empty__bind,axiom,
! [F: set_set_nat > set_set_nat] :
( ( bind_s3449802520948986007et_nat @ bot_bo7198184520161983622et_nat @ F )
= bot_bot_set_set_nat ) ).
% empty_bind
thf(fact_488_empty__bind,axiom,
! [F: set_nat > set_set_set_nat] :
( ( bind_s4332039312244270231et_nat @ bot_bot_set_set_nat @ F )
= bot_bo7198184520161983622et_nat ) ).
% empty_bind
thf(fact_489_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_490_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_491_forth__assumptions_Oapprox__neg_Ocong,axiom,
clique6623365555141101007_neg_a = clique6623365555141101007_neg_a ).
% forth_assumptions.approx_neg.cong
thf(fact_492_forth__assumptions_Odeviate__pos_Ocong,axiom,
clique3934260045859375359_pos_a = clique3934260045859375359_pos_a ).
% forth_assumptions.deviate_pos.cong
thf(fact_493_forth__assumptions_OACC__mf_Ocong,axiom,
clique4708818501384062891C_mf_a = clique4708818501384062891C_mf_a ).
% forth_assumptions.ACC_mf.cong
thf(fact_494_empty__in__Fpow,axiom,
! [A: set_set_set_nat] : ( member2946998982187404937et_nat @ bot_bo7198184520161983622et_nat @ ( finite7717622420921165910et_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_495_empty__in__Fpow,axiom,
! [A: set_set_nat] : ( member_set_set_nat @ bot_bot_set_set_nat @ ( finite_Fpow_set_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_496_empty__in__Fpow,axiom,
! [A: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_497_empty__in__Fpow,axiom,
! [A: set_nat_nat] : ( member_set_nat_nat @ bot_bot_set_nat_nat @ ( finite_Fpow_nat_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_498_Fpow__not__empty,axiom,
! [A: set_set_nat] :
( ( finite_Fpow_set_nat @ A )
!= bot_bo7198184520161983622et_nat ) ).
% Fpow_not_empty
thf(fact_499_Fpow__not__empty,axiom,
! [A: set_nat] :
( ( finite_Fpow_nat @ A )
!= bot_bot_set_set_nat ) ).
% Fpow_not_empty
thf(fact_500_the__inv__into__f__f,axiom,
! [F: a > set_nat,A: set_a,X2: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( member_a @ X2 @ A )
=> ( ( the_in5098273967424113681et_nat @ A @ F @ ( F @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_501_the__inv__into__f__eq,axiom,
! [F: a > set_nat,A: set_a,X2: a,Y: set_nat] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( ( F @ X2 )
= Y )
=> ( ( member_a @ X2 @ A )
=> ( ( the_in5098273967424113681et_nat @ A @ F @ Y )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_502_inj__on__the__inv__into,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A )
=> ( inj_on2776966659131765557et_nat @ ( the_in3575317103879706743et_nat @ A @ F ) @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_503_inj__on__the__inv__into,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat] :
( ( inj_on4164537515518332398et_nat @ F @ A )
=> ( inj_on3419524245016971886at_nat @ ( the_in6552357484096881712et_nat @ A @ F ) @ ( image_9186907679027735170et_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_504_inj__on__the__inv__into,axiom,
! [F: set_nat > a,A: set_set_nat] :
( ( inj_on_set_nat_a @ F @ A )
=> ( inj_on_a_set_nat @ ( the_in1235728082690079619_nat_a @ A @ F ) @ ( image_set_nat_a @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_505_inj__on__the__inv__into,axiom,
! [F: a > set_nat,A: set_a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( inj_on_set_nat_a @ ( the_in5098273967424113681et_nat @ A @ F ) @ ( image_a_set_nat @ F @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_506_f__the__inv__into__f,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,Y: set_set_nat] :
( ( inj_on4164537515518332398et_nat @ F @ A )
=> ( ( member_set_set_nat @ Y @ ( image_9186907679027735170et_nat @ F @ A ) )
=> ( ( F @ ( the_in6552357484096881712et_nat @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_507_f__the__inv__into__f,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,Y: set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ A )
=> ( ( member_set_nat @ Y @ ( image_5842784325960735177et_nat @ F @ A ) )
=> ( ( F @ ( the_in3575317103879706743et_nat @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_508_f__the__inv__into__f,axiom,
! [F: a > set_nat,A: set_a,Y: set_nat] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( member_set_nat @ Y @ ( image_a_set_nat @ F @ A ) )
=> ( ( F @ ( the_in5098273967424113681et_nat @ A @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_509_bind__UNION,axiom,
( bind_set_set_nat_nat
= ( ^ [A4: set_set_set_nat,F2: set_set_nat > set_nat] : ( comple7399068483239264473et_nat @ ( image_5842784325960735177et_nat @ F2 @ A4 ) ) ) ) ).
% bind_UNION
thf(fact_510_bind__UNION,axiom,
( bind_nat_nat_set_nat
= ( ^ [A4: set_nat_nat,F2: ( nat > nat ) > set_set_nat] : ( comple548664676211718543et_nat @ ( image_9186907679027735170et_nat @ F2 @ A4 ) ) ) ) ).
% bind_UNION
thf(fact_511_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_512_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_513_bot__empty__eq,axiom,
( bot_bo5536612546450143305_nat_o
= ( ^ [X: set_set_set_nat] : ( member2946998982187404937et_nat @ X @ bot_bo193956671110832956et_nat ) ) ) ).
% bot_empty_eq
thf(fact_514_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_515_bot__empty__eq,axiom,
( bot_bo6227097192321305471_nat_o
= ( ^ [X: set_set_nat] : ( member_set_set_nat @ X @ bot_bo7198184520161983622et_nat ) ) ) ).
% bot_empty_eq
thf(fact_516_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_517_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_518_bot__empty__eq,axiom,
( bot_bot_nat_nat_o
= ( ^ [X: nat > nat] : ( member_nat_nat @ X @ bot_bot_set_nat_nat ) ) ) ).
% bot_empty_eq
thf(fact_519_Collect__empty__eq__bot,axiom,
! [P: set_set_set_nat > $o] :
( ( ( collec7201453139178570183et_nat @ P )
= bot_bo193956671110832956et_nat )
= ( P = bot_bo5536612546450143305_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_520_Collect__empty__eq__bot,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( P = bot_bo6227097192321305471_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_521_Collect__empty__eq__bot,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( P = bot_bot_set_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_522_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_523_Collect__empty__eq__bot,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( P = bot_bot_nat_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_524_ACC__cf__union,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X3 ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_union
thf(fact_525_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_526_forth__assumptions_Ono__deviation_I2_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique2019076642914533763_neg_a @ L @ P2 @ K @ Pi @ monotone_FALSE_a )
= bot_bot_set_nat_nat ) ) ).
% forth_assumptions.no_deviation(2)
thf(fact_527_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_528_forth__assumptions_Odeviate__neg_Ocong,axiom,
clique2019076642914533763_neg_a = clique2019076642914533763_neg_a ).
% forth_assumptions.deviate_neg.cong
thf(fact_529_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_530_forth__assumptions_OACC__cf__mf_Ocong,axiom,
clique8961599393750669800f_mf_a = clique8961599393750669800f_mf_a ).
% forth_assumptions.ACC_cf_mf.cong
thf(fact_531_forth__assumptions_OACC__cf__mf__def,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P2 @ 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_532_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X3 ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_533_Sup__SUP__eq,axiom,
( comple2068028038703680896_nat_o
= ( ^ [S: set_set_set_nat_o,X: set_set_nat] : ( member_set_set_nat @ X @ ( comple6569609367425551173et_nat @ ( image_3164711303094801856et_nat @ collect_set_set_nat @ S ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_534_Sup__SUP__eq,axiom,
( comple3806919086088850358_nat_o
= ( ^ [S: set_set_nat_o,X: set_nat] : ( member_set_nat @ X @ ( comple548664676211718543et_nat @ ( image_4687162037615663680et_nat @ collect_set_nat @ S ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_535_Sup__SUP__eq,axiom,
( comple8312177224774716605_nat_o
= ( ^ [S: set_nat_nat_o,X: nat > nat] : ( member_nat_nat @ X @ ( comple5448282615319421384at_nat @ ( image_7977807581451749376at_nat @ collect_nat_nat @ S ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_536_Sup__SUP__eq,axiom,
( complete_Sup_Sup_a_o
= ( ^ [S: set_a_o,X: a] : ( member_a @ X @ ( comple2307003609928055243_set_a @ ( image_a_o_set_a @ collect_a @ S ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_537_Sup__SUP__eq,axiom,
( comple8839946721722594890_nat_o
= ( ^ [S: set_se8611727395572922045_nat_o,X: set_set_set_nat] : ( member2946998982187404937et_nat @ X @ ( comple5789376832584316411et_nat @ ( image_3156746147002188096et_nat @ collec7201453139178570183et_nat @ S ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_538_forth__assumptions_OACC__cf__SET_I3_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P2 @ 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_539_first__assumptions_OACC__cf__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ) ).
% first_assumptions.ACC_cf_empty
thf(fact_540_forth__assumptions_OACC__cf__SET_I2_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ 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_541_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_542__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_543__092_060A_062__simps_I2_J,axiom,
! [X2: a] :
( ( member535913909593306477mula_a @ ( monotone_Var_a @ X2 ) @ ( clique5987991184601036204th_A_a @ v ) )
= ( member_a @ X2 @ v ) ) ).
% \<A>_simps(2)
thf(fact_544_the__inv__into__into,axiom,
! [F: a > a,A: set_a,X2: a,B: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ( member_a @ X2 @ ( image_a_a @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_inv_into_a_a @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_545_the__inv__into__into,axiom,
! [F: nat > a,A: set_nat,X2: a,B: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( member_a @ X2 @ ( image_nat_a @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_inv_into_nat_a @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_546_the__inv__into__into,axiom,
! [F: a > set_nat,A: set_a,X2: set_nat,B: set_a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( member_set_nat @ X2 @ ( image_a_set_nat @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_in5098273967424113681et_nat @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_547_the__inv__into__into,axiom,
! [F: set_nat > a,A: set_set_nat,X2: a,B: set_set_nat] :
( ( inj_on_set_nat_a @ F @ A )
=> ( ( member_a @ X2 @ ( image_set_nat_a @ F @ A ) )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( member_set_nat @ ( the_in1235728082690079619_nat_a @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_548_the__inv__into__into,axiom,
! [F: nat > set_nat,A: set_nat,X2: set_nat,B: set_nat] :
( ( inj_on_nat_set_nat @ F @ A )
=> ( ( member_set_nat @ X2 @ ( image_nat_set_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_in5057678521256355851et_nat @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_549_the__inv__into__into,axiom,
! [F: a > set_set_nat,A: set_a,X2: set_set_nat,B: set_a] :
( ( inj_on_a_set_set_nat @ F @ A )
=> ( ( member_set_set_nat @ X2 @ ( image_a_set_set_nat @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_in1847442007992722631et_nat @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_550_the__inv__into__into,axiom,
! [F: a > nat > nat,A: set_a,X2: nat > nat,B: set_a] :
( ( inj_on_a_nat_nat @ F @ A )
=> ( ( member_nat_nat @ X2 @ ( image_a_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( member_a @ ( the_in1748411866455846218at_nat @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_551_the__inv__into__into,axiom,
! [F: set_set_nat > a,A: set_set_set_nat,X2: a,B: set_set_set_nat] :
( ( inj_on_set_set_nat_a @ F @ A )
=> ( ( member_a @ X2 @ ( image_set_set_nat_a @ F @ A ) )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( member_set_set_nat @ ( the_in2240265568404529229_nat_a @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_552_the__inv__into__into,axiom,
! [F: set_nat > set_nat,A: set_set_nat,X2: set_nat,B: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A )
=> ( ( member_set_nat @ X2 @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( member_set_nat @ ( the_in3610957794094371777et_nat @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_553_the__inv__into__into,axiom,
! [F: nat > set_set_nat,A: set_nat,X2: set_set_nat,B: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ A )
=> ( ( member_set_set_nat @ X2 @ ( image_2194112158459175443et_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( member_nat @ ( the_in7111554173081589953et_nat @ A @ F @ X2 ) @ B ) ) ) ) ).
