TPTP Problem File: SLH0191^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_01644_062909__16387174_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1628 ( 730 unt; 349 typ; 0 def)
% Number of atoms : 3662 (1720 equ; 0 cnn)
% Maximal formula atoms : 24 ( 2 avg)
% Number of connectives : 11909 ( 506 ~; 49 |; 221 &;9612 @)
% ( 0 <=>;1521 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 6 avg)
% Number of types : 39 ( 38 usr)
% Number of type conns : 1213 (1213 >; 0 *; 0 +; 0 <<)
% Number of symbols : 312 ( 311 usr; 48 con; 0-5 aty)
% Number of variables : 3606 ( 175 ^;3352 !; 79 ?;3606 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:50:42.092
%------------------------------------------------------------------------------
% Could-be-implicit typings (38)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_Pr3001764113791142201et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc387721731789858191et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
produc1642012749495946895et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc7767816977991823634at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc115295482908228569et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc6289966885787448281et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
produc7649650317444484569et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr7682762132356531903at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
set_Mo5210732246825857808et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc5902903927690578268at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc1367157794414377308et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Pr5488025237498180813et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
produc4045820344675478307at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc3443465069400027683et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc541613712856062755et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc1932156733058919263at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_Mo2574807150581459802et_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
monoto8535755219626829232et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_M_Eo_J_Mt__Monotone____Formula__Omformula_Itf__a_J_J,type,
produc1878458442261862582mula_a: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Mo6069479339911551325at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc7819656566062154093et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Mo5013373542560054436et_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
monoto5483634261523599098et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
set_set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
monoto8276428299528460797at_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J,type,
monoto7244996872745832772et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Monotone____Formula__Omformula_Itf__a_J_J,type,
set_Mo2626137824023173004mula_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_It__Nat__Onat_J,type,
monoto4181647612830706830la_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Monotone____Formula__Omformula_Itf__a_J,type,
monotone_mformula_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (311)
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_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
clique5528702923696243640at_nat: nat > nat > nat > set_nat_nat > ( ( nat > nat ) > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001t__Set__Oset_It__Nat__Onat_J,type,
clique522982669833463679et_nat: nat > nat > nat > set_set_nat > ( set_nat > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
clique2455256169097332789et_nat: nat > nat > nat > set_set_set_nat > ( set_set_nat > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
clique3407333501437444587et_nat: nat > nat > nat > set_set_set_set_nat > ( set_set_set_nat > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_001tf__a,type,
clique8563529963003110213ions_a: nat > nat > nat > set_a > ( a > set_nat ) > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OACC__cf__mf_001tf__a,type,
clique8961599393750669800f_mf_a: nat > ( a > set_nat ) > monotone_mformula_a > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OACC__mf_001tf__a,type,
clique4708818501384062891C_mf_a: nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OAPR_001tf__a,type,
clique3873310923663319714_APR_a: nat > nat > nat > ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OAPR__rel_001tf__a,type,
clique5870032674357670943_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OSET_001tf__a,type,
clique6509092761774629891_SET_a: ( a > set_nat ) > monotone_mformula_a > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_OSET__rel_001tf__a,type,
clique834332680210058238_rel_a: monotone_mformula_a > monotone_mformula_a > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
clique6859621968737270801at_nat: set_nat_nat > set_Mo6069479339911551325at_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001t__Set__Oset_It__Nat__Onat_J,type,
clique9181349226887787864et_nat: set_set_nat > set_Mo5013373542560054436et_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
clique7740924183492588046et_nat: set_set_set_nat > set_Mo2574807150581459802et_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
clique2555064243683067844et_nat: set_set_set_set_nat > set_Mo5210732246825857808et_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_O_092_060A_062_001tf__a,type,
clique5987991184601036204th_A_a: set_a > set_Mo2626137824023173004mula_a ).
thf(sy_c_Clique__Large__Monotone__Circuits_Oforth__assumptions_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_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_OPLU__main__sumC,type,
clique7977237465666818131n_sumC: nat > nat > nat > set_set_set_nat > produc4045820344675478307at_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cap,type,
clique1591571987438064265eg_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cup,type,
clique1591571987439376245eg_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cap,type,
clique3314026705535538693os_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cup,type,
clique3314026705536850673os_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcap,type,
clique2586627118206219037_sqcap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcup,type,
clique2586627118207531017_sqcup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__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_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on2461717442902640625at_nat: ( ( nat > nat ) > nat > nat ) > set_nat_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_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001tf__a,type,
inj_on_nat_nat_a: ( ( nat > nat ) > a ) > set_nat_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on_nat_nat_nat: ( nat > nat > nat ) > set_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,
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thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_HOL_Oundefined_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Monotone__Formula_OSUB_001tf__a,type,
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thf(sy_c_Monotone__Formula_OSUB__rel_001tf__a,type,
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thf(sy_c_Monotone__Formula_Oeval_001tf__a,type,
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thf(sy_c_Monotone__Formula_Omformula_OConj_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Monotone__Formula_Omformula_OConj_001tf__a,type,
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thf(sy_c_Monotone__Formula_Omformula_ODisj_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Monotone__Formula_Omformula_ODisj_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Monotone__Formula_Omformula_ODisj_001tf__a,type,
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thf(sy_c_Monotone__Formula_Omformula_OFALSE_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Monotone__Formula_Omformula_OFALSE_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Monotone__Formula_Omformula_OTRUE_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Monotone__Formula_Omformula_OVar_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Monotone__Formula_Omformula_Orel__mformula_001tf__a_001tf__a,type,
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thf(sy_c_Monotone__Formula_Otf__mformula_001tf__a,type,
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thf(sy_c_Monotone__Formula_Otf__mformulap_001tf__a,type,
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thf(sy_c_Monotone__Formula_Oto__tf__formula_001tf__a,type,
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thf(sy_c_Monotone__Formula_Ovars_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Monotone__Formula_Ovars__rel_001tf__a,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Relation_OField_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Relation_OField_001t__Nat__Onat,type,
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thf(sy_c_Relation_OField_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_OBex_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_OBex_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_OBex_001tf__a,type,
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thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001tf__a,type,
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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_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_5994387372328453547et_nat: ( a > set_set_set_nat ) > set_a > set_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__Monotone____Formula__Omformula_Itf__a_J,type,
insert7703626487854729094mula_a: monotone_mformula_a > set_Mo2626137824023173004mula_a > set_Mo2626137824023173004mula_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
insert8632360194428346671at_nat: produc1932156733058919263at_nat > set_Pr7682762132356531903at_nat > set_Pr7682762132356531903at_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
insert3810226134351308605et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > set_Pr5488025237498180813et_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
insert3483733655121172905et_nat: produc6289966885787448281et_nat > set_Pr3001764113791142201et_nat > set_Pr3001764113791142201et_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_Ois__singleton_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
is_singleton_nat_nat: set_nat_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
is_singleton_set_nat: set_set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
is_sin6612384548583640136et_nat: set_set_set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
is_sin2178213247671319038et_nat: set_set_set_set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
accp_M442112965984270726at_nat: ( monoto8276428299528460797at_nat > monoto8276428299528460797at_nat > $o ) > monoto8276428299528460797at_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Monotone____Formula__Omformula_It__Nat__Onat_J,type,
accp_M5007270476854070679la_nat: ( monoto4181647612830706830la_nat > monoto4181647612830706830la_nat > $o ) > monoto4181647612830706830la_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J,type,
accp_M6712534473328676429et_nat: ( monoto7244996872745832772et_nat > monoto7244996872745832772et_nat > $o ) > monoto7244996872745832772et_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
accp_M7269832456818321411et_nat: ( monoto5483634261523599098et_nat > monoto5483634261523599098et_nat > $o ) > monoto5483634261523599098et_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Monotone____Formula__Omformula_Itf__a_J,type,
accp_M6162913489380515981mula_a: ( monotone_mformula_a > monotone_mformula_a > $o ) > monotone_mformula_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
accp_set_set_set_nat: ( set_set_set_nat > set_set_set_nat > $o ) > set_set_set_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member435044527007775910at_nat: monoto8276428299528460797at_nat > set_Mo6069479339911551325at_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_It__Set__Oset_It__Nat__Onat_J_J,type,
member7623223977307079021et_nat: monoto7244996872745832772et_nat > set_Mo5013373542560054436et_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member4844836972813196067et_nat: monoto5483634261523599098et_nat > set_Mo2574807150581459802et_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
member4689220760989666777et_nat: monoto8535755219626829232et_nat > set_Mo5210732246825857808et_nat > $o ).
thf(sy_c_member_001t__Monotone____Formula__Omformula_Itf__a_J,type,
member535913909593306477mula_a: monotone_mformula_a > set_Mo2626137824023173004mula_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_DE____,type,
de: set_set_nat ).
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 (1278)
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__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_3_eval,axiom,
clique5859573001277246426al_g_a @ v @ pi @ theta @ de ).
% eval
thf(fact_4_inj__on___092_060pi_062,axiom,
inj_on_a_set_nat @ pi @ v ).
% inj_on_\<pi>
thf(fact_5_eval__gs__def,axiom,
! [Theta: a > $o,X2: set_set_set_nat] :
( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ X2 )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ X2 )
& ( clique5859573001277246426al_g_a @ v @ pi @ Theta @ X ) ) ) ) ).
% eval_gs_def
thf(fact_6_local_Oid,axiom,
( de
= ( sup_sup_set_set_nat @ d @ e ) ) ).
% local.id
thf(fact_7__092_060open_062eval__gs_A_092_060theta_062_A_IX_A_092_060odot_062_AY_J_092_060close_062,axiom,
clique835570645587132141l_gs_a @ v @ pi @ theta @ ( clique5469973757772500719t_odot @ x @ y ) ).
% \<open>eval_gs \<theta> (X \<odot> Y)\<close>
thf(fact_8_eval__gs__union,axiom,
! [Theta: a > $o,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ X2 )
| ( clique835570645587132141l_gs_a @ v @ pi @ Theta @ Y ) ) ) ).
% eval_gs_union
thf(fact_9__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062DE_O_A_092_060lbrakk_062DE_A_092_060in_062_AX_A_092_060odot_062_AY_059_Aeval__g_A_092_060theta_062_ADE_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [DE: set_set_nat] :
( ( member_set_set_nat @ DE @ ( clique5469973757772500719t_odot @ x @ y ) )
=> ~ ( clique5859573001277246426al_g_a @ v @ pi @ theta @ DE ) ) ).
% \<open>\<And>thesis. (\<And>DE. \<lbrakk>DE \<in> X \<odot> Y; eval_g \<theta> DE\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_10__092_060open_062Bex_A_IX_A_092_060odot_062_AY_J_A_Ieval__g_A_092_060theta_062_J_092_060close_062,axiom,
? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( clique5469973757772500719t_odot @ x @ y ) )
& ( clique5859573001277246426al_g_a @ v @ pi @ theta @ X3 ) ) ).
% \<open>Bex (X \<odot> Y) (eval_g \<theta>)\<close>
thf(fact_11__092_060A_062__simps_I1_J,axiom,
member535913909593306477mula_a @ monotone_FALSE_a @ ( clique5987991184601036204th_A_a @ v ) ).
% \<A>_simps(1)
thf(fact_12_second__assumptions_OPLU__main__sumC_Ocong,axiom,
clique7977237465666818131n_sumC = clique7977237465666818131n_sumC ).
% second_assumptions.PLU_main_sumC.cong
thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062D_AE_O_A_092_060lbrakk_062DE_A_061_AD_A_092_060union_062_AE_059_AD_A_092_060in_062_AX_059_AE_A_092_060in_062_AY_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [D: set_set_nat,E: set_set_nat] :
( ( de
= ( sup_sup_set_set_nat @ D @ E ) )
=> ( ( member_set_set_nat @ D @ x )
=> ~ ( member_set_set_nat @ E @ y ) ) ) ).
% \<open>\<And>thesis. (\<And>D E. \<lbrakk>DE = D \<union> E; D \<in> X; E \<in> Y\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_14_forth__assumptions_O_092_060A_062_Ocong,axiom,
clique5987991184601036204th_A_a = clique5987991184601036204th_A_a ).
% forth_assumptions.\<A>.cong
thf(fact_15_forth__assumptions_Oeval__gs_Ocong,axiom,
clique835570645587132141l_gs_a = clique835570645587132141l_gs_a ).
% forth_assumptions.eval_gs.cong
thf(fact_16_forth__assumptions_Oeval__g_Ocong,axiom,
clique5859573001277246426al_g_a = clique5859573001277246426al_g_a ).
% forth_assumptions.eval_g.cong
thf(fact_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_30_sup_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_31_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_32_sup_Oidem,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_33_sup__idem,axiom,
! [X4: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_34_sup__idem,axiom,
! [X4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_35_sup__idem,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_36_sup__idem,axiom,
! [X4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_37_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_38_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_39_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_40_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_41_sup__left__idem,axiom,
! [X4: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ X4 @ Y2 ) )
= ( sup_sup_set_set_nat @ X4 @ Y2 ) ) ).
% sup_left_idem
thf(fact_42_sup__left__idem,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X4 @ ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) ) ).
% sup_left_idem
thf(fact_43_sup__left__idem,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y2 ) )
= ( sup_sup_set_nat @ X4 @ Y2 ) ) ).
% sup_left_idem
thf(fact_44_sup__left__idem,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ ( sup_sup_set_nat_nat @ X4 @ Y2 ) )
= ( sup_sup_set_nat_nat @ X4 @ Y2 ) ) ).
% sup_left_idem
thf(fact_45_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_46_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_47_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_48_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_49_v__gs__union,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_union
thf(fact_50__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_51__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_52_sup__left__commute,axiom,
! [X4: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_53_sup__left__commute,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X4 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_54_sup__left__commute,axiom,
! [X4: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_55_sup__left__commute,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X4 @ Z ) ) ) ).
% sup_left_commute
thf(fact_56_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_57_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_58_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_59_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_60_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_61_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_62_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X ) ) ) ).
% sup_commute
thf(fact_63_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_64_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_65_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_66_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_67_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_68_sup__assoc,axiom,
! [X4: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X4 @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_69_sup__assoc,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) @ Z )
= ( sup_su4213647025997063966et_nat @ X4 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_70_sup__assoc,axiom,
! [X4: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X4 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_71_sup__assoc,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X4 @ Y2 ) @ Z )
= ( sup_sup_set_nat_nat @ X4 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) ) ) ).
% sup_assoc
thf(fact_72_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_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_inf__sup__aci_I6_J,axiom,
! [X4: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X4 @ Y2 ) @ Z )
= ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_81_inf__sup__aci_I6_J,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) @ Z )
= ( sup_su4213647025997063966et_nat @ X4 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_82_inf__sup__aci_I6_J,axiom,
! [X4: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X4 @ Y2 ) @ Z )
= ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_83_inf__sup__aci_I6_J,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X4 @ Y2 ) @ Z )
= ( sup_sup_set_nat_nat @ X4 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_84_inf__sup__aci_I7_J,axiom,
! [X4: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_85_inf__sup__aci_I7_J,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X4 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_86_inf__sup__aci_I7_J,axiom,
! [X4: set_nat,Y2: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_87_inf__sup__aci_I7_J,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ ( sup_sup_set_nat_nat @ Y2 @ Z ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X4 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_88_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_89_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_90_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_91_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_92_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_93_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_94_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_95_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_96_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_97_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_98_inf__sup__aci_I8_J,axiom,
! [X4: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ X4 @ Y2 ) )
= ( sup_sup_set_set_nat @ X4 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_99_inf__sup__aci_I8_J,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X4 @ ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_100_inf__sup__aci_I8_J,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y2 ) )
= ( sup_sup_set_nat @ X4 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_101_inf__sup__aci_I8_J,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ ( sup_sup_set_nat_nat @ X4 @ Y2 ) )
= ( sup_sup_set_nat_nat @ X4 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_113_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_114_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_115_Un__absorb,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_116_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_117_Un__absorb,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_118_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_119_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_120_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_121_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_122_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_123_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_124_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_125_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_126_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_127_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_128_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_129_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_130_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_131_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_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_Bex__def,axiom,
( bex_set_set_set_nat
= ( ^ [A4: set_set_set_set_nat,P2: set_set_set_nat > $o] :
? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A4 )
& ( P2 @ X ) ) ) ) ).
% Bex_def
thf(fact_149_Bex__def,axiom,
( bex_set_set_nat
= ( ^ [A4: set_set_set_nat,P2: set_set_nat > $o] :
? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A4 )
& ( P2 @ X ) ) ) ) ).
% Bex_def
thf(fact_150_Bex__def,axiom,
( bex_set_nat
= ( ^ [A4: set_set_nat,P2: set_nat > $o] :
? [X: set_nat] :
( ( member_set_nat @ X @ A4 )
& ( P2 @ X ) ) ) ) ).
% Bex_def
thf(fact_151_Bex__def,axiom,
( bex_nat_nat
= ( ^ [A4: set_nat_nat,P2: ( nat > nat ) > $o] :
? [X: nat > nat] :
( ( member_nat_nat @ X @ A4 )
& ( P2 @ X ) ) ) ) ).
% Bex_def
thf(fact_152_Bex__def,axiom,
( bex_a
= ( ^ [A4: set_a,P2: a > $o] :
? [X: a] :
( ( member_a @ X @ A4 )
& ( P2 @ X ) ) ) ) ).
% Bex_def
thf(fact_153_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_154_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_155_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_156_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_157_mformula_Odistinct_I11_J,axiom,
! [X41: monotone_mformula_a,X42: monotone_mformula_a] :
( monotone_FALSE_a
!= ( monotone_Conj_a @ X41 @ X42 ) ) ).
% mformula.distinct(11)
thf(fact_158_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_159_mformula_Odistinct_I19_J,axiom,
! [X41: monotone_mformula_a,X42: monotone_mformula_a,X51: monotone_mformula_a,X52: monotone_mformula_a] :
( ( monotone_Conj_a @ X41 @ X42 )
!= ( monotone_Disj_a @ X51 @ X52 ) ) ).
% mformula.distinct(19)
thf(fact_160_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_161_first__assumptions_Ov__gs__union,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ).
% first_assumptions.v_gs_union
thf(fact_162__092_060A_062__simps_I2_J,axiom,
! [X4: a] :
( ( member535913909593306477mula_a @ ( monotone_Var_a @ X4 ) @ ( clique5987991184601036204th_A_a @ v ) )
= ( member_a @ X4 @ v ) ) ).
% \<A>_simps(2)
thf(fact_163_bex__reg,axiom,
! [R: set_set_set_set_nat,P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ? [X5: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X5 @ R )
& ( P @ X5 ) )
=> ? [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ R )
& ( Q @ X3 ) ) ) ) ).
% bex_reg
thf(fact_164_bex__reg,axiom,
! [R: set_set_set_nat,P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ R )
& ( P @ X5 ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ R )
& ( Q @ X3 ) ) ) ) ).
% bex_reg
thf(fact_165_bex__reg,axiom,
! [R: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ R )
& ( P @ X5 ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ R )
& ( Q @ X3 ) ) ) ) ).
% bex_reg
thf(fact_166_bex__reg,axiom,
! [R: set_nat_nat,P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ R )
& ( P @ X5 ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ R )
& ( Q @ X3 ) ) ) ) ).
% bex_reg
thf(fact_167_bex__reg,axiom,
! [R: set_a,P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ? [X5: a] :
( ( member_a @ X5 @ R )
& ( P @ X5 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ R )
& ( Q @ X3 ) ) ) ) ).
% bex_reg
thf(fact_168_mformula_Oinject_I1_J,axiom,
! [X32: a,Y32: a] :
( ( ( monotone_Var_a @ X32 )
= ( monotone_Var_a @ Y32 ) )
= ( X32 = Y32 ) ) ).
% mformula.inject(1)
thf(fact_169_forth__assumptions_OSET_Ocong,axiom,
clique6509092761774629891_SET_a = clique6509092761774629891_SET_a ).
% forth_assumptions.SET.cong
thf(fact_170_mformula_Odistinct_I17_J,axiom,
! [X32: a,X51: monotone_mformula_a,X52: monotone_mformula_a] :
( ( monotone_Var_a @ X32 )
!= ( monotone_Disj_a @ X51 @ X52 ) ) ).
% mformula.distinct(17)
thf(fact_171_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_172_mformula_Odistinct_I9_J,axiom,
! [X32: a] :
( monotone_FALSE_a
!= ( monotone_Var_a @ X32 ) ) ).
% mformula.distinct(9)
thf(fact_173_first__assumptions_Ov__union,axiom,
! [L: nat,P3: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ 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_174_SET_Osimps_I1_J,axiom,
( ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% SET.simps(1)
thf(fact_175_approx__pos_Ocases,axiom,
! [X4: monotone_mformula_a] :
( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X4
!= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ( X4 != monotone_TRUE_a )
=> ( ( X4 != monotone_FALSE_a )
=> ( ! [V: a] :
( X4
!= ( monotone_Var_a @ V ) )
=> ~ ! [V: monotone_mformula_a,Va: monotone_mformula_a] :
( X4
!= ( monotone_Disj_a @ V @ Va ) ) ) ) ) ) ).
% approx_pos.cases
thf(fact_176_approx__neg_Ocases,axiom,
! [X4: monotone_mformula_a] :
( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X4
!= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X4
!= ( monotone_Disj_a @ Phi2 @ Psi2 ) )
=> ( ( X4 != monotone_TRUE_a )
=> ( ( X4 != monotone_FALSE_a )
=> ~ ! [V: a] :
( X4
!= ( monotone_Var_a @ V ) ) ) ) ) ) ).
% approx_neg.cases
thf(fact_177_SET_Ocases,axiom,
! [X4: monotone_mformula_a] :
( ( X4 != monotone_FALSE_a )
=> ( ! [X3: a] :
( X4
!= ( monotone_Var_a @ X3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( X4 = monotone_TRUE_a ) ) ) ) ) ).
% SET.cases
thf(fact_178_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_179_tf__mformulap_Osimps,axiom,
( monoto8740115226577600765ulap_a
= ( ^ [A3: monotone_mformula_a] :
( ( A3 = monotone_FALSE_a )
| ? [X: a] :
( A3
= ( monotone_Var_a @ X ) )
| ? [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( A3
= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
& ( monoto8740115226577600765ulap_a @ Phi4 )
& ( monoto8740115226577600765ulap_a @ Psi4 ) )
| ? [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( A3
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
& ( monoto8740115226577600765ulap_a @ Phi4 )
& ( monoto8740115226577600765ulap_a @ Psi4 ) ) ) ) ) ).