% the_inv_into_into
thf(fact_554_forth__assumptions_Oapprox__neg_Osimps_I3_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique6623365555141101007_neg_a @ L @ P2 @ K @ Pi @ monotone_TRUE_a )
= bot_bot_set_nat_nat ) ) ).
% forth_assumptions.approx_neg.simps(3)
thf(fact_555_forth__assumptions_Oapprox__pos_Osimps_I2_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P2 @ K @ Pi @ monotone_TRUE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(2)
thf(fact_556_forth__assumptions_OAPR_Osimps_I1_J,axiom,
! [L: nat,P2: nat,K: nat,V: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P2 @ K @ V @ Pi )
=> ( ( clique3873310923663319714_APR_a @ L @ P2 @ K @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.APR.simps(1)
thf(fact_557_the__elem__eq,axiom,
! [X2: set_set_nat] :
( ( the_elem_set_set_nat @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_558_the__elem__eq,axiom,
! [X2: set_nat] :
( ( the_elem_set_nat @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_559_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_560_the__elem__eq,axiom,
! [X2: nat > nat] :
( ( the_elem_nat_nat @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_561_v__gs__mono,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X3 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ).
% v_gs_mono
thf(fact_562_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_563_ACC__cf__mono,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X3 ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_mono
thf(fact_564_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_565_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_566_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_567_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_568_dual__order_Orefl,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_569_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_570_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_571_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_572_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_573_order__refl,axiom,
! [X2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_574_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_575_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_576_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_577_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_578_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_579_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_580_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_581_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_582_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_583_inf__right__idem,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ Y )
= ( inf_in5711780100303410308et_nat @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_584_inf__right__idem,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ Y )
= ( inf_inf_set_nat_nat @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_585_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_586_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_587_inf__left__idem,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) )
= ( inf_in5711780100303410308et_nat @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_588_inf__left__idem,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ X2 @ Y ) )
= ( inf_inf_set_nat_nat @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_589_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_590_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_591_inf__idem,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_592_inf__idem,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_593_inf_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_594_inf_Oidem,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_595_insert__absorb2,axiom,
! [X2: set_set_nat,A: set_set_set_nat] :
( ( insert_set_set_nat @ X2 @ ( insert_set_set_nat @ X2 @ A ) )
= ( insert_set_set_nat @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_596_insert__absorb2,axiom,
! [X2: set_nat,A: set_set_nat] :
( ( insert_set_nat @ X2 @ ( insert_set_nat @ X2 @ A ) )
= ( insert_set_nat @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_597_insert__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,A: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ A2 @ ( insert3687027775829606434et_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member2946998982187404937et_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_598_insert__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ A2 @ ( insert_set_set_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_set_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_599_insert__iff,axiom,
! [A2: set_nat,B2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_set_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_600_insert__iff,axiom,
! [A2: nat > nat,B2: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_601_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_602_insertCI,axiom,
! [A2: set_set_set_nat,B: set_set_set_set_nat,B2: set_set_set_nat] :
( ( ~ ( member2946998982187404937et_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member2946998982187404937et_nat @ A2 @ ( insert3687027775829606434et_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_603_insertCI,axiom,
! [A2: set_set_nat,B: set_set_set_nat,B2: set_set_nat] :
( ( ~ ( member_set_set_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_set_nat @ A2 @ ( insert_set_set_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_604_insertCI,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( ~ ( member_set_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_605_insertCI,axiom,
! [A2: nat > nat,B: set_nat_nat,B2: nat > nat] :
( ( ~ ( member_nat_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_606_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_607_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_608_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_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_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_617_approx__pos_Ocases,axiom,
! [X2: monotone_mformula_a] :
( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X2
!= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ( X2 != monotone_TRUE_a )
=> ( ( X2 != monotone_FALSE_a )
=> ( ! [V3: a] :
( X2
!= ( monotone_Var_a @ V3 ) )
=> ~ ! [V3: monotone_mformula_a,Va2: monotone_mformula_a] :
( X2
!= ( monotone_Disj_a @ V3 @ Va2 ) ) ) ) ) ) ).
% approx_pos.cases
thf(fact_618_approx__neg_Ocases,axiom,
! [X2: monotone_mformula_a] :
( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X2
!= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X2
!= ( monotone_Disj_a @ Phi2 @ Psi2 ) )
=> ( ( X2 != monotone_TRUE_a )
=> ( ( X2 != monotone_FALSE_a )
=> ~ ! [V3: a] :
( X2
!= ( monotone_Var_a @ V3 ) ) ) ) ) ) ).
% approx_neg.cases
thf(fact_619_SET_Ocases,axiom,
! [X2: monotone_mformula_a] :
( ( X2 != monotone_FALSE_a )
=> ( ! [X4: a] :
( X2
!= ( monotone_Var_a @ X4 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X2
!= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X2
!= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( X2 = monotone_TRUE_a ) ) ) ) ) ).
% SET.cases
thf(fact_620_mformula_Oinject_I2_J,axiom,
! [X41: monotone_mformula_a,X42: monotone_mformula_a,Y41: monotone_mformula_a,Y42: monotone_mformula_a] :
( ( ( monotone_Conj_a @ X41 @ X42 )
= ( monotone_Conj_a @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% mformula.inject(2)
thf(fact_621_mformula_Oinject_I1_J,axiom,
! [X32: a,Y32: a] :
( ( ( monotone_Var_a @ X32 )
= ( monotone_Var_a @ Y32 ) )
= ( X32 = Y32 ) ) ).
% mformula.inject(1)
thf(fact_622_SET_Osimps_I2_J,axiom,
! [X2: a] :
( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Var_a @ X2 ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X2 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).
% SET.simps(2)
thf(fact_623_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_624_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_625_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_626_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_627_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_628_inf_Obounded__iff,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat_nat @ A2 @ B2 )
& ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_629_le__inf__iff,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z ) )
= ( ( ord_le9131159989063066194et_nat @ X2 @ Y )
& ( ord_le9131159989063066194et_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_630_le__inf__iff,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y @ Z ) )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Y )
& ( ord_le6893508408891458716et_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_631_le__inf__iff,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y )
& ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_632_le__inf__iff,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z ) )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Y )
& ( ord_le9059583361652607317at_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_633_le__inf__iff,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_634_le__inf__iff,axiom,
! [X2: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat_nat @ X2 @ Y )
& ( ord_less_eq_nat_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_635_empty__subsetI,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% empty_subsetI
thf(fact_636_empty__subsetI,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% empty_subsetI
thf(fact_637_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_638_empty__subsetI,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% empty_subsetI
thf(fact_639_subset__empty,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_640_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_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_sup_Obounded__iff,axiom,
! [B2: nat > nat,C: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat_nat @ B2 @ A2 )
& ( ord_less_eq_nat_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_649_le__sup__iff,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ Z )
= ( ( ord_le9131159989063066194et_nat @ X2 @ Z )
& ( ord_le9131159989063066194et_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_650_le__sup__iff,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Z )
& ( ord_le6893508408891458716et_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_651_le__sup__iff,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X2 @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_652_le__sup__iff,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ Z )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Z )
& ( ord_le9059583361652607317at_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_653_le__sup__iff,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_654_le__sup__iff,axiom,
! [X2: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_nat_nat @ X2 @ Z )
& ( ord_less_eq_nat_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_655_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_656_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_657_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_658_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_659_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= bot_bo7198184520161983622et_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_660_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_661_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_662_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_663_inf__bot__right,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_right
thf(fact_664_inf__bot__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_665_inf__bot__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_666_inf__bot__right,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% inf_bot_right
thf(fact_667_inf__bot__left,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_left
thf(fact_668_inf__bot__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_669_inf__bot__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_670_inf__bot__left,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= bot_bot_set_nat_nat ) ).
% inf_bot_left
thf(fact_671_insert__image,axiom,
! [X2: set_set_set_nat,A: set_set_set_set_nat,F: set_set_set_nat > set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_2225960715480453173et_nat @ F @ A ) )
= ( image_2225960715480453173et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_672_insert__image,axiom,
! [X2: set_set_set_nat,A: set_set_set_set_nat,F: set_set_set_nat > set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_7149431738526707583et_nat @ F @ A ) )
= ( image_7149431738526707583et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_673_insert__image,axiom,
! [X2: set_set_nat,A: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_7884819252390400639et_nat @ F @ A ) )
= ( image_7884819252390400639et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_674_insert__image,axiom,
! [X2: set_set_nat,A: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_5842784325960735177et_nat @ F @ A ) )
= ( image_5842784325960735177et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_675_insert__image,axiom,
! [X2: set_nat,A: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_6725021117256019401et_nat @ F @ A ) )
= ( image_6725021117256019401et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_676_insert__image,axiom,
! [X2: set_nat,A: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A ) )
= ( image_7916887816326733075et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_677_insert__image,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > set_set_nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_9186907679027735170et_nat @ F @ A ) )
= ( image_9186907679027735170et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_678_insert__image,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_7432509271690132940et_nat @ F @ A ) )
= ( image_7432509271690132940et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_679_insert__image,axiom,
! [X2: a,A: set_a,F: a > set_set_nat] :
( ( member_a @ X2 @ A )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_a_set_set_nat @ F @ A ) )
= ( image_a_set_set_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_680_insert__image,axiom,
! [X2: a,A: set_a,F: a > set_nat] :
( ( member_a @ X2 @ A )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_a_set_nat @ F @ A ) )
= ( image_a_set_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_681_image__insert,axiom,
! [F: ( nat > nat ) > set_set_nat,A2: nat > nat,B: set_nat_nat] :
( ( image_9186907679027735170et_nat @ F @ ( insert_nat_nat @ A2 @ B ) )
= ( insert_set_set_nat @ ( F @ A2 ) @ ( image_9186907679027735170et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_682_image__insert,axiom,
! [F: set_set_nat > set_set_nat,A2: set_set_nat,B: set_set_set_nat] :
( ( image_7884819252390400639et_nat @ F @ ( insert_set_set_nat @ A2 @ B ) )
= ( insert_set_set_nat @ ( F @ A2 ) @ ( image_7884819252390400639et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_683_image__insert,axiom,
! [F: set_set_nat > set_nat,A2: set_set_nat,B: set_set_set_nat] :
( ( image_5842784325960735177et_nat @ F @ ( insert_set_set_nat @ A2 @ B ) )
= ( insert_set_nat @ ( F @ A2 ) @ ( image_5842784325960735177et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_684_image__insert,axiom,
! [F: set_nat > set_set_nat,A2: set_nat,B: set_set_nat] :
( ( image_6725021117256019401et_nat @ F @ ( insert_set_nat @ A2 @ B ) )
= ( insert_set_set_nat @ ( F @ A2 ) @ ( image_6725021117256019401et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_685_image__insert,axiom,
! [F: set_nat > set_nat,A2: set_nat,B: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( insert_set_nat @ A2 @ B ) )
= ( insert_set_nat @ ( F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_686_insert__subset,axiom,
! [X2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ ( insert3687027775829606434et_nat @ X2 @ A ) @ B )
= ( ( member2946998982187404937et_nat @ X2 @ B )
& ( ord_le572741076514265352et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_687_insert__subset,axiom,
! [X2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ X2 @ A ) @ B )
= ( ( member_set_set_nat @ X2 @ B )
& ( ord_le9131159989063066194et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_688_insert__subset,axiom,
! [X2: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A ) @ B )
= ( ( member_a @ X2 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_689_insert__subset,axiom,
! [X2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ A ) @ B )
= ( ( member_set_nat @ X2 @ B )
& ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_690_insert__subset,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A ) @ B )
= ( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_691_insert__subset,axiom,
! [X2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( insert_nat_nat @ X2 @ A ) @ B )
= ( ( member_nat_nat @ X2 @ B )
& ( ord_le9059583361652607317at_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_692_singletonI,axiom,
! [A2: set_set_set_nat] : ( member2946998982187404937et_nat @ A2 @ ( insert3687027775829606434et_nat @ A2 @ bot_bo193956671110832956et_nat ) ) ).