% tf_mformulap.simps
thf(fact_180_image__eqI,axiom,
! [B2: a,F: a > a,X4: a,A: set_a] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_181_image__eqI,axiom,
! [B2: a,F: set_nat > a,X4: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_set_nat @ X4 @ A )
=> ( member_a @ B2 @ ( image_set_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_182_image__eqI,axiom,
! [B2: set_nat,F: a > set_nat,X4: a,A: set_a] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_set_nat @ B2 @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_183_image__eqI,axiom,
! [B2: a,F: set_set_nat > a,X4: set_set_nat,A: set_set_set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_set_set_nat @ X4 @ A )
=> ( member_a @ B2 @ ( image_set_set_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_184_image__eqI,axiom,
! [B2: set_nat,F: set_nat > set_nat,X4: set_nat,A: set_set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_185_image__eqI,axiom,
! [B2: a,F: ( nat > nat ) > a,X4: nat > nat,A: set_nat_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_nat_nat @ X4 @ A )
=> ( member_a @ B2 @ ( image_nat_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_186_image__eqI,axiom,
! [B2: set_set_nat,F: a > set_set_nat,X4: a,A: set_a] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_set_set_nat @ B2 @ ( image_a_set_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_187_image__eqI,axiom,
! [B2: nat > nat,F: a > nat > nat,X4: a,A: set_a] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_nat_nat @ B2 @ ( image_a_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_188_image__eqI,axiom,
! [B2: a,F: set_set_set_nat > a,X4: set_set_set_nat,A: set_set_set_set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member2946998982187404937et_nat @ X4 @ A )
=> ( member_a @ B2 @ ( image_3422112407882505029_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_189_image__eqI,axiom,
! [B2: set_nat,F: set_set_nat > set_nat,X4: set_set_nat,A: set_set_set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_set_set_nat @ X4 @ A )
=> ( member_set_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_190_empty__iff,axiom,
! [C: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ C @ bot_bo193956671110832956et_nat ) ).
% empty_iff
thf(fact_191_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_192_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_193_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_194_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_195_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_196_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_197_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_198_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_199_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_200_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_201_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_202_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_203_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_204_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_205_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_206_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_207_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_208_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_209_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_210_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_211_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_212_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_213_image__empty,axiom,
! [F: set_set_nat > nat] :
( ( image_1454916318497077779at_nat @ F @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_214_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_215_image__empty,axiom,
! [F: nat > set_set_nat] :
( ( image_2194112158459175443et_nat @ F @ bot_bot_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_216_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_217_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_218_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_219_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_220_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_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_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_232_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_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_sup__bot_Oright__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_241_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_242_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_243_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_244_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_245_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_246_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_247_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_248_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_249_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_250_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_251_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_252_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_253_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_254_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_255_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_256_sup__eq__bot__iff,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ X4 @ Y2 )
= bot_bo7198184520161983622et_nat )
= ( ( X4 = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_257_sup__eq__bot__iff,axiom,
! [X4: set_set_nat,Y2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X4 @ Y2 )
= bot_bot_set_set_nat )
= ( ( X4 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_258_sup__eq__bot__iff,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X4 @ Y2 )
= bot_bot_set_nat )
= ( ( X4 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_259_sup__eq__bot__iff,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ X4 @ Y2 )
= bot_bot_set_nat_nat )
= ( ( X4 = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_260_bot__eq__sup__iff,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) )
= ( ( X4 = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_261_bot__eq__sup__iff,axiom,
! [X4: set_set_nat,Y2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X4 @ Y2 ) )
= ( ( X4 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_262_bot__eq__sup__iff,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X4 @ Y2 ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_263_bot__eq__sup__iff,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ X4 @ Y2 ) )
= ( ( X4 = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_264_sup__bot__right,axiom,
! [X4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X4 @ bot_bo7198184520161983622et_nat )
= X4 ) ).
% sup_bot_right
thf(fact_265_sup__bot__right,axiom,
! [X4: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ bot_bot_set_set_nat )
= X4 ) ).
% sup_bot_right
thf(fact_266_sup__bot__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% sup_bot_right
thf(fact_267_sup__bot__right,axiom,
! [X4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ bot_bot_set_nat_nat )
= X4 ) ).
% sup_bot_right
thf(fact_268_sup__bot__left,axiom,
! [X4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_269_sup__bot__left,axiom,
! [X4: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_270_sup__bot__left,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_271_sup__bot__left,axiom,
! [X4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_272_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_273_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_274_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_275_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_276_bex__empty,axiom,
! [P: set_set_nat > $o] :
~ ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ bot_bo7198184520161983622et_nat )
& ( P @ X5 ) ) ).
% bex_empty
thf(fact_277_bex__empty,axiom,
! [P: set_nat > $o] :
~ ? [X5: set_nat] :
( ( member_set_nat @ X5 @ bot_bot_set_set_nat )
& ( P @ X5 ) ) ).
% bex_empty
thf(fact_278_bex__empty,axiom,
! [P: nat > $o] :
~ ? [X5: nat] :
( ( member_nat @ X5 @ bot_bot_set_nat )
& ( P @ X5 ) ) ).
% bex_empty
thf(fact_279_bex__empty,axiom,
! [P: ( nat > nat ) > $o] :
~ ? [X5: nat > nat] :
( ( member_nat_nat @ X5 @ bot_bot_set_nat_nat )
& ( P @ X5 ) ) ).
% bex_empty
thf(fact_280_emptyE,axiom,
! [A2: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ A2 @ bot_bo193956671110832956et_nat ) ).
% emptyE
thf(fact_281_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_282_emptyE,axiom,
! [A2: set_set_nat] :
~ ( member_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_283_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_284_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_285_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_286_imageI,axiom,
! [X4: a,A: set_a,F: a > a] :
( ( member_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_287_imageI,axiom,
! [X4: set_nat,A: set_set_nat,F: set_nat > a] :
( ( member_set_nat @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_set_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_288_imageI,axiom,
! [X4: a,A: set_a,F: a > set_nat] :
( ( member_a @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ ( image_a_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_289_imageI,axiom,
! [X4: set_set_nat,A: set_set_set_nat,F: set_set_nat > a] :
( ( member_set_set_nat @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_set_set_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_290_imageI,axiom,
! [X4: set_nat,A: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_291_imageI,axiom,
! [X4: nat > nat,A: set_nat_nat,F: ( nat > nat ) > a] :
( ( member_nat_nat @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_nat_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_292_imageI,axiom,
! [X4: a,A: set_a,F: a > set_set_nat] :
( ( member_a @ X4 @ A )
=> ( member_set_set_nat @ ( F @ X4 ) @ ( image_a_set_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_293_imageI,axiom,
! [X4: a,A: set_a,F: a > nat > nat] :
( ( member_a @ X4 @ A )
=> ( member_nat_nat @ ( F @ X4 ) @ ( image_a_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_294_imageI,axiom,
! [X4: set_set_set_nat,A: set_set_set_set_nat,F: set_set_set_nat > a] :
( ( member2946998982187404937et_nat @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_3422112407882505029_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_295_imageI,axiom,
! [X4: set_set_nat,A: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( member_set_nat @ ( F @ X4 ) @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_296_equals0D,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( A = bot_bo193956671110832956et_nat )
=> ~ ( member2946998982187404937et_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_297_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_298_equals0D,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( A = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_299_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_300_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_301_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_302_equals0I,axiom,
! [A: set_set_set_set_nat] :
( ! [Y4: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ Y4 @ A )
=> ( A = bot_bo193956671110832956et_nat ) ) ).
% equals0I
thf(fact_303_equals0I,axiom,
! [A: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_304_equals0I,axiom,
! [A: set_set_set_nat] :
( ! [Y4: set_set_nat] :
~ ( member_set_set_nat @ Y4 @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_305_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y4: set_nat] :
~ ( member_set_nat @ Y4 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_306_equals0I,axiom,
! [A: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_307_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y4: nat > nat] :
~ ( member_nat_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_308_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_309_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_310_bex__imageD,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,P: set_nat > $o] :
( ? [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_5842784325960735177et_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_311_bex__imageD,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_set_nat > $o] :
( ? [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ ( image_9186907679027735170et_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_312_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_313_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_314_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_315_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_316_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_317_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_318_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 )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_5842784325960735177et_nat @ F @ M )
= ( image_5842784325960735177et_nat @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_319_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 )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image_9186907679027735170et_nat @ F @ M )
= ( image_9186907679027735170et_nat @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_320_ball__imageD,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( image_5842784325960735177et_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X5: set_set_nat] :
( ( member_set_set_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_321_ball__imageD,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( image_9186907679027735170et_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X5: nat > nat] :
( ( member_nat_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_322_rev__image__eqI,axiom,
! [X4: a,A: set_a,B2: a,F: a > a] :
( ( member_a @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_a @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_323_rev__image__eqI,axiom,
! [X4: set_nat,A: set_set_nat,B2: a,F: set_nat > a] :
( ( member_set_nat @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_a @ B2 @ ( image_set_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_324_rev__image__eqI,axiom,
! [X4: a,A: set_a,B2: set_nat,F: a > set_nat] :
( ( member_a @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_set_nat @ B2 @ ( image_a_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_325_rev__image__eqI,axiom,
! [X4: set_set_nat,A: set_set_set_nat,B2: a,F: set_set_nat > a] :
( ( member_set_set_nat @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_a @ B2 @ ( image_set_set_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_326_rev__image__eqI,axiom,
! [X4: set_nat,A: set_set_nat,B2: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_327_rev__image__eqI,axiom,
! [X4: nat > nat,A: set_nat_nat,B2: a,F: ( nat > nat ) > a] :
( ( member_nat_nat @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_a @ B2 @ ( image_nat_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_328_rev__image__eqI,axiom,
! [X4: a,A: set_a,B2: set_set_nat,F: a > set_set_nat] :
( ( member_a @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_set_set_nat @ B2 @ ( image_a_set_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_329_rev__image__eqI,axiom,
! [X4: a,A: set_a,B2: nat > nat,F: a > nat > nat] :
( ( member_a @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_nat_nat @ B2 @ ( image_a_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_330_rev__image__eqI,axiom,
! [X4: set_set_set_nat,A: set_set_set_set_nat,B2: a,F: set_set_set_nat > a] :
( ( member2946998982187404937et_nat @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_a @ B2 @ ( image_3422112407882505029_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_331_rev__image__eqI,axiom,
! [X4: set_set_nat,A: set_set_set_nat,B2: set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_set_nat @ B2 @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_332_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_333_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_334_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_335_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_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_mformula_Odistinct_I7_J,axiom,
! [X51: monotone_mformula_a,X52: monotone_mformula_a] :
( monotone_TRUE_a
!= ( monotone_Disj_a @ X51 @ X52 ) ) ).
% mformula.distinct(7)
thf(fact_344_mformula_Odistinct_I3_J,axiom,
! [X32: a] :
( monotone_TRUE_a
!= ( monotone_Var_a @ X32 ) ) ).
% mformula.distinct(3)
thf(fact_345_mformula_Odistinct_I1_J,axiom,
monotone_TRUE_a != monotone_FALSE_a ).
% mformula.distinct(1)
thf(fact_346_Un__empty__right,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ bot_bo7198184520161983622et_nat )
= A ) ).
% Un_empty_right
thf(fact_347_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_348_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_349_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_350_Un__empty__left,axiom,
! [B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_351_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_352_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_353_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_354_tf__mformulap_Otf__Disj,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monoto8740115226577600765ulap_a @ Phi )
=> ( ( monoto8740115226577600765ulap_a @ Psi )
=> ( monoto8740115226577600765ulap_a @ ( monotone_Disj_a @ Phi @ Psi ) ) ) ) ).
% tf_mformulap.tf_Disj
thf(fact_355_tf__mformulap_Otf__Conj,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monoto8740115226577600765ulap_a @ Phi )
=> ( ( monoto8740115226577600765ulap_a @ Psi )
=> ( monoto8740115226577600765ulap_a @ ( monotone_Conj_a @ Phi @ Psi ) ) ) ) ).
% tf_mformulap.tf_Conj
thf(fact_356_tf__mformulap_Otf__Var,axiom,
! [X4: a] : ( monoto8740115226577600765ulap_a @ ( monotone_Var_a @ X4 ) ) ).
% tf_mformulap.tf_Var
thf(fact_357_tf__mformulap_Otf__False,axiom,
monoto8740115226577600765ulap_a @ monotone_FALSE_a ).
% tf_mformulap.tf_False
thf(fact_358_first__assumptions_Ov__gs__def,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique8462013130872731469t_v_gs @ X2 )
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ X2 ) ) ) ).
% first_assumptions.v_gs_def
thf(fact_359_SUB_Ocases,axiom,
! [X4: monotone_mformula_a] :
( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ! [X3: a] :
( X4
!= ( monotone_Var_a @ X3 ) )
=> ( ( X4 != monotone_FALSE_a )
=> ( X4 = monotone_TRUE_a ) ) ) ) ) ).
% SUB.cases
thf(fact_360_vars_Ocases,axiom,
! [X4: monotone_mformula_a] :
( ! [X3: a] :
( X4
!= ( monotone_Var_a @ X3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( X4 != monotone_FALSE_a )
=> ( X4 = monotone_TRUE_a ) ) ) ) ) ).
% vars.cases
thf(fact_361_mformula_Oexhaust,axiom,
! [Y2: monotone_mformula_a] :
( ( Y2 != monotone_TRUE_a )
=> ( ( Y2 != monotone_FALSE_a )
=> ( ! [X33: a] :
( Y2
!= ( monotone_Var_a @ X33 ) )
=> ( ! [X412: monotone_mformula_a,X422: monotone_mformula_a] :
( Y2
!= ( monotone_Conj_a @ X412 @ X422 ) )
=> ~ ! [X512: monotone_mformula_a,X522: monotone_mformula_a] :
( Y2
!= ( monotone_Disj_a @ X512 @ X522 ) ) ) ) ) ) ).
% mformula.exhaust
thf(fact_362_to__tf__formula_Ocases,axiom,
! [X4: monotone_mformula_a] :
( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X4
!= ( monotone_Disj_a @ Phi2 @ Psi2 ) )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X4
!= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ( X4 != monotone_TRUE_a )
=> ( ( X4 != monotone_FALSE_a )
=> ~ ! [V: a] :
( X4
!= ( monotone_Var_a @ V ) ) ) ) ) ) ).
% to_tf_formula.cases
thf(fact_363_tf__mformulap_Ocases,axiom,
! [A2: monotone_mformula_a] :
( ( monoto8740115226577600765ulap_a @ A2 )
=> ( ( A2 != monotone_FALSE_a )
=> ( ! [X3: a] :
( A2
!= ( monotone_Var_a @ X3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( A2
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( monoto8740115226577600765ulap_a @ Phi3 )
=> ~ ( monoto8740115226577600765ulap_a @ Psi3 ) ) )
=> ~ ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( A2
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( monoto8740115226577600765ulap_a @ Phi3 )
=> ~ ( monoto8740115226577600765ulap_a @ Psi3 ) ) ) ) ) ) ) ).
% tf_mformulap.cases
thf(fact_364_inj__on__empty,axiom,
! [F: a > set_nat] : ( inj_on_a_set_nat @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_365_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_366_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_367_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_368_inj__on__image__iff,axiom,
! [A: set_a,G2: a > set_nat,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ! [Xa: a] :
( ( member_a @ Xa @ A )
=> ( ( ( G2 @ ( F @ X3 ) )
= ( G2 @ ( F @ Xa ) ) )
= ( ( G2 @ X3 )
= ( 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_369_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X4 @ bot_bo7198184520161983622et_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_370_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ bot_bot_set_set_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_371_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_372_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ bot_bot_set_nat_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_373_tf__mformula_Ocases,axiom,
! [A2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ A2 @ monoto4877036962378694605mula_a )
=> ( ( A2 != monotone_FALSE_a )
=> ( ! [X3: a] :
( A2
!= ( monotone_Var_a @ X3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( A2
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( member535913909593306477mula_a @ Phi3 @ monoto4877036962378694605mula_a )
=> ~ ( member535913909593306477mula_a @ Psi3 @ monoto4877036962378694605mula_a ) ) )
=> ~ ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( A2
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( member535913909593306477mula_a @ Phi3 @ monoto4877036962378694605mula_a )
=> ~ ( member535913909593306477mula_a @ Psi3 @ monoto4877036962378694605mula_a ) ) ) ) ) ) ) ).
% tf_mformula.cases
thf(fact_374_tf__mformula_Osimps,axiom,
! [A2: monotone_mformula_a] :
( ( member535913909593306477mula_a @ A2 @ monoto4877036962378694605mula_a )
= ( ( A2 = monotone_FALSE_a )
| ? [X: a] :
( A2
= ( monotone_Var_a @ X ) )
| ? [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( A2
= ( monotone_Disj_a @ Phi4 @ Psi4 ) )
& ( member535913909593306477mula_a @ Phi4 @ monoto4877036962378694605mula_a )
& ( member535913909593306477mula_a @ Psi4 @ monoto4877036962378694605mula_a ) )
| ? [Phi4: monotone_mformula_a,Psi4: monotone_mformula_a] :
( ( A2
= ( monotone_Conj_a @ Phi4 @ Psi4 ) )
& ( member535913909593306477mula_a @ Phi4 @ monoto4877036962378694605mula_a )
& ( member535913909593306477mula_a @ Psi4 @ monoto4877036962378694605mula_a ) ) ) ) ).
% tf_mformula.simps
thf(fact_375_eval_Oelims_I1_J,axiom,
! [X4: a > $o,Xa2: monotone_mformula_a,Y2: $o] :
( ( ( monotone_eval_a @ X4 @ Xa2 )
= Y2 )
=> ( ( ( Xa2 = monotone_FALSE_a )
=> Y2 )
=> ( ( ( Xa2 = monotone_TRUE_a )
=> ~ Y2 )
=> ( ! [X3: a] :
( ( Xa2
= ( monotone_Var_a @ X3 ) )
=> ( Y2
= ( ~ ( X4 @ X3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( Xa2
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y2
= ( ~ ( ( monotone_eval_a @ X4 @ Phi3 )
| ( monotone_eval_a @ X4 @ Psi3 ) ) ) ) )
=> ~ ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( Xa2
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y2
= ( ~ ( ( monotone_eval_a @ X4 @ Phi3 )
& ( monotone_eval_a @ X4 @ Psi3 ) ) ) ) ) ) ) ) ) ) ).
% eval.elims(1)
thf(fact_376_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_377_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_378_to__tf__formula,axiom,
! [Phi: monotone_mformula_a] :
( ( ( monoto1040866505217576201mula_a @ Phi )
!= monotone_TRUE_a )
=> ( member535913909593306477mula_a @ ( monoto1040866505217576201mula_a @ Phi ) @ monoto4877036962378694605mula_a ) ) ).
% to_tf_formula
thf(fact_379_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_380_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_381_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_382_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_383_eval_Osimps_I5_J,axiom,
! [Theta: a > $o,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_eval_a @ Theta @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( ( monotone_eval_a @ Theta @ Phi )
& ( monotone_eval_a @ Theta @ Psi ) ) ) ).
% eval.simps(5)
thf(fact_384_eval_Osimps_I4_J,axiom,
! [Theta: a > $o,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_eval_a @ Theta @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( ( monotone_eval_a @ Theta @ Phi )
| ( monotone_eval_a @ Theta @ Psi ) ) ) ).
% eval.simps(4)
thf(fact_385_eval_Osimps_I3_J,axiom,
! [Theta: a > $o,X4: a] :
( ( monotone_eval_a @ Theta @ ( monotone_Var_a @ X4 ) )
= ( Theta @ X4 ) ) ).
% eval.simps(3)
thf(fact_386_eval_Osimps_I1_J,axiom,
! [Theta: a > $o] :
~ ( monotone_eval_a @ Theta @ monotone_FALSE_a ) ).
% eval.simps(1)
thf(fact_387_eval_Osimps_I2_J,axiom,
! [Theta: a > $o] : ( monotone_eval_a @ Theta @ monotone_TRUE_a ) ).
% eval.simps(2)
thf(fact_388_first__assumptions_Ov__empty,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_389_to__tf__formula_Osimps_I5_J,axiom,
! [V2: a] :
( ( monoto1040866505217576201mula_a @ ( monotone_Var_a @ V2 ) )
= ( monotone_Var_a @ V2 ) ) ).
% to_tf_formula.simps(5)
thf(fact_390_tf__mformula_Otf__Conj,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Psi @ monoto4877036962378694605mula_a )
=> ( member535913909593306477mula_a @ ( monotone_Conj_a @ Phi @ Psi ) @ monoto4877036962378694605mula_a ) ) ) ).
% tf_mformula.tf_Conj
thf(fact_391_tf__mformula_Otf__Disj,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Psi @ monoto4877036962378694605mula_a )
=> ( member535913909593306477mula_a @ ( monotone_Disj_a @ Phi @ Psi ) @ monoto4877036962378694605mula_a ) ) ) ).
% tf_mformula.tf_Disj
thf(fact_392_tf__mformula_Otf__Var,axiom,
! [X4: a] : ( member535913909593306477mula_a @ ( monotone_Var_a @ X4 ) @ monoto4877036962378694605mula_a ) ).
% tf_mformula.tf_Var
thf(fact_393_to__tf__formula_Osimps_I4_J,axiom,
( ( monoto1040866505217576201mula_a @ monotone_FALSE_a )
= monotone_FALSE_a ) ).
% to_tf_formula.simps(4)
thf(fact_394_to__tf__formula_Osimps_I3_J,axiom,
( ( monoto1040866505217576201mula_a @ monotone_TRUE_a )
= monotone_TRUE_a ) ).
% to_tf_formula.simps(3)
thf(fact_395_tf__mformula_Otf__False,axiom,
member535913909593306477mula_a @ monotone_FALSE_a @ monoto4877036962378694605mula_a ).
% tf_mformula.tf_False
thf(fact_396_tf__mformulap__tf__mformula__eq,axiom,
( monoto8740115226577600765ulap_a
= ( ^ [X: monotone_mformula_a] : ( member535913909593306477mula_a @ X @ monoto4877036962378694605mula_a ) ) ) ).
% tf_mformulap_tf_mformula_eq
thf(fact_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_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_405_inj__onD,axiom,
! [F: a > set_nat,A: set_a,X4: a,Y2: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( ( F @ X4 )
= ( F @ Y2 ) )
=> ( ( member_a @ X4 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( X4 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_406_inj__onI,axiom,
! [A: set_a,F: a > set_nat] :
( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y4 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y4 ) )
=> ( X3 = Y4 ) ) ) )
=> ( inj_on_a_set_nat @ F @ A ) ) ).