% singletonI
thf(fact_693_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_694_singletonI,axiom,
! [A2: set_set_nat] : ( member_set_set_nat @ A2 @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ) ).
% singletonI
thf(fact_695_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_696_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_697_singletonI,axiom,
! [A2: nat > nat] : ( member_nat_nat @ A2 @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) ).
% singletonI
thf(fact_698_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_699_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_700_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_701_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_702_Int__insert__right__if1,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ A2 @ A )
=> ( ( inf_in2396666505901392698et_nat @ A @ ( insert3687027775829606434et_nat @ A2 @ B ) )
= ( insert3687027775829606434et_nat @ A2 @ ( inf_in2396666505901392698et_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_703_Int__insert__right__if1,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_704_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_705_Int__insert__right__if1,axiom,
! [A2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ A2 @ A )
=> ( ( inf_in5711780100303410308et_nat @ A @ ( insert_set_set_nat @ A2 @ B ) )
= ( insert_set_set_nat @ A2 @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_706_Int__insert__right__if1,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_707_Int__insert__right__if0,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ~ ( member2946998982187404937et_nat @ A2 @ A )
=> ( ( inf_in2396666505901392698et_nat @ A @ ( insert3687027775829606434et_nat @ A2 @ B ) )
= ( inf_in2396666505901392698et_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_708_Int__insert__right__if0,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B ) )
= ( inf_inf_set_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_709_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_710_Int__insert__right__if0,axiom,
! [A2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ A2 @ A )
=> ( ( inf_in5711780100303410308et_nat @ A @ ( insert_set_set_nat @ A2 @ B ) )
= ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_711_Int__insert__right__if0,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ A2 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
= ( inf_inf_set_nat_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_712_insert__inter__insert,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ ( insert_set_nat @ A2 @ B ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_713_insert__inter__insert,axiom,
! [A2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A2 @ A ) @ ( insert_set_set_nat @ A2 @ B ) )
= ( insert_set_set_nat @ A2 @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_714_insert__inter__insert,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ A ) @ ( insert_nat_nat @ A2 @ B ) )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_715_Int__insert__left__if1,axiom,
! [A2: set_set_set_nat,C2: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ A2 @ C2 )
=> ( ( inf_in2396666505901392698et_nat @ ( insert3687027775829606434et_nat @ A2 @ B ) @ C2 )
= ( insert3687027775829606434et_nat @ A2 @ ( inf_in2396666505901392698et_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_716_Int__insert__left__if1,axiom,
! [A2: set_nat,C2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B ) @ C2 )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_717_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_718_Int__insert__left__if1,axiom,
! [A2: set_set_nat,C2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ A2 @ C2 )
=> ( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A2 @ B ) @ C2 )
= ( insert_set_set_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_719_Int__insert__left__if1,axiom,
! [A2: nat > nat,C2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C2 )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_720_Int__insert__left__if0,axiom,
! [A2: set_set_set_nat,C2: set_set_set_set_nat,B: set_set_set_set_nat] :
( ~ ( member2946998982187404937et_nat @ A2 @ C2 )
=> ( ( inf_in2396666505901392698et_nat @ ( insert3687027775829606434et_nat @ A2 @ B ) @ C2 )
= ( inf_in2396666505901392698et_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_721_Int__insert__left__if0,axiom,
! [A2: set_nat,C2: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_722_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_723_Int__insert__left__if0,axiom,
! [A2: set_set_nat,C2: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ A2 @ C2 )
=> ( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A2 @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_724_Int__insert__left__if0,axiom,
! [A2: nat > nat,C2: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_725_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_726_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_727_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_728_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_729_sup__inf__absorb,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_730_sup__inf__absorb,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_731_sup__inf__absorb,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_732_sup__inf__absorb,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_733_inf__sup__absorb,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_734_inf__sup__absorb,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_735_inf__sup__absorb,axiom,
! [X2: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_736_inf__sup__absorb,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_737_Un__insert__right,axiom,
! [A: set_set_nat,A2: set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( insert_set_nat @ A2 @ B ) )
= ( insert_set_nat @ A2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_738_Un__insert__right,axiom,
! [A: set_set_set_nat,A2: set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( insert_set_set_nat @ A2 @ B ) )
= ( insert_set_set_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_739_Un__insert__right,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_740_Un__insert__right,axiom,
! [A: set_nat_nat,A2: nat > nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
= ( insert_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_741_Un__insert__left,axiom,
! [A2: set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( insert_set_nat @ A2 @ B ) @ C2 )
= ( insert_set_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_742_Un__insert__left,axiom,
! [A2: set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( insert_set_set_nat @ A2 @ B ) @ C2 )
= ( insert_set_set_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_743_Un__insert__left,axiom,
! [A2: nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_744_Un__insert__left,axiom,
! [A2: nat > nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C2 )
= ( insert_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_745_Un__Int__eq_I1_J,axiom,
! [S2: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S2 @ T ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_746_Un__Int__eq_I1_J,axiom,
! [S2: set_set_set_nat,T: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_747_Un__Int__eq_I1_J,axiom,
! [S2: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S2 @ T ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_748_Un__Int__eq_I1_J,axiom,
! [S2: set_nat_nat,T: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S2 @ T ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_749_Un__Int__eq_I2_J,axiom,
! [S2: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S2 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_750_Un__Int__eq_I2_J,axiom,
! [S2: set_set_set_nat,T: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_751_Un__Int__eq_I2_J,axiom,
! [S2: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S2 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_752_Un__Int__eq_I2_J,axiom,
! [S2: set_nat_nat,T: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S2 @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_753_Un__Int__eq_I3_J,axiom,
! [S2: set_set_nat,T: set_set_nat] :
( ( inf_inf_set_set_nat @ S2 @ ( sup_sup_set_set_nat @ S2 @ T ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_754_Un__Int__eq_I3_J,axiom,
! [S2: set_set_set_nat,T: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ S2 @ ( sup_su4213647025997063966et_nat @ S2 @ T ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_755_Un__Int__eq_I3_J,axiom,
! [S2: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ S2 @ ( sup_sup_set_nat @ S2 @ T ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_756_Un__Int__eq_I3_J,axiom,
! [S2: set_nat_nat,T: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S2 @ ( sup_sup_set_nat_nat @ S2 @ T ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_757_Un__Int__eq_I4_J,axiom,
! [T: set_set_nat,S2: set_set_nat] :
( ( inf_inf_set_set_nat @ T @ ( sup_sup_set_set_nat @ S2 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_758_Un__Int__eq_I4_J,axiom,
! [T: set_set_set_nat,S2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ T @ ( sup_su4213647025997063966et_nat @ S2 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_759_Un__Int__eq_I4_J,axiom,
! [T: set_nat,S2: set_nat] :
( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S2 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_760_Un__Int__eq_I4_J,axiom,
! [T: set_nat_nat,S2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T @ ( sup_sup_set_nat_nat @ S2 @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_761_Int__Un__eq_I1_J,axiom,
! [S2: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S2 @ T ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_762_Int__Un__eq_I1_J,axiom,
! [S2: set_set_set_nat,T: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S2 @ T ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_763_Int__Un__eq_I1_J,axiom,
! [S2: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S2 @ T ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_764_Int__Un__eq_I1_J,axiom,
! [S2: set_nat_nat,T: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S2 @ T ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_765_Int__Un__eq_I2_J,axiom,
! [S2: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S2 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_766_Int__Un__eq_I2_J,axiom,
! [S2: set_set_set_nat,T: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S2 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_767_Int__Un__eq_I2_J,axiom,
! [S2: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S2 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_768_Int__Un__eq_I2_J,axiom,
! [S2: set_nat_nat,T: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S2 @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_769_Int__Un__eq_I3_J,axiom,
! [S2: set_set_nat,T: set_set_nat] :
( ( sup_sup_set_set_nat @ S2 @ ( inf_inf_set_set_nat @ S2 @ T ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_770_Int__Un__eq_I3_J,axiom,
! [S2: set_set_set_nat,T: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ S2 @ ( inf_in5711780100303410308et_nat @ S2 @ T ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_771_Int__Un__eq_I3_J,axiom,
! [S2: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ S2 @ ( inf_inf_set_nat @ S2 @ T ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_772_Int__Un__eq_I3_J,axiom,
! [S2: set_nat_nat,T: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S2 @ ( inf_inf_set_nat_nat @ S2 @ T ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_773_Int__Un__eq_I4_J,axiom,
! [T: set_set_nat,S2: set_set_nat] :
( ( sup_sup_set_set_nat @ T @ ( inf_inf_set_set_nat @ S2 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_774_Int__Un__eq_I4_J,axiom,
! [T: set_set_set_nat,S2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ T @ ( inf_in5711780100303410308et_nat @ S2 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_775_Int__Un__eq_I4_J,axiom,
! [T: set_nat,S2: set_nat] :
( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S2 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_776_Int__Un__eq_I4_J,axiom,
! [T: set_nat_nat,S2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T @ ( inf_inf_set_nat_nat @ S2 @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_777_singleton__insert__inj__eq_H,axiom,
! [A2: set_set_nat,A: set_set_set_nat,B2: set_set_nat] :
( ( ( insert_set_set_nat @ A2 @ A )
= ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) )
= ( ( A2 = B2 )
& ( ord_le9131159989063066194et_nat @ A @ ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_778_singleton__insert__inj__eq_H,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_nat] :
( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) )
= ( ( A2 = B2 )
& ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_779_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B2: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_780_singleton__insert__inj__eq_H,axiom,
! [A2: nat > nat,A: set_nat_nat,B2: nat > nat] :
( ( ( insert_nat_nat @ A2 @ A )
= ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) )
= ( ( A2 = B2 )
& ( ord_le9059583361652607317at_nat @ A @ ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_781_singleton__insert__inj__eq,axiom,
! [B2: set_set_nat,A2: set_set_nat,A: set_set_set_nat] :
( ( ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat )
= ( insert_set_set_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le9131159989063066194et_nat @ A @ ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_782_singleton__insert__inj__eq,axiom,
! [B2: set_nat,A2: set_nat,A: set_set_nat] :
( ( ( insert_set_nat @ B2 @ bot_bot_set_set_nat )
= ( insert_set_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_783_singleton__insert__inj__eq,axiom,
! [B2: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_784_singleton__insert__inj__eq,axiom,
! [B2: nat > nat,A2: nat > nat,A: set_nat_nat] :
( ( ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat )
= ( insert_nat_nat @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_le9059583361652607317at_nat @ A @ ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_785_insert__disjoint_I1_J,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ( inf_in2396666505901392698et_nat @ ( insert3687027775829606434et_nat @ A2 @ A ) @ B )
= bot_bo193956671110832956et_nat )
= ( ~ ( member2946998982187404937et_nat @ A2 @ B )
& ( ( inf_in2396666505901392698et_nat @ A @ B )
= bot_bo193956671110832956et_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_786_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_787_insert__disjoint_I1_J,axiom,
! [A2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A2 @ A ) @ B )
= bot_bo7198184520161983622et_nat )
= ( ~ ( member_set_set_nat @ A2 @ B )
& ( ( inf_in5711780100303410308et_nat @ A @ B )
= bot_bo7198184520161983622et_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_788_insert__disjoint_I1_J,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ B )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A2 @ B )
& ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_789_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_790_insert__disjoint_I1_J,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ A ) @ B )
= bot_bot_set_nat_nat )
= ( ~ ( member_nat_nat @ A2 @ B )
& ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_791_insert__disjoint_I2_J,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( bot_bo193956671110832956et_nat
= ( inf_in2396666505901392698et_nat @ ( insert3687027775829606434et_nat @ A2 @ A ) @ B ) )
= ( ~ ( member2946998982187404937et_nat @ A2 @ B )
& ( bot_bo193956671110832956et_nat
= ( inf_in2396666505901392698et_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_792_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_793_insert__disjoint_I2_J,axiom,
! [A2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_set_set_nat @ A2 @ B )
& ( bot_bo7198184520161983622et_nat
= ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_794_insert__disjoint_I2_J,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_set_nat @ A2 @ B )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_795_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_796_insert__disjoint_I2_J,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat_nat @ A2 @ B )
& ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_797_disjoint__insert_I1_J,axiom,
! [B: set_set_set_set_nat,A2: set_set_set_nat,A: set_set_set_set_nat] :
( ( ( inf_in2396666505901392698et_nat @ B @ ( insert3687027775829606434et_nat @ A2 @ A ) )
= bot_bo193956671110832956et_nat )
= ( ~ ( member2946998982187404937et_nat @ A2 @ B )
& ( ( inf_in2396666505901392698et_nat @ B @ A )
= bot_bo193956671110832956et_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_798_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_799_disjoint__insert_I1_J,axiom,
! [B: set_set_set_nat,A2: set_set_nat,A: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ B @ ( insert_set_set_nat @ A2 @ A ) )
= bot_bo7198184520161983622et_nat )
= ( ~ ( member_set_set_nat @ A2 @ B )
& ( ( inf_in5711780100303410308et_nat @ B @ A )
= bot_bo7198184520161983622et_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_800_disjoint__insert_I1_J,axiom,
! [B: set_set_nat,A2: set_nat,A: set_set_nat] :
( ( ( inf_inf_set_set_nat @ B @ ( insert_set_nat @ A2 @ A ) )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A2 @ B )
& ( ( inf_inf_set_set_nat @ B @ A )
= bot_bot_set_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_801_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_802_disjoint__insert_I1_J,axiom,
! [B: set_nat_nat,A2: nat > nat,A: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ B @ ( insert_nat_nat @ A2 @ A ) )
= bot_bot_set_nat_nat )
= ( ~ ( member_nat_nat @ A2 @ B )
& ( ( inf_inf_set_nat_nat @ B @ A )
= bot_bot_set_nat_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_803_disjoint__insert_I2_J,axiom,
! [A: set_set_set_set_nat,B2: set_set_set_nat,B: set_set_set_set_nat] :
( ( bot_bo193956671110832956et_nat
= ( inf_in2396666505901392698et_nat @ A @ ( insert3687027775829606434et_nat @ B2 @ B ) ) )
= ( ~ ( member2946998982187404937et_nat @ B2 @ A )
& ( bot_bo193956671110832956et_nat
= ( inf_in2396666505901392698et_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_804_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_805_disjoint__insert_I2_J,axiom,
! [A: set_set_set_nat,B2: set_set_nat,B: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( inf_in5711780100303410308et_nat @ A @ ( insert_set_set_nat @ B2 @ B ) ) )
= ( ~ ( member_set_set_nat @ B2 @ A )
& ( bot_bo7198184520161983622et_nat
= ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_806_disjoint__insert_I2_J,axiom,
! [A: set_set_nat,B2: set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ B2 @ B ) ) )
= ( ~ ( member_set_nat @ B2 @ A )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_807_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_808_disjoint__insert_I2_J,axiom,
! [A: set_nat_nat,B2: nat > nat,B: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ B2 @ B ) ) )
= ( ~ ( member_nat_nat @ B2 @ A )
& ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_809_complete__lattice__class_OSup__insert,axiom,
! [A2: set_set_nat,A: set_set_set_nat] :
( ( comple548664676211718543et_nat @ ( insert_set_set_nat @ A2 @ A ) )
= ( sup_sup_set_set_nat @ A2 @ ( comple548664676211718543et_nat @ A ) ) ) ).
% complete_lattice_class.Sup_insert
thf(fact_810_complete__lattice__class_OSup__insert,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat] :
( ( comple6569609367425551173et_nat @ ( insert3687027775829606434et_nat @ A2 @ A ) )
= ( sup_su4213647025997063966et_nat @ A2 @ ( comple6569609367425551173et_nat @ A ) ) ) ).
% complete_lattice_class.Sup_insert
thf(fact_811_complete__lattice__class_OSup__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( insert_set_nat @ A2 @ A ) )
= ( sup_sup_set_nat @ A2 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% complete_lattice_class.Sup_insert
thf(fact_812_complete__lattice__class_OSup__insert,axiom,
! [A2: set_nat_nat,A: set_set_nat_nat] :
( ( comple5448282615319421384at_nat @ ( insert_set_nat_nat @ A2 @ A ) )
= ( sup_sup_set_nat_nat @ A2 @ ( comple5448282615319421384at_nat @ A ) ) ) ).
% complete_lattice_class.Sup_insert
thf(fact_813_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_814_Un__Int__assoc__eq,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) )
= ( ord_le9131159989063066194et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_815_Un__Int__assoc__eq,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) )
= ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_816_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_817_Un__Int__assoc__eq,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) )
= ( ord_le9059583361652607317at_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_818_distrib__inf__le,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z ) ) @ ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_819_distrib__inf__le,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y ) @ ( inf_inf_set_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_820_distrib__inf__le,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y ) @ ( inf_inf_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_821_distrib__inf__le,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ ( inf_inf_set_nat_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_822_distrib__inf__le,axiom,
! [X2: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y ) @ ( inf_inf_nat @ X2 @ Z ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_823_distrib__inf__le,axiom,
! [X2: nat > nat,Y: nat > nat,Z: nat > nat] : ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y ) @ ( inf_inf_nat_nat @ X2 @ Z ) ) @ ( inf_inf_nat_nat @ X2 @ ( sup_sup_nat_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_824_distrib__sup__le,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z ) ) @ ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_825_distrib__sup__le,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y @ Z ) ) @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ ( sup_sup_set_set_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_826_distrib__sup__le,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_827_distrib__sup__le,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z ) ) @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ ( sup_sup_set_nat_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_828_distrib__sup__le,axiom,
! [X2: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y ) @ ( sup_sup_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_829_distrib__sup__le,axiom,
! [X2: nat > nat,Y: nat > nat,Z: nat > nat] : ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y @ Z ) ) @ ( inf_inf_nat_nat @ ( sup_sup_nat_nat @ X2 @ Y ) @ ( sup_sup_nat_nat @ X2 @ Z ) ) ) ).
% distrib_sup_le
thf(fact_830_Sup__inter__less__eq,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( comple6569609367425551173et_nat @ ( inf_in2396666505901392698et_nat @ A @ B ) ) @ ( inf_in5711780100303410308et_nat @ ( comple6569609367425551173et_nat @ A ) @ ( comple6569609367425551173et_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_831_Sup__inter__less__eq,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) ) @ ( inf_inf_set_set_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_832_Sup__inter__less__eq,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_833_Sup__inter__less__eq,axiom,
! [A: set_set_nat_nat,B: set_set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( inf_in710756014367367485at_nat @ A @ B ) ) @ ( inf_inf_set_nat_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_834_Union__Int__subset,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( comple6569609367425551173et_nat @ ( inf_in2396666505901392698et_nat @ A @ B ) ) @ ( inf_in5711780100303410308et_nat @ ( comple6569609367425551173et_nat @ A ) @ ( comple6569609367425551173et_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_835_Union__Int__subset,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) ) @ ( inf_inf_set_set_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_836_Union__Int__subset,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_837_Union__Int__subset,axiom,
! [A: set_set_nat_nat,B: set_set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ ( inf_in710756014367367485at_nat @ A @ B ) ) @ ( inf_inf_set_nat_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_838_order__antisym__conv,axiom,
! [Y: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_839_order__antisym__conv,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_840_order__antisym__conv,axiom,
! [Y: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_841_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_842_order__antisym__conv,axiom,
! [Y: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_843_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_844_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_845_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_846_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_847_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_848_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_849_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_850_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_851_ord__le__eq__subst,axiom,
! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > nat,C: nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_852_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_853_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_set_nat,C: set_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_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_854_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_855_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_856_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_857_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_858_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_859_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_860_ord__eq__le__subst,axiom,
! [A2: nat > nat,F: nat > 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_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_861_ord__eq__le__subst,axiom,
! [A2: nat,F: ( nat > nat ) > nat,B2: nat > nat,C: nat > nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ! [X4: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_862_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_863_ord__eq__le__subst,axiom,
! [A2: set_set_nat,F: set_nat > set_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_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_864_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_865_order__eq__refl,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( X2 = Y )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_866_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_867_order__eq__refl,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( X2 = Y )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_868_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_869_order__eq__refl,axiom,
! [X2: nat > nat,Y: nat > nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_870_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_871_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_872_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_873_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_874_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_875_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_876_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_877_order__subst2,axiom,
! [A2: nat > nat,B2: nat > nat,F: ( nat > nat ) > nat,C: nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_878_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_879_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ ( F @ 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 @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_880_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_881_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_882_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_883_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_884_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_885_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_886_order__subst1,axiom,
! [A2: nat,F: ( nat > nat ) > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ! [X4: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_887_order__subst1,axiom,
! [A2: nat > nat,F: nat > nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_888_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_889_order__subst1,axiom,
! [A2: set_nat,F: set_set_nat > set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_less_eq_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 ) ) ) ) ) ).
% order_subst1
thf(fact_890_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_nat,Z2: set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_891_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_892_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_893_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_894_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat > nat,Z2: nat > nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat > nat,B3: nat > nat] :
( ( ord_less_eq_nat_nat @ A3 @ B3 )
& ( ord_less_eq_nat_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_895_le__fun__def,axiom,
( ord_less_eq_nat_nat
= ( ^ [F2: nat > nat,G3: nat > nat] :
! [X: nat] : ( ord_less_eq_nat @ ( F2 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_fun_def
thf(fact_896_le__funI,axiom,
! [F: nat > nat,G2: nat > nat] :
( ! [X4: nat] : ( ord_less_eq_nat @ ( F @ X4 ) @ ( G2 @ X4 ) )
=> ( ord_less_eq_nat_nat @ F @ G2 ) ) ).