% inj_onI
thf(fact_407_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_408_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_409_inj__on__eq__iff,axiom,
! [F: a > set_nat,A: set_a,X4: a,Y2: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( member_a @ X4 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( ( F @ X4 )
= ( F @ Y2 ) )
= ( X4 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_410_inj__on__contraD,axiom,
! [F: a > set_nat,A: set_a,X4: a,Y2: a] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ( X4 != Y2 )
=> ( ( member_a @ X4 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( F @ X4 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_411_inj__on__inverseI,axiom,
! [A: set_a,G2: set_nat > a,F: a > set_nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( G2 @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_a_set_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_412_first__assumptions_Ov__gs__empty,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ) ).
% first_assumptions.v_gs_empty
thf(fact_413_eval_Oelims_I3_J,axiom,
! [X4: a > $o,Xa2: monotone_mformula_a] :
( ~ ( monotone_eval_a @ X4 @ Xa2 )
=> ( ( Xa2 != monotone_FALSE_a )
=> ( ! [X3: a] :
( ( Xa2
= ( monotone_Var_a @ X3 ) )
=> ( X4 @ X3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( Xa2
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( monotone_eval_a @ X4 @ Phi3 )
| ( monotone_eval_a @ X4 @ Psi3 ) ) )
=> ~ ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( Xa2
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( monotone_eval_a @ X4 @ Phi3 )
& ( monotone_eval_a @ X4 @ Psi3 ) ) ) ) ) ) ) ).
% eval.elims(3)
thf(fact_414_eval_Oelims_I2_J,axiom,
! [X4: a > $o,Xa2: monotone_mformula_a] :
( ( monotone_eval_a @ X4 @ Xa2 )
=> ( ( Xa2 != monotone_TRUE_a )
=> ( ! [X3: a] :
( ( Xa2
= ( monotone_Var_a @ X3 ) )
=> ~ ( X4 @ X3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( Xa2
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ~ ( ( monotone_eval_a @ X4 @ Phi3 )
| ( monotone_eval_a @ X4 @ Psi3 ) ) )
=> ~ ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( Xa2
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ~ ( ( monotone_eval_a @ X4 @ Phi3 )
& ( monotone_eval_a @ X4 @ Psi3 ) ) ) ) ) ) ) ).
% eval.elims(2)
thf(fact_415_eval_Ocases,axiom,
! [X4: produc1878458442261862582mula_a] :
( ! [Theta2: a > $o] :
( X4
!= ( produc6524915681399349230mula_a @ Theta2 @ monotone_FALSE_a ) )
=> ( ! [Theta2: a > $o] :
( X4
!= ( produc6524915681399349230mula_a @ Theta2 @ monotone_TRUE_a ) )
=> ( ! [Theta2: a > $o,X3: a] :
( X4
!= ( produc6524915681399349230mula_a @ Theta2 @ ( monotone_Var_a @ X3 ) ) )
=> ( ! [Theta2: a > $o,Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( produc6524915681399349230mula_a @ Theta2 @ ( monotone_Disj_a @ Phi3 @ Psi3 ) ) )
=> ~ ! [Theta2: a > $o,Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( produc6524915681399349230mula_a @ Theta2 @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) ) ) ) ) ).
% eval.cases
thf(fact_416_SET_Oelims,axiom,
! [X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique6509092761774629891_SET_a @ pi @ X4 )
= Y2 )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( Y2
!= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X3 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi3 ) ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( Y2 != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ).
% SET.elims
thf(fact_417_the__elem__image__unique,axiom,
! [A: set_set_set_nat,F: set_set_nat > set_nat,X4: set_set_nat] :
( ( A != bot_bo7198184520161983622et_nat )
=> ( ! [Y4: set_set_nat] :
( ( member_set_set_nat @ Y4 @ A )
=> ( ( F @ Y4 )
= ( F @ X4 ) ) )
=> ( ( the_elem_set_nat @ ( image_5842784325960735177et_nat @ F @ A ) )
= ( F @ X4 ) ) ) ) ).
% the_elem_image_unique
thf(fact_418_the__elem__image__unique,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > set_set_nat,X4: nat > nat] :
( ( A != bot_bot_set_nat_nat )
=> ( ! [Y4: nat > nat] :
( ( member_nat_nat @ Y4 @ A )
=> ( ( F @ Y4 )
= ( F @ X4 ) ) )
=> ( ( the_elem_set_set_nat @ ( image_9186907679027735170et_nat @ F @ A ) )
= ( F @ X4 ) ) ) ) ).
% the_elem_image_unique
thf(fact_419_forth__assumptions_Oapprox__pos_Ocases,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X4
!= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ( X4 != monotone_TRUE_a )
=> ( ( X4 != monotone_FALSE_a )
=> ( ! [V: a] :
( X4
!= ( monotone_Var_a @ V ) )
=> ~ ! [V: monotone_mformula_a,Va: monotone_mformula_a] :
( X4
!= ( monotone_Disj_a @ V @ Va ) ) ) ) ) ) ) ).
% forth_assumptions.approx_pos.cases
thf(fact_420_forth__assumptions_Oapprox__neg_Ocases,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X4
!= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( X4
!= ( monotone_Disj_a @ Phi2 @ Psi2 ) )
=> ( ( X4 != monotone_TRUE_a )
=> ( ( X4 != monotone_FALSE_a )
=> ~ ! [V: a] :
( X4
!= ( monotone_Var_a @ V ) ) ) ) ) ) ) ).
% forth_assumptions.approx_neg.cases
thf(fact_421_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_422_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_423_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_424_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_425_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_426_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_427_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_428_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_429_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_430_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_431_insert__absorb2,axiom,
! [X4: set_set_nat,A: set_set_set_nat] :
( ( insert_set_set_nat @ X4 @ ( insert_set_set_nat @ X4 @ A ) )
= ( insert_set_set_nat @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_432_insert__absorb2,axiom,
! [X4: set_nat,A: set_set_nat] :
( ( insert_set_nat @ X4 @ ( insert_set_nat @ X4 @ A ) )
= ( insert_set_nat @ X4 @ A ) ) ).
% insert_absorb2
thf(fact_433_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_434_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_435_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_436_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_437_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_438_insert__image,axiom,
! [X4: set_set_set_nat,A: set_set_set_set_nat,F: set_set_set_nat > set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
=> ( ( insert_set_set_nat @ ( F @ X4 ) @ ( image_2225960715480453173et_nat @ F @ A ) )
= ( image_2225960715480453173et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_439_insert__image,axiom,
! [X4: set_set_set_nat,A: set_set_set_set_nat,F: set_set_set_nat > set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
=> ( ( insert_set_nat @ ( F @ X4 ) @ ( image_7149431738526707583et_nat @ F @ A ) )
= ( image_7149431738526707583et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_440_insert__image,axiom,
! [X4: set_set_nat,A: set_set_set_nat,F: set_set_nat > set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( ( insert_set_set_nat @ ( F @ X4 ) @ ( image_7884819252390400639et_nat @ F @ A ) )
= ( image_7884819252390400639et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_441_insert__image,axiom,
! [X4: set_set_nat,A: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( ( insert_set_nat @ ( F @ X4 ) @ ( image_5842784325960735177et_nat @ F @ A ) )
= ( image_5842784325960735177et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_442_insert__image,axiom,
! [X4: set_nat,A: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( insert_set_set_nat @ ( F @ X4 ) @ ( image_6725021117256019401et_nat @ F @ A ) )
= ( image_6725021117256019401et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_443_insert__image,axiom,
! [X4: set_nat,A: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( insert_set_nat @ ( F @ X4 ) @ ( image_7916887816326733075et_nat @ F @ A ) )
= ( image_7916887816326733075et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_444_insert__image,axiom,
! [X4: nat > nat,A: set_nat_nat,F: ( nat > nat ) > set_set_nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ( insert_set_set_nat @ ( F @ X4 ) @ ( image_9186907679027735170et_nat @ F @ A ) )
= ( image_9186907679027735170et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_445_insert__image,axiom,
! [X4: nat > nat,A: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ( insert_set_nat @ ( F @ X4 ) @ ( image_7432509271690132940et_nat @ F @ A ) )
= ( image_7432509271690132940et_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_446_insert__image,axiom,
! [X4: a,A: set_a,F: a > set_set_nat] :
( ( member_a @ X4 @ A )
=> ( ( insert_set_set_nat @ ( F @ X4 ) @ ( image_a_set_set_nat @ F @ A ) )
= ( image_a_set_set_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_447_insert__image,axiom,
! [X4: a,A: set_a,F: a > set_nat] :
( ( member_a @ X4 @ A )
=> ( ( insert_set_nat @ ( F @ X4 ) @ ( image_a_set_nat @ F @ A ) )
= ( image_a_set_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_448_singletonI,axiom,
! [A2: set_set_set_nat] : ( member2946998982187404937et_nat @ A2 @ ( insert3687027775829606434et_nat @ A2 @ bot_bo193956671110832956et_nat ) ) ).
% singletonI
thf(fact_449_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_450_singletonI,axiom,
! [A2: set_set_nat] : ( member_set_set_nat @ A2 @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ) ).
% singletonI
thf(fact_451_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_452_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_453_singletonI,axiom,
! [A2: nat > nat] : ( member_nat_nat @ A2 @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) ).
% singletonI
thf(fact_454_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_455_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_456_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_457_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_458_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_459_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_460_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_461_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_462_SET_Osimps_I2_J,axiom,
! [X4: a] :
( ( clique6509092761774629891_SET_a @ pi @ ( monotone_Var_a @ X4 ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).
% SET.simps(2)
thf(fact_463_the__elem__eq,axiom,
! [X4: set_set_nat] :
( ( the_elem_set_set_nat @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_464_the__elem__eq,axiom,
! [X4: set_nat] :
( ( the_elem_set_nat @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_465_the__elem__eq,axiom,
! [X4: nat] :
( ( the_elem_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_466_the__elem__eq,axiom,
! [X4: nat > nat] :
( ( the_elem_nat_nat @ ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) )
= X4 ) ).
% the_elem_eq
thf(fact_467_insertE,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 ) ) ) ).
% insertE
thf(fact_468_insertE,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 ) ) ) ).
% insertE
thf(fact_469_insertE,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 ) ) ) ).
% insertE
thf(fact_470_insertE,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 ) ) ) ).