% le_funI
thf(fact_897_le__funE,axiom,
! [F: nat > nat,G2: nat > nat,X2: nat] :
( ( ord_less_eq_nat_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_898_le__funD,axiom,
! [F: nat > nat,G2: nat > nat,X2: nat] :
( ( ord_less_eq_nat_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_899_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_900_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_901_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_902_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_903_antisym,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_904_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_905_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_906_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_907_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_908_dual__order_Otrans,axiom,
! [B2: nat > nat,A2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ C @ B2 )
=> ( ord_less_eq_nat_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_909_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_910_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_911_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_912_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_913_dual__order_Oantisym,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_914_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_nat,Z2: set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
& ( ord_le6893508408891458716et_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_915_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_916_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ A3 )
& ( ord_le9059583361652607317at_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_917_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_918_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat > nat,Z2: nat > nat] : ( Y5 = Z2 ) )
= ( ^ [A3: nat > nat,B3: nat > nat] :
( ( ord_less_eq_nat_nat @ B3 @ A3 )
& ( ord_less_eq_nat_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_919_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_920_order__trans,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Z )
=> ( ord_le6893508408891458716et_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_921_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_922_order__trans,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ Z )
=> ( ord_le9059583361652607317at_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_923_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_924_order__trans,axiom,
! [X2: nat > nat,Y: nat > nat,Z: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat_nat @ Y @ Z )
=> ( ord_less_eq_nat_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_925_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_926_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_927_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_928_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_929_order_Otrans,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_930_order__antisym,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_931_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_932_order__antisym,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_933_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_934_order__antisym,axiom,
! [X2: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_935_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_936_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_937_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_938_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_939_ord__le__eq__trans,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_940_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_941_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_942_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_943_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_944_ord__eq__le__trans,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_945_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_nat,Z2: set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [X: set_set_nat,Y3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X @ Y3 )
& ( ord_le6893508408891458716et_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_946_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_947_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [X: set_nat_nat,Y3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X @ Y3 )
& ( ord_le9059583361652607317at_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_948_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_949_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat > nat,Z2: nat > nat] : ( Y5 = Z2 ) )
= ( ^ [X: nat > nat,Y3: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ Y3 )
& ( ord_less_eq_nat_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_950_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_951_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_952_mformula_Odistinct_I3_J,axiom,
! [X32: a] :
( monotone_TRUE_a
!= ( monotone_Var_a @ X32 ) ) ).
% mformula.distinct(3)
thf(fact_953_mformula_Odistinct_I5_J,axiom,
! [X41: monotone_mformula_a,X42: monotone_mformula_a] :
( monotone_TRUE_a
!= ( monotone_Conj_a @ X41 @ X42 ) ) ).
% mformula.distinct(5)
thf(fact_954_mformula_Odistinct_I15_J,axiom,
! [X32: a,X41: monotone_mformula_a,X42: monotone_mformula_a] :
( ( monotone_Var_a @ X32 )
!= ( monotone_Conj_a @ X41 @ X42 ) ) ).
% mformula.distinct(15)
thf(fact_955_image__Int__subset,axiom,
! [F: set_set_nat > set_set_nat,A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( image_7884819252390400639et_nat @ F @ ( inf_in5711780100303410308et_nat @ A @ B ) ) @ ( inf_in5711780100303410308et_nat @ ( image_7884819252390400639et_nat @ F @ A ) @ ( image_7884819252390400639et_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_956_image__Int__subset,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,B: set_nat_nat] : ( ord_le9131159989063066194et_nat @ ( image_9186907679027735170et_nat @ F @ ( inf_inf_set_nat_nat @ A @ B ) ) @ ( inf_in5711780100303410308et_nat @ ( image_9186907679027735170et_nat @ F @ A ) @ ( image_9186907679027735170et_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_957_image__Int__subset,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,B: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ ( inf_in5711780100303410308et_nat @ A @ B ) ) @ ( inf_inf_set_set_nat @ ( image_5842784325960735177et_nat @ F @ A ) @ ( image_5842784325960735177et_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_958_image__Int__subset,axiom,
! [F: ( nat > nat ) > set_nat,A: set_nat_nat,B: set_nat_nat] : ( ord_le6893508408891458716et_nat @ ( image_7432509271690132940et_nat @ F @ ( inf_inf_set_nat_nat @ A @ B ) ) @ ( inf_inf_set_set_nat @ ( image_7432509271690132940et_nat @ F @ A ) @ ( image_7432509271690132940et_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_959_image__Int__subset,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat,B: set_set_set_nat] : ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ ( inf_in5711780100303410308et_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_1454916318497077779at_nat @ F @ A ) @ ( image_1454916318497077779at_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_960_image__Int__subset,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,B: set_nat_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ ( inf_inf_set_nat_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat_nat @ F @ A ) @ ( image_nat_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_961_image__Int__subset,axiom,
! [F: set_set_nat > nat > nat,A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( image_8441894408526374658at_nat @ F @ ( inf_in5711780100303410308et_nat @ A @ B ) ) @ ( inf_inf_set_nat_nat @ ( image_8441894408526374658at_nat @ F @ A ) @ ( image_8441894408526374658at_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_962_image__Int__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( image_3205354838064109189at_nat @ F @ ( inf_inf_set_nat_nat @ A @ B ) ) @ ( inf_inf_set_nat_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_963_Sup__subset__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_964_Sup__subset__mono,axiom,
! [A: set_set_nat_nat,B: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_965_Sup__subset__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_966_Union__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( comple548664676211718543et_nat @ A ) @ ( comple548664676211718543et_nat @ B ) ) ) ).
% Union_mono
thf(fact_967_Union__mono,axiom,
! [A: set_set_nat_nat,B: set_set_nat_nat] :
( ( ord_le4954213926817602059at_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ ( comple5448282615319421384at_nat @ A ) @ ( comple5448282615319421384at_nat @ B ) ) ) ).
% Union_mono
thf(fact_968_Union__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_mono
thf(fact_969_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_set_set_nat,K: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( B
= ( inf_in5711780100303410308et_nat @ K @ B2 ) )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B )
= ( inf_in5711780100303410308et_nat @ K @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_970_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_nat_nat,K: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( B
= ( inf_inf_set_nat_nat @ K @ B2 ) )
=> ( ( inf_inf_set_nat_nat @ A2 @ B )
= ( inf_inf_set_nat_nat @ K @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_971_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_set_set_nat,K: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A
= ( inf_in5711780100303410308et_nat @ K @ A2 ) )
=> ( ( inf_in5711780100303410308et_nat @ A @ B2 )
= ( inf_in5711780100303410308et_nat @ K @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_972_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_nat_nat,K: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( A
= ( inf_inf_set_nat_nat @ K @ A2 ) )
=> ( ( inf_inf_set_nat_nat @ A @ B2 )
= ( inf_inf_set_nat_nat @ K @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_973_forth__assumptions_OAPR_Ocong,axiom,
clique3873310923663319714_APR_a = clique3873310923663319714_APR_a ).
% forth_assumptions.APR.cong
thf(fact_974_mk__disjoint__insert,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ A2 @ A )
=> ? [B6: set_set_set_set_nat] :
( ( A
= ( insert3687027775829606434et_nat @ A2 @ B6 ) )
& ~ ( member2946998982187404937et_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_975_mk__disjoint__insert,axiom,
! [A2: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ A2 @ A )
=> ? [B6: set_set_set_nat] :
( ( A
= ( insert_set_set_nat @ A2 @ B6 ) )
& ~ ( member_set_set_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_976_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ? [B6: set_set_nat] :
( ( A
= ( insert_set_nat @ A2 @ B6 ) )
& ~ ( member_set_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_977_mk__disjoint__insert,axiom,
! [A2: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ A2 @ A )
=> ? [B6: set_nat_nat] :
( ( A
= ( insert_nat_nat @ A2 @ B6 ) )
& ~ ( member_nat_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_978_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B6: set_a] :
( ( A
= ( insert_a @ A2 @ B6 ) )
& ~ ( member_a @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_979_Int__left__commute,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( inf_in5711780100303410308et_nat @ B @ ( inf_in5711780100303410308et_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_980_Int__left__commute,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ B @ ( inf_inf_set_nat_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_981_Int__insert__right,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ( member2946998982187404937et_nat @ A2 @ A )
=> ( ( inf_in2396666505901392698et_nat @ A @ ( insert3687027775829606434et_nat @ A2 @ B ) )
= ( insert3687027775829606434et_nat @ A2 @ ( inf_in2396666505901392698et_nat @ A @ B ) ) ) )
& ( ~ ( member2946998982187404937et_nat @ A2 @ A )
=> ( ( inf_in2396666505901392698et_nat @ A @ ( insert3687027775829606434et_nat @ A2 @ B ) )
= ( inf_in2396666505901392698et_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_982_Int__insert__right,axiom,
! [A2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) )
& ( ~ ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B ) )
= ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_983_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_984_Int__insert__right,axiom,
! [A2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ( member_set_set_nat @ A2 @ A )
=> ( ( inf_in5711780100303410308et_nat @ A @ ( insert_set_set_nat @ A2 @ B ) )
= ( insert_set_set_nat @ A2 @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) )
& ( ~ ( member_set_set_nat @ A2 @ A )
=> ( ( inf_in5711780100303410308et_nat @ A @ ( insert_set_set_nat @ A2 @ B ) )
= ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_985_Int__insert__right,axiom,
! [A2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( ( member_nat_nat @ A2 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) )
& ( ~ ( member_nat_nat @ A2 @ A )
=> ( ( inf_inf_set_nat_nat @ A @ ( insert_nat_nat @ A2 @ B ) )
= ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_986_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_987_Int__Collect__mono,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ! [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le572741076514265352et_nat @ ( inf_in2396666505901392698et_nat @ A @ ( collec7201453139178570183et_nat @ P ) ) @ ( inf_in2396666505901392698et_nat @ B @ ( collec7201453139178570183et_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_988_Int__Collect__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ ( collect_set_set_nat @ P ) ) @ ( inf_in5711780100303410308et_nat @ B @ ( collect_set_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_989_Int__Collect__mono,axiom,
! [A: set_set_nat,B: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_990_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_991_Int__Collect__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ ( collect_nat_nat @ P ) ) @ ( inf_inf_set_nat_nat @ B @ ( collect_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_992_Collect__mono__iff,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P ) @ ( collec7201453139178570183et_nat @ Q ) )
= ( ! [X: set_set_set_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_993_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_994_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_995_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_996_subset__insertI2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,B2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ A @ ( insert_set_set_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_997_subset__insertI2,axiom,
! [A: set_set_nat,B: set_set_nat,B2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_998_subset__insertI2,axiom,
! [A: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_999_subset__insertI2,axiom,
! [A: set_nat_nat,B: set_nat_nat,B2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ ( insert_nat_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1000_Int__left__absorb,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( inf_in5711780100303410308et_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_1001_Int__left__absorb,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( inf_inf_set_nat_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_1002_Int__insert__left,axiom,
! [A2: set_set_set_nat,C2: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ( member2946998982187404937et_nat @ A2 @ C2 )
=> ( ( inf_in2396666505901392698et_nat @ ( insert3687027775829606434et_nat @ A2 @ B ) @ C2 )
= ( insert3687027775829606434et_nat @ A2 @ ( inf_in2396666505901392698et_nat @ B @ C2 ) ) ) )
& ( ~ ( member2946998982187404937et_nat @ A2 @ C2 )
=> ( ( inf_in2396666505901392698et_nat @ ( insert3687027775829606434et_nat @ A2 @ B ) @ C2 )
= ( inf_in2396666505901392698et_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1003_Int__insert__left,axiom,
! [A2: set_nat,C2: set_set_nat,B: set_set_nat] :
( ( ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B ) @ C2 )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1004_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1005_Int__insert__left,axiom,
! [A2: set_set_nat,C2: set_set_set_nat,B: set_set_set_nat] :
( ( ( member_set_set_nat @ A2 @ C2 )
=> ( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A2 @ B ) @ C2 )
= ( insert_set_set_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) ) )
& ( ~ ( member_set_set_nat @ A2 @ C2 )
=> ( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A2 @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1006_Int__insert__left,axiom,
! [A2: nat > nat,C2: set_nat_nat,B: set_nat_nat] :
( ( ( member_nat_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C2 )
= ( insert_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1007_subset__insertI,axiom,
! [B: set_set_set_nat,A2: set_set_nat] : ( ord_le9131159989063066194et_nat @ B @ ( insert_set_set_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_1008_subset__insertI,axiom,
! [B: set_set_nat,A2: set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( insert_set_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_1009_subset__insertI,axiom,
! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_1010_subset__insertI,axiom,
! [B: set_nat_nat,A2: nat > nat] : ( ord_le9059583361652607317at_nat @ B @ ( insert_nat_nat @ A2 @ B ) ) ).
% subset_insertI
thf(fact_1011_insert__commute,axiom,
! [X2: set_set_nat,Y: set_set_nat,A: set_set_set_nat] :
( ( insert_set_set_nat @ X2 @ ( insert_set_set_nat @ Y @ A ) )
= ( insert_set_set_nat @ Y @ ( insert_set_set_nat @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_1012_insert__commute,axiom,
! [X2: set_nat,Y: set_nat,A: set_set_nat] :
( ( insert_set_nat @ X2 @ ( insert_set_nat @ Y @ A ) )
= ( insert_set_nat @ Y @ ( insert_set_nat @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_1013_subset__insert,axiom,
! [X2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ~ ( member2946998982187404937et_nat @ X2 @ A )
=> ( ( ord_le572741076514265352et_nat @ A @ ( insert3687027775829606434et_nat @ X2 @ B ) )
= ( ord_le572741076514265352et_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1014_subset__insert,axiom,
! [X2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A )
=> ( ( ord_le9131159989063066194et_nat @ A @ ( insert_set_set_nat @ X2 @ B ) )
= ( ord_le9131159989063066194et_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1015_subset__insert,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X2 @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_1016_subset__insert,axiom,
! [X2: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A )
=> ( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X2 @ B ) )
= ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1017_subset__insert,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X2 @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1018_subset__insert,axiom,
! [X2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ X2 @ A )
=> ( ( ord_le9059583361652607317at_nat @ A @ ( insert_nat_nat @ X2 @ B ) )
= ( ord_le9059583361652607317at_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_1019_set__eq__subset,axiom,
( ( ^ [Y5: set_set_nat,Z2: set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_1020_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_1021_set__eq__subset,axiom,
( ( ^ [Y5: set_nat_nat,Z2: set_nat_nat] : ( Y5 = Z2 ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_1022_insert__eq__iff,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat,B2: set_set_set_nat,B: set_set_set_set_nat] :
( ~ ( member2946998982187404937et_nat @ A2 @ A )
=> ( ~ ( member2946998982187404937et_nat @ B2 @ B )
=> ( ( ( insert3687027775829606434et_nat @ A2 @ A )
= ( insert3687027775829606434et_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_set_set_nat] :
( ( A
= ( insert3687027775829606434et_nat @ B2 @ C3 ) )
& ~ ( member2946998982187404937et_nat @ B2 @ C3 )
& ( B
= ( insert3687027775829606434et_nat @ A2 @ C3 ) )
& ~ ( member2946998982187404937et_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1023_insert__eq__iff,axiom,
! [A2: set_set_nat,A: set_set_set_nat,B2: set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ A2 @ A )
=> ( ~ ( member_set_set_nat @ B2 @ B )
=> ( ( ( insert_set_set_nat @ A2 @ A )
= ( insert_set_set_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_set_nat] :
( ( A
= ( insert_set_set_nat @ B2 @ C3 ) )
& ~ ( member_set_set_nat @ B2 @ C3 )
& ( B
= ( insert_set_set_nat @ A2 @ C3 ) )
& ~ ( member_set_set_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1024_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ~ ( member_set_nat @ B2 @ B )
=> ( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_set_nat] :
( ( A
= ( insert_set_nat @ B2 @ C3 ) )
& ~ ( member_set_nat @ B2 @ C3 )
& ( B
= ( insert_set_nat @ A2 @ C3 ) )
& ~ ( member_set_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1025_insert__eq__iff,axiom,
! [A2: nat > nat,A: set_nat_nat,B2: nat > nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ A2 @ A )
=> ( ~ ( member_nat_nat @ B2 @ B )
=> ( ( ( insert_nat_nat @ A2 @ A )
= ( insert_nat_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat_nat] :
( ( A
= ( insert_nat_nat @ B2 @ C3 ) )
& ~ ( member_nat_nat @ B2 @ C3 )
& ( B
= ( insert_nat_nat @ A2 @ C3 ) )
& ~ ( member_nat_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1026_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1027_insert__absorb,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ A2 @ A )
=> ( ( insert3687027775829606434et_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1028_insert__absorb,axiom,
! [A2: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ A2 @ A )
=> ( ( insert_set_set_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1029_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( insert_set_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1030_insert__absorb,axiom,
! [A2: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ A2 @ A )
=> ( ( insert_nat_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1031_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1032_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_1033_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_1034_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_1035_insert__ident,axiom,
! [X2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ~ ( member2946998982187404937et_nat @ X2 @ A )
=> ( ~ ( member2946998982187404937et_nat @ X2 @ B )
=> ( ( ( insert3687027775829606434et_nat @ X2 @ A )
= ( insert3687027775829606434et_nat @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1036_insert__ident,axiom,
! [X2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A )
=> ( ~ ( member_set_set_nat @ X2 @ B )
=> ( ( ( insert_set_set_nat @ X2 @ A )
= ( insert_set_set_nat @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1037_insert__ident,axiom,
! [X2: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A )
=> ( ~ ( member_set_nat @ X2 @ B )
=> ( ( ( insert_set_nat @ X2 @ A )
= ( insert_set_nat @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1038_insert__ident,axiom,
! [X2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ X2 @ A )
=> ( ~ ( member_nat_nat @ X2 @ B )
=> ( ( ( insert_nat_nat @ X2 @ A )
= ( insert_nat_nat @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1039_insert__ident,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A )
=> ( ~ ( member_a @ X2 @ B )
=> ( ( ( insert_a @ X2 @ A )
= ( insert_a @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_1040_Int__greatest,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A )
=> ( ( ord_le9131159989063066194et_nat @ C2 @ B )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1041_Int__greatest,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ B )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1042_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1043_Int__greatest,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ B )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_1044_Collect__mono,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X4: set_set_set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P ) @ ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1045_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1046_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1047_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1048_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_1049_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_1050_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_1051_insert__mono,axiom,
! [C2: set_set_set_nat,D2: set_set_set_nat,A2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ A2 @ C2 ) @ ( insert_set_set_nat @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_1052_insert__mono,axiom,
! [C2: set_set_nat,D2: set_set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ A2 @ C2 ) @ ( insert_set_nat @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_1053_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_1054_insert__mono,axiom,
! [C2: set_nat_nat,D2: set_nat_nat,A2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( insert_nat_nat @ A2 @ C2 ) @ ( insert_nat_nat @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_1055_Int__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_1056_Int__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( inf_inf_set_nat_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_1057_Int__absorb2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( inf_in5711780100303410308et_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1058_Int__absorb2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( inf_inf_set_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1059_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1060_Int__absorb2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_1061_Int__absorb1,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( inf_in5711780100303410308et_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1062_Int__absorb1,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( inf_inf_set_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1063_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1064_Int__absorb1,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_1065_subset__iff,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A4: set_set_set_set_nat,B4: set_set_set_set_nat] :
! [T2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ T2 @ A4 )
=> ( member2946998982187404937et_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1066_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
! [T2: set_set_nat] :
( ( member_set_set_nat @ T2 @ A4 )
=> ( member_set_set_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1067_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A4 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1068_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [T2: set_nat] :
( ( member_set_nat @ T2 @ A4 )
=> ( member_set_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1069_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A4 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1070_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [T2: nat > nat] :
( ( member_nat_nat @ T2 @ A4 )
=> ( member_nat_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1071_Set_Oset__insert,axiom,
! [X2: set_set_set_nat,A: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ A )
=> ~ ! [B6: set_set_set_set_nat] :
( ( A
= ( insert3687027775829606434et_nat @ X2 @ B6 ) )
=> ( member2946998982187404937et_nat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1072_Set_Oset__insert,axiom,
! [X2: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ~ ! [B6: set_set_set_nat] :
( ( A
= ( insert_set_set_nat @ X2 @ B6 ) )
=> ( member_set_set_nat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1073_Set_Oset__insert,axiom,
! [X2: set_nat,A: set_set_nat] :
( ( member_set_nat @ X2 @ A )
=> ~ ! [B6: set_set_nat] :
( ( A
= ( insert_set_nat @ X2 @ B6 ) )
=> ( member_set_nat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1074_Set_Oset__insert,axiom,
! [X2: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ X2 @ A )
=> ~ ! [B6: set_nat_nat] :
( ( A
= ( insert_nat_nat @ X2 @ B6 ) )
=> ( member_nat_nat @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1075_Set_Oset__insert,axiom,
! [X2: a,A: set_a] :
( ( member_a @ X2 @ A )
=> ~ ! [B6: set_a] :
( ( A
= ( insert_a @ X2 @ B6 ) )
=> ( member_a @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_1076_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_1077_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_1078_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_1079_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_1080_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_1081_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_1082_Int__lower2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1083_Int__lower2,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1084_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1085_Int__lower2,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_1086_Int__lower1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1087_Int__lower1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1088_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1089_Int__lower1,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_1090_Int__absorb,axiom,
! [A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_1091_Int__absorb,axiom,
! [A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_1092_subset__eq,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A4: set_set_set_set_nat,B4: set_set_set_set_nat] :
! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A4 )
=> ( member2946998982187404937et_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1093_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A4 )
=> ( member_set_set_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1094_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X: a] :
( ( member_a @ X @ A4 )
=> ( member_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1095_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ A4 )
=> ( member_set_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1096_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1097_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [X: nat > nat] :
( ( member_nat_nat @ X @ A4 )
=> ( member_nat_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1098_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_1099_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_1100_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_1101_Int__assoc,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_1102_Int__assoc,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_1103_insertI2,axiom,
! [A2: set_set_set_nat,B: set_set_set_set_nat,B2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A2 @ B )
=> ( member2946998982187404937et_nat @ A2 @ ( insert3687027775829606434et_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1104_insertI2,axiom,
! [A2: set_set_nat,B: set_set_set_nat,B2: set_set_nat] :
( ( member_set_set_nat @ A2 @ B )
=> ( member_set_set_nat @ A2 @ ( insert_set_set_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1105_insertI2,axiom,
! [A2: set_nat,B: set_set_nat,B2: set_nat] :
( ( member_set_nat @ A2 @ B )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1106_insertI2,axiom,
! [A2: nat > nat,B: set_nat_nat,B2: nat > nat] :
( ( member_nat_nat @ A2 @ B )
=> ( member_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1107_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1108_insertI1,axiom,
! [A2: set_set_set_nat,B: set_set_set_set_nat] : ( member2946998982187404937et_nat @ A2 @ ( insert3687027775829606434et_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_1109_insertI1,axiom,
! [A2: set_set_nat,B: set_set_set_nat] : ( member_set_set_nat @ A2 @ ( insert_set_set_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_1110_insertI1,axiom,
! [A2: set_nat,B: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_1111_insertI1,axiom,
! [A2: nat > nat,B: set_nat_nat] : ( member_nat_nat @ A2 @ ( insert_nat_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_1112_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_1113_Int__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 @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1114_Int__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 @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1115_Int__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 @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_1116_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_1117_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X3 ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_1118_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X3 ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_1119_first__assumptions_Ov__mono,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_1120_first__assumptions_OCLIQUE__NEG,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K ) @ ( clique3210737375870294875st_NEG @ K ) )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.CLIQUE_NEG
thf(fact_1121_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X7: set_set_set_nat,G4: set_set_nat] :
? [X: set_set_nat] :
( ( member_set_set_nat @ X @ X7 )
& ( ord_le6893508408891458716et_nat @ X @ G4 ) ) ) ) ).
% accepts_def
thf(fact_1122_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_1123_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_1124_ACC__odot,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X3 ) @ ( clique3210737319928189260st_ACC @ k @ Y2 ) ) ) ).
% ACC_odot
thf(fact_1125_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_1126_ACC__cf__odot,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X3 ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_odot
thf(fact_1127_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_1128_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_1129_second__assumptions_Odeviate__pos__cap_Ocong,axiom,
clique3314026705535538693os_cap = clique3314026705535538693os_cap ).