% insertE
thf(fact_471_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_472_insertI1,axiom,
! [A2: set_set_set_nat,B: set_set_set_set_nat] : ( member2946998982187404937et_nat @ A2 @ ( insert3687027775829606434et_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_473_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_474_insertI1,axiom,
! [A2: set_nat,B: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_475_insertI1,axiom,
! [A2: nat > nat,B: set_nat_nat] : ( member_nat_nat @ A2 @ ( insert_nat_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_476_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_477_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_478_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_479_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_480_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_481_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_482_Set_Oset__insert,axiom,
! [X4: set_set_set_nat,A: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
=> ~ ! [B5: set_set_set_set_nat] :
( ( A
= ( insert3687027775829606434et_nat @ X4 @ B5 ) )
=> ( member2946998982187404937et_nat @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_483_Set_Oset__insert,axiom,
! [X4: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ~ ! [B5: set_set_set_nat] :
( ( A
= ( insert_set_set_nat @ X4 @ B5 ) )
=> ( member_set_set_nat @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_484_Set_Oset__insert,axiom,
! [X4: set_nat,A: set_set_nat] :
( ( member_set_nat @ X4 @ A )
=> ~ ! [B5: set_set_nat] :
( ( A
= ( insert_set_nat @ X4 @ B5 ) )
=> ( member_set_nat @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_485_Set_Oset__insert,axiom,
! [X4: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ X4 @ A )
=> ~ ! [B5: set_nat_nat] :
( ( A
= ( insert_nat_nat @ X4 @ B5 ) )
=> ( member_nat_nat @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_486_Set_Oset__insert,axiom,
! [X4: a,A: set_a] :
( ( member_a @ X4 @ A )
=> ~ ! [B5: set_a] :
( ( A
= ( insert_a @ X4 @ B5 ) )
=> ( member_a @ X4 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_487_insert__ident,axiom,
! [X4: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ~ ( member2946998982187404937et_nat @ X4 @ A )
=> ( ~ ( member2946998982187404937et_nat @ X4 @ B )
=> ( ( ( insert3687027775829606434et_nat @ X4 @ A )
= ( insert3687027775829606434et_nat @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_488_insert__ident,axiom,
! [X4: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ X4 @ A )
=> ( ~ ( member_set_set_nat @ X4 @ B )
=> ( ( ( insert_set_set_nat @ X4 @ A )
= ( insert_set_set_nat @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_489_insert__ident,axiom,
! [X4: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ~ ( member_set_nat @ X4 @ B )
=> ( ( ( insert_set_nat @ X4 @ A )
= ( insert_set_nat @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_490_insert__ident,axiom,
! [X4: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ X4 @ A )
=> ( ~ ( member_nat_nat @ X4 @ B )
=> ( ( ( insert_nat_nat @ X4 @ A )
= ( insert_nat_nat @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_491_insert__ident,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ~ ( member_a @ X4 @ B )
=> ( ( ( insert_a @ X4 @ A )
= ( insert_a @ X4 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_492_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_493_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_494_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_495_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_496_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_497_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_498_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_499_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_500_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_501_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_502_insert__commute,axiom,
! [X4: set_set_nat,Y2: set_set_nat,A: set_set_set_nat] :
( ( insert_set_set_nat @ X4 @ ( insert_set_set_nat @ Y2 @ A ) )
= ( insert_set_set_nat @ Y2 @ ( insert_set_set_nat @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_503_insert__commute,axiom,
! [X4: set_nat,Y2: set_nat,A: set_set_nat] :
( ( insert_set_nat @ X4 @ ( insert_set_nat @ Y2 @ A ) )
= ( insert_set_nat @ Y2 @ ( insert_set_nat @ X4 @ A ) ) ) ).
% insert_commute
thf(fact_504_mk__disjoint__insert,axiom,
! [A2: set_set_set_nat,A: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ A2 @ A )
=> ? [B5: set_set_set_set_nat] :
( ( A
= ( insert3687027775829606434et_nat @ A2 @ B5 ) )
& ~ ( member2946998982187404937et_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_505_mk__disjoint__insert,axiom,
! [A2: set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ A2 @ A )
=> ? [B5: set_set_set_nat] :
( ( A
= ( insert_set_set_nat @ A2 @ B5 ) )
& ~ ( member_set_set_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_506_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ? [B5: set_set_nat] :
( ( A
= ( insert_set_nat @ A2 @ B5 ) )
& ~ ( member_set_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_507_mk__disjoint__insert,axiom,
! [A2: nat > nat,A: set_nat_nat] :
( ( member_nat_nat @ A2 @ A )
=> ? [B5: set_nat_nat] :
( ( A
= ( insert_nat_nat @ A2 @ B5 ) )
& ~ ( member_nat_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_508_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B5: set_a] :
( ( A
= ( insert_a @ A2 @ B5 ) )
& ~ ( member_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_509_forth__assumptions_OSET_Osimps_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique6509092761774629891_SET_a @ Pi @ ( monotone_Var_a @ X4 ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ) ).
% forth_assumptions.SET.simps(2)
thf(fact_510_singletonD,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ B2 @ ( insert3687027775829606434et_nat @ A2 @ bot_bo193956671110832956et_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_511_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_512_singletonD,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( member_set_set_nat @ B2 @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_513_singletonD,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat @ B2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_514_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_515_singletonD,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( member_nat_nat @ B2 @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_516_singleton__iff,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( member2946998982187404937et_nat @ B2 @ ( insert3687027775829606434et_nat @ A2 @ bot_bo193956671110832956et_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_517_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_518_singleton__iff,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( member_set_set_nat @ B2 @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_519_singleton__iff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( member_set_nat @ B2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_520_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_521_singleton__iff,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( member_nat_nat @ B2 @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_522_doubleton__eq__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat,D2: set_set_nat] :
( ( ( insert_set_set_nat @ A2 @ ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) )
= ( insert_set_set_nat @ C @ ( insert_set_set_nat @ D2 @ bot_bo7198184520161983622et_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_523_doubleton__eq__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat,D2: set_nat] :
( ( ( insert_set_nat @ A2 @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) )
= ( insert_set_nat @ C @ ( insert_set_nat @ D2 @ bot_bot_set_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_524_doubleton__eq__iff,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_525_doubleton__eq__iff,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat,D2: nat > nat] :
( ( ( insert_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) )
= ( insert_nat_nat @ C @ ( insert_nat_nat @ D2 @ bot_bot_set_nat_nat ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_526_insert__not__empty,axiom,
! [A2: set_set_nat,A: set_set_set_nat] :
( ( insert_set_set_nat @ A2 @ A )
!= bot_bo7198184520161983622et_nat ) ).
% insert_not_empty
thf(fact_527_insert__not__empty,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( insert_set_nat @ A2 @ A )
!= bot_bot_set_set_nat ) ).
% insert_not_empty
thf(fact_528_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_529_insert__not__empty,axiom,
! [A2: nat > nat,A: set_nat_nat] :
( ( insert_nat_nat @ A2 @ A )
!= bot_bot_set_nat_nat ) ).
% insert_not_empty
thf(fact_530_singleton__inject,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_531_singleton__inject,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( insert_set_nat @ A2 @ bot_bot_set_set_nat )
= ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_532_singleton__inject,axiom,
! [A2: nat,B2: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_533_singleton__inject,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat )
= ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_534_forth__assumptions_Oeval__g__def,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Theta: a > $o,G: set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique5859573001277246426al_g_a @ V3 @ Pi @ Theta @ G )
= ( ! [X: a] :
( ( member_a @ X @ V3 )
=> ( ( member_set_nat @ ( Pi @ X ) @ G )
=> ( Theta @ X ) ) ) ) ) ) ).
% forth_assumptions.eval_g_def
thf(fact_535_forth__assumptions_Oinj__on___092_060pi_062,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( inj_on_a_set_nat @ Pi @ V3 ) ) ).
% forth_assumptions.inj_on_\<pi>
thf(fact_536_forth__assumptions_Oeval__simps_I7_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Theta: a > $o,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( monotone_eval_a @ Theta @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( ( monotone_eval_a @ Theta @ Phi )
& ( monotone_eval_a @ Theta @ Psi ) ) ) ) ).
% forth_assumptions.eval_simps(7)
thf(fact_537_forth__assumptions_Oeval__simps_I6_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Theta: a > $o,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( monotone_eval_a @ Theta @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( ( monotone_eval_a @ Theta @ Phi )
| ( monotone_eval_a @ Theta @ Psi ) ) ) ) ).
% forth_assumptions.eval_simps(6)
thf(fact_538_forth__assumptions_Oeval__simps_I5_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Theta: a > $o,X4: a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( monotone_eval_a @ Theta @ ( monotone_Var_a @ X4 ) )
= ( Theta @ X4 ) ) ) ).
% forth_assumptions.eval_simps(5)
thf(fact_539_forth__assumptions_Oeval__simps_I3_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Theta: a > $o] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ~ ( monotone_eval_a @ Theta @ monotone_FALSE_a ) ) ).
% forth_assumptions.eval_simps(3)
thf(fact_540_forth__assumptions_Oeval__simps_I4_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Theta: a > $o] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( monotone_eval_a @ Theta @ monotone_TRUE_a ) ) ).
% forth_assumptions.eval_simps(4)
thf(fact_541_forth__assumptions_O_092_060A_062__simps_I3_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member535913909593306477mula_a @ ( monotone_Conj_a @ Phi @ Psi ) @ ( clique5987991184601036204th_A_a @ V3 ) )
= ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ V3 ) )
& ( member535913909593306477mula_a @ Psi @ ( clique5987991184601036204th_A_a @ V3 ) ) ) ) ) ).
% forth_assumptions.\<A>_simps(3)
thf(fact_542_forth__assumptions_O_092_060A_062__simps_I4_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member535913909593306477mula_a @ ( monotone_Disj_a @ Phi @ Psi ) @ ( clique5987991184601036204th_A_a @ V3 ) )
= ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ V3 ) )
& ( member535913909593306477mula_a @ Psi @ ( clique5987991184601036204th_A_a @ V3 ) ) ) ) ) ).
% forth_assumptions.\<A>_simps(4)
thf(fact_543_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_set_set_set_nat,Pi: set_set_set_nat > set_nat,X4: set_set_set_nat] :
( ( clique3407333501437444587et_nat @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member4689220760989666777et_nat @ ( monoto7822445266502226924et_nat @ X4 ) @ ( clique2555064243683067844et_nat @ V3 ) )
= ( member2946998982187404937et_nat @ X4 @ V3 ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_544_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_set_set_nat,Pi: set_set_nat > set_nat,X4: set_set_nat] :
( ( clique2455256169097332789et_nat @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member4844836972813196067et_nat @ ( monoto3251651810667535926et_nat @ X4 ) @ ( clique7740924183492588046et_nat @ V3 ) )
= ( member_set_set_nat @ X4 @ V3 ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_545_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_set_nat,Pi: set_nat > set_nat,X4: set_nat] :
( ( clique522982669833463679et_nat @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member7623223977307079021et_nat @ ( monotone_Var_set_nat @ X4 ) @ ( clique9181349226887787864et_nat @ V3 ) )
= ( member_set_nat @ X4 @ V3 ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_546_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_nat_nat,Pi: ( nat > nat ) > set_nat,X4: nat > nat] :
( ( clique5528702923696243640at_nat @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member435044527007775910at_nat @ ( monotone_Var_nat_nat @ X4 ) @ ( clique6859621968737270801at_nat @ V3 ) )
= ( member_nat_nat @ X4 @ V3 ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_547_forth__assumptions_O_092_060A_062__simps_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member535913909593306477mula_a @ ( monotone_Var_a @ X4 ) @ ( clique5987991184601036204th_A_a @ V3 ) )
= ( member_a @ X4 @ V3 ) ) ) ).
% forth_assumptions.\<A>_simps(2)
thf(fact_548_forth__assumptions_O_092_060A_062__simps_I1_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( member535913909593306477mula_a @ monotone_FALSE_a @ ( clique5987991184601036204th_A_a @ V3 ) ) ) ).
% forth_assumptions.\<A>_simps(1)
thf(fact_549_insert__is__Un,axiom,
( insert_set_set_nat
= ( ^ [A3: set_set_nat] : ( sup_su4213647025997063966et_nat @ ( insert_set_set_nat @ A3 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% insert_is_Un
thf(fact_550_insert__is__Un,axiom,
( insert_set_nat
= ( ^ [A3: set_nat] : ( sup_sup_set_set_nat @ ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_551_insert__is__Un,axiom,
( insert_nat
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_552_insert__is__Un,axiom,
( insert_nat_nat
= ( ^ [A3: nat > nat] : ( sup_sup_set_nat_nat @ ( insert_nat_nat @ A3 @ bot_bot_set_nat_nat ) ) ) ) ).
% insert_is_Un
thf(fact_553_Un__singleton__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X4: set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A @ B )
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
= ( ( ( A = bot_bo7198184520161983622et_nat )
& ( B
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) )
| ( ( A
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
& ( B = bot_bo7198184520161983622et_nat ) )
| ( ( A
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
& ( B
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_554_Un__singleton__iff,axiom,
! [A: set_set_nat,B: set_set_nat,X4: set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B )
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
= ( ( ( A = bot_bot_set_set_nat )
& ( B
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) )
| ( ( A
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
& ( B = bot_bot_set_set_nat ) )
| ( ( A
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
& ( B
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_555_Un__singleton__iff,axiom,
! [A: set_nat,B: set_nat,X4: nat] :
( ( ( sup_sup_set_nat @ A @ B )
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_556_Un__singleton__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat,X4: nat > nat] :
( ( ( sup_sup_set_nat_nat @ A @ B )
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) )
= ( ( ( A = bot_bot_set_nat_nat )
& ( B
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) )
| ( ( A
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) )
& ( B = bot_bot_set_nat_nat ) )
| ( ( A
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) )
& ( B
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_557_singleton__Un__iff,axiom,
! [X4: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat )
= ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( ( ( A = bot_bo7198184520161983622et_nat )
& ( B
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) )
| ( ( A
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
& ( B = bot_bo7198184520161983622et_nat ) )
| ( ( A
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) )
& ( B
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_558_singleton__Un__iff,axiom,
! [X4: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ( insert_set_nat @ X4 @ bot_bot_set_set_nat )
= ( sup_sup_set_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_set_nat )
& ( B
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) )
| ( ( A
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
& ( B = bot_bot_set_set_nat ) )
| ( ( A
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
& ( B
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_559_singleton__Un__iff,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ( ( insert_nat @ X4 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_560_singleton__Un__iff,axiom,
! [X4: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat )
= ( sup_sup_set_nat_nat @ A @ B ) )
= ( ( ( A = bot_bot_set_nat_nat )
& ( B
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) )
| ( ( A
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) )
& ( B = bot_bot_set_nat_nat ) )
| ( ( A
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) )
& ( B
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_561_inj__img__insertE,axiom,
! [F: a > a,A: set_a,X4: a,B: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ~ ( member_a @ X4 @ B )
=> ( ( ( insert_a @ X4 @ B )
= ( image_a_a @ F @ A ) )
=> ~ ! [X6: a,A6: set_a] :
( ~ ( member_a @ X6 @ A6 )
=> ( ( A
= ( insert_a @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_a_a @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_562_inj__img__insertE,axiom,
! [F: a > set_nat,A: set_a,X4: set_nat,B: set_set_nat] :
( ( inj_on_a_set_nat @ F @ A )
=> ( ~ ( member_set_nat @ X4 @ B )
=> ( ( ( insert_set_nat @ X4 @ B )
= ( image_a_set_nat @ F @ A ) )
=> ~ ! [X6: a,A6: set_a] :
( ~ ( member_a @ X6 @ A6 )
=> ( ( A
= ( insert_a @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_a_set_nat @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_563_inj__img__insertE,axiom,
! [F: set_nat > a,A: set_set_nat,X4: a,B: set_a] :
( ( inj_on_set_nat_a @ F @ A )
=> ( ~ ( member_a @ X4 @ B )
=> ( ( ( insert_a @ X4 @ B )
= ( image_set_nat_a @ F @ A ) )
=> ~ ! [X6: set_nat,A6: set_set_nat] :
( ~ ( member_set_nat @ X6 @ A6 )
=> ( ( A
= ( insert_set_nat @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_set_nat_a @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_564_inj__img__insertE,axiom,
! [F: a > set_set_nat,A: set_a,X4: set_set_nat,B: set_set_set_nat] :
( ( inj_on_a_set_set_nat @ F @ A )
=> ( ~ ( member_set_set_nat @ X4 @ B )
=> ( ( ( insert_set_set_nat @ X4 @ B )
= ( image_a_set_set_nat @ F @ A ) )
=> ~ ! [X6: a,A6: set_a] :
( ~ ( member_a @ X6 @ A6 )
=> ( ( A
= ( insert_a @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_a_set_set_nat @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_565_inj__img__insertE,axiom,
! [F: set_nat > set_nat,A: set_set_nat,X4: set_nat,B: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ A )
=> ( ~ ( member_set_nat @ X4 @ B )
=> ( ( ( insert_set_nat @ X4 @ B )
= ( image_7916887816326733075et_nat @ F @ A ) )
=> ~ ! [X6: set_nat,A6: set_set_nat] :
( ~ ( member_set_nat @ X6 @ A6 )
=> ( ( A
= ( insert_set_nat @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_7916887816326733075et_nat @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_566_inj__img__insertE,axiom,
! [F: a > nat > nat,A: set_a,X4: nat > nat,B: set_nat_nat] :
( ( inj_on_a_nat_nat @ F @ A )
=> ( ~ ( member_nat_nat @ X4 @ B )
=> ( ( ( insert_nat_nat @ X4 @ B )
= ( image_a_nat_nat @ F @ A ) )
=> ~ ! [X6: a,A6: set_a] :
( ~ ( member_a @ X6 @ A6 )
=> ( ( A
= ( insert_a @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_a_nat_nat @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_567_inj__img__insertE,axiom,
! [F: set_set_nat > a,A: set_set_set_nat,X4: a,B: set_a] :
( ( inj_on_set_set_nat_a @ F @ A )
=> ( ~ ( member_a @ X4 @ B )
=> ( ( ( insert_a @ X4 @ B )
= ( image_set_set_nat_a @ F @ A ) )
=> ~ ! [X6: set_set_nat,A6: set_set_set_nat] :
( ~ ( member_set_set_nat @ X6 @ A6 )
=> ( ( A
= ( insert_set_set_nat @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_set_set_nat_a @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_568_inj__img__insertE,axiom,
! [F: ( nat > nat ) > a,A: set_nat_nat,X4: a,B: set_a] :
( ( inj_on_nat_nat_a @ F @ A )
=> ( ~ ( member_a @ X4 @ B )
=> ( ( ( insert_a @ X4 @ B )
= ( image_nat_nat_a @ F @ A ) )
=> ~ ! [X6: nat > nat,A6: set_nat_nat] :
( ~ ( member_nat_nat @ X6 @ A6 )
=> ( ( A
= ( insert_nat_nat @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_nat_nat_a @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_569_inj__img__insertE,axiom,
! [F: a > set_set_set_nat,A: set_a,X4: set_set_set_nat,B: set_set_set_set_nat] :
( ( inj_on2720386165175993151et_nat @ F @ A )
=> ( ~ ( member2946998982187404937et_nat @ X4 @ B )
=> ( ( ( insert3687027775829606434et_nat @ X4 @ B )
= ( image_5994387372328453547et_nat @ F @ A ) )
=> ~ ! [X6: a,A6: set_a] :
( ~ ( member_a @ X6 @ A6 )
=> ( ( A
= ( insert_a @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_5994387372328453547et_nat @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_570_inj__img__insertE,axiom,
! [F: set_nat > set_set_nat,A: set_set_nat,X4: set_set_nat,B: set_set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ A )
=> ( ~ ( member_set_set_nat @ X4 @ B )
=> ( ( ( insert_set_set_nat @ X4 @ B )
= ( image_6725021117256019401et_nat @ F @ A ) )
=> ~ ! [X6: set_nat,A6: set_set_nat] :
( ~ ( member_set_nat @ X6 @ A6 )
=> ( ( A
= ( insert_set_nat @ X6 @ A6 ) )
=> ( ( X4
= ( F @ X6 ) )
=> ( B
!= ( image_6725021117256019401et_nat @ F @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_571_forth__assumptions_OSET_Oelims,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( ( clique6509092761774629891_SET_a @ Pi @ X4 )
= Y2 )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( Y2
!= ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X3 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_su4213647025997063966et_nat @ ( clique6509092761774629891_SET_a @ Pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ Pi @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ Pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ Pi @ Psi3 ) ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( Y2 != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ) ).
% forth_assumptions.SET.elims
thf(fact_572_forth__assumptions_Oeval__gs__union,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Theta: a > $o,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique835570645587132141l_gs_a @ V3 @ Pi @ Theta @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( ( clique835570645587132141l_gs_a @ V3 @ Pi @ Theta @ X2 )
| ( clique835570645587132141l_gs_a @ V3 @ Pi @ Theta @ Y ) ) ) ) ).
% forth_assumptions.eval_gs_union
thf(fact_573_forth__assumptions_Oeval__gs__def,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Theta: a > $o,X2: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique835570645587132141l_gs_a @ V3 @ Pi @ Theta @ X2 )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ X2 )
& ( clique5859573001277246426al_g_a @ V3 @ Pi @ Theta @ X ) ) ) ) ) ).
% forth_assumptions.eval_gs_def
thf(fact_574_forth__assumptions_OSET_Osimps_I1_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique6509092761774629891_SET_a @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.SET.simps(1)
thf(fact_575_forth__assumptions_OSET_Osimps_I3_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ 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_576_forth__assumptions_OSET_Osimps_I4_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique6509092761774629891_SET_a @ Pi @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ Pi @ Phi ) @ ( clique6509092761774629891_SET_a @ Pi @ Psi ) ) ) ) ).
% forth_assumptions.SET.simps(4)
thf(fact_577_Sup_OSUP__cong,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat > set_nat,D3: set_set_nat > set_nat,Sup: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_5842784325960735177et_nat @ C2 @ A ) )
= ( Sup @ ( image_5842784325960735177et_nat @ D3 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_578_Sup_OSUP__cong,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: ( nat > nat ) > set_set_nat,D3: ( nat > nat ) > set_set_nat,Sup: set_set_set_nat > set_set_nat] :
( ( A = B )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image_9186907679027735170et_nat @ C2 @ A ) )
= ( Sup @ ( image_9186907679027735170et_nat @ D3 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_579_Inf_OINF__cong,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat > set_nat,D3: set_set_nat > set_nat,Inf: set_set_nat > set_nat] :
( ( A = B )
=> ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_5842784325960735177et_nat @ C2 @ A ) )
= ( Inf @ ( image_5842784325960735177et_nat @ D3 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_580_Inf_OINF__cong,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: ( nat > nat ) > set_set_nat,D3: ( nat > nat ) > set_set_nat,Inf: set_set_set_nat > set_set_nat] :
( ( A = B )
=> ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
=> ( ( C2 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image_9186907679027735170et_nat @ C2 @ A ) )
= ( Inf @ ( image_9186907679027735170et_nat @ D3 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_581_forth__assumptions_OSET_Ocases,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( X4 != monotone_FALSE_a )
=> ( ! [X3: a] :
( X4
!= ( monotone_Var_a @ X3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( X4
!= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( X4 = monotone_TRUE_a ) ) ) ) ) ) ).