% second_assumptions.deviate_pos_cap.cong
thf(fact_1130_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_1131_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_1132_first__assumptions_OACC__cf__odot,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X3 ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_odot
thf(fact_1133_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3686358387679108662ccepts @ X3 @ G )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ X3 )
& ( ord_le6893508408891458716et_nat @ X @ G ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_1134_first__assumptions_OacceptsI,axiom,
! [L: nat,P2: nat,K: nat,D2: set_set_nat,G: set_set_nat,X3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ G )
=> ( ( member_set_set_nat @ D2 @ X3 )
=> ( clique3686358387679108662ccepts @ X3 @ G ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_1135_first__assumptions_OPOS__sub__CLIQUE,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_sub_CLIQUE
thf(fact_1136_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X3 ) @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_1137_first__assumptions_OACC__odot,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K @ X3 ) @ ( clique3210737319928189260st_ACC @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_odot
thf(fact_1138_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_1139_SET_Oelims,axiom,
! [X2: monotone_mformula_a,Y: set_set_set_nat] :
( ( ( clique6509092761774629891_SET_a @ pi @ X2 )
= Y )
=> ( ( ( X2 = monotone_FALSE_a )
=> ( Y != bot_bo7198184520161983622et_nat ) )
=> ( ! [X4: a] :
( ( X2
= ( monotone_Var_a @ X4 ) )
=> ( Y
!= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X2
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y
!= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X2
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y
!= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi3 ) ) ) )
=> ~ ( ( X2 = monotone_TRUE_a )
=> ( Y != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ).
% SET.elims
thf(fact_1140_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_1141_ACC__cf__I,axiom,
! [F3: nat > nat,X3: set_set_set_nat] :
( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ k ) )
=> ( ( clique3686358387679108662ccepts @ X3 @ ( clique5033774636164728462irst_C @ k @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ k @ X3 ) ) ) ) ).
% ACC_cf_I
thf(fact_1142_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1143_first__assumptions_Ofinite__POS__NEG,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737375870294875st_NEG @ K ) ) ) ) ).
% first_assumptions.finite_POS_NEG
thf(fact_1144_first__assumptions_ONEG__def,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737375870294875st_NEG @ K )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K ) @ ( clique2971579238625216137irst_F @ K ) ) ) ) ).
% first_assumptions.NEG_def
thf(fact_1145_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P2: nat,K: nat,F3: nat > nat,X3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ K ) )
=> ( ( clique3686358387679108662ccepts @ X3 @ ( clique5033774636164728462irst_C @ K @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ K @ X3 ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_1146_SET_Opelims,axiom,
! [X2: monotone_mformula_a,Y: set_set_set_nat] :
( ( ( clique6509092761774629891_SET_a @ pi @ X2 )
= Y )
=> ( ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ X2 )
=> ( ( ( X2 = monotone_FALSE_a )
=> ( ( Y = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X4: a] :
( ( X2
= ( monotone_Var_a @ X4 ) )
=> ( ( Y
= ( 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 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X2
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( Y
= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Disj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X2
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y
= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
=> ~ ( ( X2 = monotone_TRUE_a )
=> ( ( Y = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% SET.pelims
thf(fact_1147_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_1148_finite__ACC,axiom,
! [X3: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X3 ) ) ).
% finite_ACC
thf(fact_1149_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K @ X3 ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_1150_first__assumptions_Ofinite___092_060F_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.finite_\<F>
thf(fact_1151_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_1152_second__assumptions_Odeviate__pos__cup_Ocong,axiom,
clique3314026705536850673os_cup = clique3314026705536850673os_cup ).
% second_assumptions.deviate_pos_cup.cong
thf(fact_1153_first__assumptions_OPOS__CLIQUE,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_CLIQUE
thf(fact_1154_second__assumptions_Odeviate__neg__cup_Ocong,axiom,
clique1591571987439376245eg_cup = clique1591571987439376245eg_cup ).
% second_assumptions.deviate_neg_cup.cong
thf(fact_1155_second__assumptions_Odeviate__neg__cap_Ocong,axiom,
clique1591571987438064265eg_cap = clique1591571987438064265eg_cap ).
% second_assumptions.deviate_neg_cap.cong
thf(fact_1156_second__assumptions_Osqcap_Ocong,axiom,
clique2586627118206219037_sqcap = clique2586627118206219037_sqcap ).
% second_assumptions.sqcap.cong
thf(fact_1157_second__assumptions_Osqcup_Ocong,axiom,
clique2586627118207531017_sqcup = clique2586627118207531017_sqcup ).
% second_assumptions.sqcup.cong
thf(fact_1158_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_1159_pointwise__minimal__pointwise__maximal_I1_J,axiom,
! [S3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat_nat )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S3 )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S3 )
=> ( ( ord_less_eq_nat_nat @ X4 @ Xa )
| ( ord_less_eq_nat_nat @ Xa @ X4 ) ) ) )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S3 )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S3 )
=> ( ord_less_eq_nat_nat @ X4 @ Xa2 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(1)
thf(fact_1160_pointwise__minimal__pointwise__maximal_I2_J,axiom,
! [S3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat_nat )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S3 )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S3 )
=> ( ( ord_less_eq_nat_nat @ X4 @ Xa )
| ( ord_less_eq_nat_nat @ Xa @ X4 ) ) ) )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S3 )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S3 )
=> ( ord_less_eq_nat_nat @ Xa2 @ X4 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(2)
thf(fact_1161_second__assumptions_Odeviate__pos__cup__def,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique3314026705536850673os_cup @ L @ P2 @ K @ X3 @ Y2 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X3 @ Y2 ) ) ) ) ) ).
% second_assumptions.deviate_pos_cup_def
thf(fact_1162_second__assumptions_Odeviate__pos__cap__def,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique3314026705535538693os_cap @ L @ P2 @ K @ X3 @ Y2 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X3 @ Y2 ) ) ) ) ) ).
% second_assumptions.deviate_pos_cap_def
thf(fact_1163_second__assumptions_Odeviate__neg__cup__def,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique1591571987439376245eg_cup @ L @ P2 @ K @ X3 @ Y2 )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X3 @ Y2 ) ) @ ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) ) ) ) ).
% second_assumptions.deviate_neg_cup_def
thf(fact_1164_second__assumptions_Odeviate__neg__cap__def,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique1591571987438064265eg_cap @ L @ P2 @ K @ X3 @ Y2 )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X3 @ Y2 ) ) @ ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) ) ) ) ) ).
% second_assumptions.deviate_neg_cap_def
thf(fact_1165_second__assumptions_Ov__sameprod__subset,axiom,
! [L: nat,P2: nat,K: nat,Vs: set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ) ).
% second_assumptions.v_sameprod_subset
thf(fact_1166_second__assumptions_Osqcup__sub,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X3 @ Y2 ) ) ) ) ) ) ).
% second_assumptions.sqcup_sub
thf(fact_1167_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_1168_second__assumptions_Osqcup,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X3 @ Y2 ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).
% second_assumptions.sqcup
thf(fact_1169_second__assumptions_Osqcap,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X3 @ Y2 ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).
% second_assumptions.sqcap
thf(fact_1170_second__assumptions_Odeviate__pos__cup,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( clique3314026705536850673os_cup @ L @ P2 @ K @ X3 @ Y2 )
= bot_bo7198184520161983622et_nat ) ) ) ) ).
% second_assumptions.deviate_pos_cup
thf(fact_1171_APR_Opelims,axiom,
! [X2: monotone_mformula_a,Y: set_set_set_nat] :
( ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ X2 )
= Y )
=> ( ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ X2 )
=> ( ( ( X2 = monotone_FALSE_a )
=> ( ( Y = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X4: a] :
( ( X2
= ( monotone_Var_a @ X4 ) )
=> ( ( Y
= ( 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 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X2
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( Y
= ( clique2586627118207531017_sqcup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Disj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X2
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y
= ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
=> ~ ( ( X2 = monotone_TRUE_a )
=> ( ( Y = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% APR.pelims
thf(fact_1172_second__assumptions_Oaxioms_I1_J,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( assump5453534214990993103ptions @ L @ P2 @ K ) ) ).
% second_assumptions.axioms(1)
thf(fact_1173_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_1174_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_1175_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_1176_forth__assumptions__axioms,axiom,
clique8563529963003110213ions_a @ l @ p @ k @ v @ pi ).
% forth_assumptions_axioms
thf(fact_1177_L0,axiom,
ord_less_eq_nat @ assumptions_and_L0 @ l ).
% L0
thf(fact_1178_L0_H,axiom,
ord_less_eq_nat @ assumptions_and_L02 @ l ).
% L0'
thf(fact_1179_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_1180_second__assumptions__axioms,axiom,
assump2881078719466019805ptions @ l @ p @ k ).
% second_assumptions_axioms
thf(fact_1181_sqcup,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y2 ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% sqcup
thf(fact_1182_sqcap,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X3 @ Y2 ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% sqcap
thf(fact_1183_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_1184_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_1185_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_1186_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_1187_deviate__finite_I2_J,axiom,
! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) ).
% deviate_finite(2)
thf(fact_1188_finite__approx__neg,axiom,
! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ Phi ) ) ).
% finite_approx_neg
thf(fact_1189_deviate__finite_I1_J,axiom,
! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) ).
% deviate_finite(1)
thf(fact_1190_finite__approx__pos,axiom,
! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ Phi ) ) ).
% finite_approx_pos
thf(fact_1191_deviate__pos__cup,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( clique3314026705536850673os_cup @ l @ p @ k @ X3 @ Y2 )
= bot_bo7198184520161983622et_nat ) ) ) ).
% deviate_pos_cup
thf(fact_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_APR_Osimps_I1_J,axiom,
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% APR.simps(1)
thf(fact_1200_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_1201_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_1202_approx__pos_Osimps_I1_J,axiom,
! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi4 @ Psi4 ) )
= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) ) ).
% approx_pos.simps(1)
thf(fact_1203_approx__neg_Osimps_I2_J,axiom,
! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi4 @ Psi4 ) )
= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) ) ).
% approx_neg.simps(2)
thf(fact_1204_approx__neg_Osimps_I1_J,axiom,
! [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi4 @ Psi4 ) )
= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi4 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi4 ) ) ) ).
% approx_neg.simps(1)
thf(fact_1205_deviate__neg__cup__def,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique1591571987439376245eg_cup @ l @ p @ k @ X3 @ Y2 )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y2 ) ) @ ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) ) ) ).
% deviate_neg_cup_def
thf(fact_1206_deviate__neg__cap__def,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique1591571987438064265eg_cap @ l @ p @ k @ X3 @ Y2 )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X3 @ Y2 ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) ) ) ) ).
% deviate_neg_cap_def
thf(fact_1207_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_1208_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_1209_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_1210_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_1211_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_1212_sqcup__sub,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y2 ) ) ) ) ) ).
% sqcup_sub
thf(fact_1213_APR_Osimps_I2_J,axiom,
! [X2: a] :
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X2 ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X2 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).
% APR.simps(2)
thf(fact_1214_deviate__pos__cap__def,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3314026705535538693os_cap @ l @ p @ k @ X3 @ Y2 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X3 @ Y2 ) ) ) ) ).
% deviate_pos_cap_def
thf(fact_1215_deviate__pos__cup__def,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3314026705536850673os_cup @ l @ p @ k @ X3 @ Y2 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y2 ) ) ) ) ).
% deviate_pos_cup_def
thf(fact_1216_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_1217_approx__pos_Oelims,axiom,
! [X2: monotone_mformula_a,Y: set_set_set_nat] :
( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X2 )
= Y )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X2
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( Y
!= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) )
=> ( ( ( X2 = monotone_TRUE_a )
=> ( Y != bot_bo7198184520161983622et_nat ) )
=> ( ( ( X2 = monotone_FALSE_a )
=> ( Y != bot_bo7198184520161983622et_nat ) )
=> ( ( ? [V3: a] :
( X2
= ( monotone_Var_a @ V3 ) )
=> ( Y != bot_bo7198184520161983622et_nat ) )
=> ~ ( ? [V3: monotone_mformula_a,Va2: monotone_mformula_a] :
( X2
= ( monotone_Disj_a @ V3 @ Va2 ) )
=> ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ).
% approx_pos.elims
thf(fact_1218_approx__pos_Opelims,axiom,
! [X2: monotone_mformula_a,Y: set_set_set_nat] :
( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X2 )
= Y )
=> ( ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ X2 )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X2
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Conj_a @ Phi2 @ Psi2 ) ) ) )
=> ( ( ( X2 = monotone_TRUE_a )
=> ( ( Y = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_TRUE_a ) ) )
=> ( ( ( X2 = monotone_FALSE_a )
=> ( ( Y = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [V3: a] :
( ( X2
= ( monotone_Var_a @ V3 ) )
=> ( ( Y = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Var_a @ V3 ) ) ) )
=> ~ ! [V3: monotone_mformula_a,Va2: monotone_mformula_a] :
( ( X2
= ( monotone_Disj_a @ V3 @ Va2 ) )
=> ( ( Y = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Disj_a @ V3 @ Va2 ) ) ) ) ) ) ) ) ) ) ).
% approx_pos.pelims
thf(fact_1219_approx__neg_Oelims,axiom,
! [X2: monotone_mformula_a,Y: set_nat_nat] :
( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X2 )
= Y )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X2
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( Y
!= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X2
= ( monotone_Disj_a @ Phi2 @ Psi2 ) )
=> ( Y
!= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) )
=> ( ( ( X2 = monotone_TRUE_a )
=> ( Y != bot_bot_set_nat_nat ) )
=> ( ( ( X2 = monotone_FALSE_a )
=> ( Y != bot_bot_set_nat_nat ) )
=> ~ ( ? [V3: a] :
( X2
= ( monotone_Var_a @ V3 ) )
=> ( Y != bot_bot_set_nat_nat ) ) ) ) ) ) ) ).
% approx_neg.elims
thf(fact_1220_approx__neg_Opelims,axiom,
! [X2: monotone_mformula_a,Y: set_nat_nat] :
( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X2 )
= Y )
=> ( ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ X2 )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X2
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Conj_a @ Phi2 @ Psi2 ) ) ) )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X2
= ( monotone_Disj_a @ Phi2 @ Psi2 ) )
=> ( ( Y
= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Disj_a @ Phi2 @ Psi2 ) ) ) )
=> ( ( ( X2 = monotone_TRUE_a )
=> ( ( Y = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_TRUE_a ) ) )
=> ( ( ( X2 = monotone_FALSE_a )
=> ( ( Y = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_FALSE_a ) ) )
=> ~ ! [V3: a] :
( ( X2
= ( monotone_Var_a @ V3 ) )
=> ( ( Y = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Var_a @ V3 ) ) ) ) ) ) ) ) ) ) ).
% approx_neg.pelims
thf(fact_1221_APR_Oelims,axiom,
! [X2: monotone_mformula_a,Y: set_set_set_nat] :
( ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ X2 )
= Y )
=> ( ( ( X2 = monotone_FALSE_a )
=> ( Y != bot_bo7198184520161983622et_nat ) )
=> ( ! [X4: a] :
( ( X2
= ( monotone_Var_a @ X4 ) )
=> ( Y
!= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X2
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y
!= ( clique2586627118207531017_sqcup @ 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] :
( ( X2
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y
!= ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) ) )
=> ~ ( ( X2 = monotone_TRUE_a )
=> ( Y != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ).
% APR.elims
thf(fact_1222_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_1223_no__deviation_I3_J,axiom,
! [X2: a] :
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X2 ) )
= bot_bo7198184520161983622et_nat ) ).
% no_deviation(3)
thf(fact_1224_no__deviation_I1_J,axiom,
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% no_deviation(1)
thf(fact_1225_no__deviation_I4_J,axiom,
! [X2: a] :
( ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X2 ) )
= bot_bot_set_nat_nat ) ).
% no_deviation(4)
thf(fact_1226_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_1227__092_060pi_062__singleton_I2_J,axiom,
! [X2: a] :
( ( member_a @ X2 @ v )
=> ( member2946998982187404937et_nat @ ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X2 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ).
% \<pi>_singleton(2)
thf(fact_1228_joinl__join,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X3 @ Y2 ) @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) ) ).
% joinl_join
thf(fact_1229_first__assumptions_Opl,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ P2 ) ) ).
% first_assumptions.pl
thf(fact_1230_first__assumptions_Okp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ K ) ) ).
% first_assumptions.kp
thf(fact_1231_first__assumptions_Ok,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_1232_sqcup__def,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique2586627118207531017_sqcup @ l @ p @ k @ X3 @ Y2 )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) ) ).
% sqcup_def
thf(fact_1233_PLU__union,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% PLU_union
thf(fact_1234_PLU__joinl,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X3 @ Y2 ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% PLU_joinl
thf(fact_1235_sqcap__def,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique2586627118206219037_sqcap @ l @ p @ k @ X3 @ Y2 )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X3 @ Y2 ) ) ) ).
% sqcap_def
thf(fact_1236_second__assumptions_Osqcap__def,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X3 @ Y2 )
= ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( clique7966186356931407165_odotl @ L @ K @ X3 @ Y2 ) ) ) ) ).
% second_assumptions.sqcap_def
thf(fact_1237_second__assumptions_OPLU_Ocong,axiom,
clique2699557479641037314nd_PLU = clique2699557479641037314nd_PLU ).
% second_assumptions.PLU.cong
thf(fact_1238_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_1239_second__assumptions_OPLU__joinl,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( clique7966186356931407165_odotl @ L @ K @ X3 @ Y2 ) ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).
% second_assumptions.PLU_joinl
thf(fact_1240_second__assumptions_OPLU__union,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X3 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y2 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).
% second_assumptions.PLU_union
thf(fact_1241_second__assumptions_Osqcup__def,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X3 @ Y2 )
= ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( sup_su4213647025997063966et_nat @ X3 @ Y2 ) ) ) ) ).
% second_assumptions.sqcup_def
thf(fact_1242_first__assumptions_Ojoinl__join,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K @ X3 @ Y2 ) @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) ) ) ).
% first_assumptions.joinl_join
thf(fact_1243_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_1244_third__assumptions__axioms,axiom,
assump2119784843035796504ptions @ l @ p @ k ).
% third_assumptions_axioms
thf(fact_1245_odotl__def,axiom,
! [X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k @ X3 @ Y2 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).
% odotl_def
thf(fact_1246_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_1247_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_1248_first__assumptions_Ofinite__v__gs__Gl,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) ) ) ).
% first_assumptions.finite_v_gs_Gl
thf(fact_1249_first__assumptions_Oodotl__def,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique7966186356931407165_odotl @ L @ K @ X3 @ Y2 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X3 @ Y2 ) @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ).
% first_assumptions.odotl_def
thf(fact_1250_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_1251_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_1252_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_1253_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_1254_Lm,axiom,
ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lm
thf(fact_1255_M0,axiom,
ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ k ) ).
% M0
thf(fact_1256_M0_H,axiom,
ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ k ) ).
% M0'
thf(fact_1257_first__assumptions_Okml,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_1258_first__assumptions_Omp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.mp
thf(fact_1259_first__assumptions_Okm,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.km
thf(fact_1260_plucking__step_I3_J,axiom,
! [X3: set_set_set_nat,Y2: 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 ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ p @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ X3 ) ) @ ( clique3210737319928189260st_ACC @ k @ Y2 ) ) ) ) ) ).
% plucking_step(3)
thf(fact_1261_plucking__step_I5_J,axiom,
! [X3: set_set_set_nat,Y2: 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 ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ p @ X3 ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) ) ) ) ).
% plucking_step(5)
thf(fact_1262_plucking__step_I2_J,axiom,
! [X3: set_set_set_nat,Y2: 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 ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ p @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ Y2 @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ) ) ).
% plucking_step(2)
thf(fact_1263_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_1264_second__assumptions_Oplucking__step_I2_J,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ P2 @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ Y2 @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ) ) ).
% second_assumptions.plucking_step(2)
thf(fact_1265_second__assumptions_Oplucking__step_I5_J,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ P2 @ X3 ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% second_assumptions.plucking_step(5)
thf(fact_1266_second__assumptions_Oplucking__step_I3_J,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ P2 @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ X3 ) ) @ ( clique3210737319928189260st_ACC @ K @ Y2 ) ) ) ) ) ) ).
% second_assumptions.plucking_step(3)
thf(fact_1267_PLU__main_Opinduct,axiom,
! [A0: set_set_set_nat,P: set_set_set_nat > $o] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k ) @ A0 )
=> ( ! [X5: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k ) @ X5 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ l @ k ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) )
=> ( P @ ( clique4095374090462327202g_step @ p @ X5 ) ) )
=> ( P @ X5 ) ) )
=> ( P @ A0 ) ) ) ).
% PLU_main.pinduct
thf(fact_1268_plucking__step_I1_J,axiom,
! [X3: set_set_set_nat,Y2: 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 ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ p @ X3 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y2 ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) @ p ) @ one_one_nat ) ) ) ) ) ).
% plucking_step(1)
thf(fact_1269_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1270_first__assumptions_Olm,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.lm
thf(fact_1271_second__assumptions_Oplucking__step_I1_J,axiom,
! [L: nat,P2: nat,K: nat,X3: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X3 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ P2 @ X3 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y2 ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X3 ) ) @ P2 ) @ one_one_nat ) ) ) ) ) ) ).
% second_assumptions.plucking_step(1)
thf(fact_1272_second__assumptions_OPLU__main_Opinduct,axiom,
! [L: nat,P2: nat,K: nat,A0: set_set_set_nat,P: set_set_set_nat > $o] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P2 @ K ) @ A0 )
=> ( ! [X5: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P2 @ K ) @ X5 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ L @ K ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) )
=> ( P @ ( clique4095374090462327202g_step @ P2 @ X5 ) ) )
=> ( P @ X5 ) ) )
=> ( P @ A0 ) ) ) ) ).
% second_assumptions.PLU_main.pinduct
thf(fact_1273_card__POS,axiom,
( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ k ) )
= ( binomial @ ( assump1710595444109740334irst_m @ k ) @ k ) ) ).
% card_POS
thf(fact_1274_first__assumptions_Ocard__POS,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ K ) )
= ( binomial @ ( assump1710595444109740334irst_m @ K ) @ K ) ) ) ).
% first_assumptions.card_POS
thf(fact_1275__092_060pi_062m2,axiom,
! [X2: a] :
( ( member_a @ X2 @ v )
=> ( member_set_nat @ ( pi @ X2 ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ).
% \<pi>m2
thf(fact_1276__092_060P_062L_092_060G_062l__def,axiom,
( ( clique2294137941332549862_L_G_l @ l @ p @ k )
= ( collec7201453139178570183et_nat
@ ^ [X7: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X7 @ ( clique7840962075309931874st_G_l @ l @ k ) )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X7 ) ) @ ( assump1710595444109740301irst_L @ l @ p ) ) ) ) ) ).
% \<P>L\<G>l_def
% Conjectures (1)
thf(conj_0,conjecture,
! [X4: a] :
( ( member_a @ X4 @ v )
=> ( ( member_set_nat @ ( pi @ X4 ) @ ( sup_sup_set_set_nat @ d @ e ) )
=> ( theta @ X4 ) ) ) ).
%------------------------------------------------------------------------------