% forth_assumptions.SET.cases
thf(fact_582_sup__Pair__Pair,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat,D2: set_nat] :
( ( sup_su3606485054733231425et_nat @ ( produc4532415448927165861et_nat @ A2 @ B2 ) @ ( produc4532415448927165861et_nat @ C @ D2 ) )
= ( produc4532415448927165861et_nat @ ( sup_sup_set_nat @ A2 @ C ) @ ( sup_sup_set_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_583_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_nat,C: set_set_nat,D2: set_nat] :
( ( sup_su2207333250037597687et_nat @ ( produc1634951652945875547et_nat @ A2 @ B2 ) @ ( produc1634951652945875547et_nat @ C @ D2 ) )
= ( produc1634951652945875547et_nat @ ( sup_sup_set_set_nat @ A2 @ C ) @ ( sup_sup_set_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_584_sup__Pair__Pair,axiom,
! [A2: set_nat,B2: set_set_nat,C: set_nat,D2: set_set_nat] :
( ( sup_su8528853930348408567et_nat @ ( produc2517188444241159771et_nat @ A2 @ B2 ) @ ( produc2517188444241159771et_nat @ C @ D2 ) )
= ( produc2517188444241159771et_nat @ ( sup_sup_set_nat @ A2 @ C ) @ ( sup_sup_set_set_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_585_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat,D2: set_set_nat] :
( ( sup_su8202740707494180781et_nat @ ( produc9057842353944101649et_nat @ A2 @ B2 ) @ ( produc9057842353944101649et_nat @ C @ D2 ) )
= ( produc9057842353944101649et_nat @ ( sup_sup_set_set_nat @ A2 @ C ) @ ( sup_sup_set_set_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_586_sup__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_nat,C: set_set_set_nat,D2: set_nat] :
( ( sup_su2028069304614961069et_nat @ ( produc8322454840080408593et_nat @ A2 @ B2 ) @ ( produc8322454840080408593et_nat @ C @ D2 ) )
= ( produc8322454840080408593et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ C ) @ ( sup_sup_set_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_587_sup__Pair__Pair,axiom,
! [A2: set_nat,B2: set_set_set_nat,C: set_nat,D2: set_set_set_nat] :
( ( sup_su339052102296441261et_nat @ ( produc5756764756359792657et_nat @ A2 @ B2 ) @ ( produc5756764756359792657et_nat @ C @ D2 ) )
= ( produc5756764756359792657et_nat @ ( sup_sup_set_nat @ A2 @ C ) @ ( sup_su4213647025997063966et_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_588_sup__Pair__Pair,axiom,
! [A2: set_nat,B2: set_nat_nat,C: set_nat,D2: set_nat_nat] :
( ( sup_su7748964192236511024at_nat @ ( produc1465592063598779540at_nat @ A2 @ B2 ) @ ( produc1465592063598779540at_nat @ C @ D2 ) )
= ( produc1465592063598779540at_nat @ ( sup_sup_set_nat @ A2 @ C ) @ ( sup_sup_set_nat_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_589_sup__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_nat,C: set_nat_nat,D2: set_nat] :
( ( sup_su3213218058960310064et_nat @ ( produc1342248241601784596et_nat @ A2 @ B2 ) @ ( produc1342248241601784596et_nat @ C @ D2 ) )
= ( produc1342248241601784596et_nat @ ( sup_sup_set_nat_nat @ A2 @ C ) @ ( sup_sup_set_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_590_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_set_set_nat,C: set_set_nat,D2: set_set_set_nat] :
( ( sup_su8797241888131514979et_nat @ ( produc7315026656311086279et_nat @ A2 @ B2 ) @ ( produc7315026656311086279et_nat @ C @ D2 ) )
= ( produc7315026656311086279et_nat @ ( sup_sup_set_set_nat @ A2 @ C ) @ ( sup_su4213647025997063966et_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_591_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_nat_nat,C: set_set_nat,D2: set_nat_nat] :
( ( sup_su6693998056113403622at_nat @ ( produc8981877611209197642at_nat @ A2 @ B2 ) @ ( produc8981877611209197642at_nat @ C @ D2 ) )
= ( produc8981877611209197642at_nat @ ( sup_sup_set_set_nat @ A2 @ C ) @ ( sup_sup_set_nat_nat @ B2 @ D2 ) ) ) ).
% sup_Pair_Pair
thf(fact_592_SET_Opelims,axiom,
! [X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique6509092761774629891_SET_a @ pi @ X4 )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ X4 )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( ( Y2
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X3 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Var_a @ X3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( 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] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( ( Y2 = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% SET.pelims
thf(fact_593_vars_Oelims,axiom,
! [X4: monoto5483634261523599098et_nat,Y2: set_set_set_nat] :
( ( ( monoto3765512064276419502et_nat @ X4 )
= Y2 )
=> ( ! [X3: set_set_nat] :
( ( X4
= ( monoto3251651810667535926et_nat @ X3 ) )
=> ( Y2
!= ( insert_set_set_nat @ X3 @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi3: monoto5483634261523599098et_nat,Psi3: monoto5483634261523599098et_nat] :
( ( X4
= ( monoto78699555928797203et_nat @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_su4213647025997063966et_nat @ ( monoto3765512064276419502et_nat @ Phi3 ) @ ( monoto3765512064276419502et_nat @ Psi3 ) ) ) )
=> ( ! [Phi3: monoto5483634261523599098et_nat,Psi3: monoto5483634261523599098et_nat] :
( ( X4
= ( monoto5378078033476056327et_nat @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_su4213647025997063966et_nat @ ( monoto3765512064276419502et_nat @ Phi3 ) @ ( monoto3765512064276419502et_nat @ Psi3 ) ) ) )
=> ( ( ( X4 = monoto6214072352461320530et_nat )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ~ ( ( X4 = monoto5104785069271071961et_nat )
=> ( Y2 != bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ).
% vars.elims
thf(fact_594_vars_Oelims,axiom,
! [X4: monoto7244996872745832772et_nat,Y2: set_set_nat] :
( ( ( monoto8378391831928444664et_nat @ X4 )
= Y2 )
=> ( ! [X3: set_nat] :
( ( X4
= ( monotone_Var_set_nat @ X3 ) )
=> ( Y2
!= ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) )
=> ( ! [Phi3: monoto7244996872745832772et_nat,Psi3: monoto7244996872745832772et_nat] :
( ( X4
= ( monoto3675431328128845661et_nat @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_sup_set_set_nat @ ( monoto8378391831928444664et_nat @ Phi3 ) @ ( monoto8378391831928444664et_nat @ Psi3 ) ) ) )
=> ( ! [Phi3: monoto7244996872745832772et_nat,Psi3: monoto7244996872745832772et_nat] :
( ( X4
= ( monoto2996447309290675281et_nat @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_sup_set_set_nat @ ( monoto8378391831928444664et_nat @ Phi3 ) @ ( monoto8378391831928444664et_nat @ Psi3 ) ) ) )
=> ( ( ( X4 = monoto2388303931541111964et_nat )
=> ( Y2 != bot_bot_set_set_nat ) )
=> ~ ( ( X4 = monoto7549873196617247779et_nat )
=> ( Y2 != bot_bot_set_set_nat ) ) ) ) ) ) ) ).
% vars.elims
thf(fact_595_vars_Oelims,axiom,
! [X4: monoto4181647612830706830la_nat,Y2: set_nat] :
( ( ( monotone_vars_nat @ X4 )
= Y2 )
=> ( ! [X3: nat] :
( ( X4
= ( monotone_Var_nat @ X3 ) )
=> ( Y2
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
=> ( ! [Phi3: monoto4181647612830706830la_nat,Psi3: monoto4181647612830706830la_nat] :
( ( X4
= ( monotone_Conj_nat @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_sup_set_nat @ ( monotone_vars_nat @ Phi3 ) @ ( monotone_vars_nat @ Psi3 ) ) ) )
=> ( ! [Phi3: monoto4181647612830706830la_nat,Psi3: monoto4181647612830706830la_nat] :
( ( X4
= ( monotone_Disj_nat @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_sup_set_nat @ ( monotone_vars_nat @ Phi3 ) @ ( monotone_vars_nat @ Psi3 ) ) ) )
=> ( ( ( X4 = monotone_FALSE_nat )
=> ( Y2 != bot_bot_set_nat ) )
=> ~ ( ( X4 = monotone_TRUE_nat )
=> ( Y2 != bot_bot_set_nat ) ) ) ) ) ) ) ).
% vars.elims
thf(fact_596_vars_Oelims,axiom,
! [X4: monoto8276428299528460797at_nat,Y2: set_nat_nat] :
( ( ( monoto4799612099597528497at_nat @ X4 )
= Y2 )
=> ( ! [X3: nat > nat] :
( ( X4
= ( monotone_Var_nat_nat @ X3 ) )
=> ( Y2
!= ( insert_nat_nat @ X3 @ bot_bot_set_nat_nat ) ) )
=> ( ! [Phi3: monoto8276428299528460797at_nat,Psi3: monoto8276428299528460797at_nat] :
( ( X4
= ( monoto3849437543655978646at_nat @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_sup_set_nat_nat @ ( monoto4799612099597528497at_nat @ Phi3 ) @ ( monoto4799612099597528497at_nat @ Psi3 ) ) ) )
=> ( ! [Phi3: monoto8276428299528460797at_nat,Psi3: monoto8276428299528460797at_nat] :
( ( X4
= ( monoto6252637820927860106at_nat @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_sup_set_nat_nat @ ( monoto4799612099597528497at_nat @ Phi3 ) @ ( monoto4799612099597528497at_nat @ Psi3 ) ) ) )
=> ( ( ( X4 = monoto3735746735589746005at_nat )
=> ( Y2 != bot_bot_set_nat_nat ) )
=> ~ ( ( X4 = monoto581402444256252508at_nat )
=> ( Y2 != bot_bot_set_nat_nat ) ) ) ) ) ) ) ).
% vars.elims
thf(fact_597_vars_Oelims,axiom,
! [X4: monotone_mformula_a,Y2: set_a] :
( ( ( monotone_vars_a @ X4 )
= Y2 )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( Y2
!= ( insert_a @ X3 @ bot_bot_set_a ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_sup_set_a @ ( monotone_vars_a @ Phi3 ) @ ( monotone_vars_a @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_sup_set_a @ ( monotone_vars_a @ Phi3 ) @ ( monotone_vars_a @ Psi3 ) ) ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( Y2 != bot_bot_set_a ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( Y2 != bot_bot_set_a ) ) ) ) ) ) ) ).
% vars.elims
thf(fact_598_forth__assumptions_OSET_Opelims,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( ( clique6509092761774629891_SET_a @ Pi @ X4 )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ X4 )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( ( Y2
= ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X3 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Var_a @ X3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( 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] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( clique5469973757772500719t_odot @ ( clique6509092761774629891_SET_a @ Pi @ Phi3 ) @ ( clique6509092761774629891_SET_a @ Pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( ( Y2 = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique834332680210058238_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ) ).
% forth_assumptions.SET.pelims
thf(fact_599_is__singleton__the__elem,axiom,
( is_sin6612384548583640136et_nat
= ( ^ [A4: set_set_set_nat] :
( A4
= ( insert_set_set_nat @ ( the_elem_set_set_nat @ A4 ) @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_600_is__singleton__the__elem,axiom,
( is_singleton_set_nat
= ( ^ [A4: set_set_nat] :
( A4
= ( insert_set_nat @ ( the_elem_set_nat @ A4 ) @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_601_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
( A4
= ( insert_nat @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_602_is__singleton__the__elem,axiom,
( is_singleton_nat_nat
= ( ^ [A4: set_nat_nat] :
( A4
= ( insert_nat_nat @ ( the_elem_nat_nat @ A4 ) @ bot_bot_set_nat_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_603_bot__empty__eq,axiom,
( bot_bo5536612546450143305_nat_o
= ( ^ [X: set_set_set_nat] : ( member2946998982187404937et_nat @ X @ bot_bo193956671110832956et_nat ) ) ) ).
% bot_empty_eq
thf(fact_604_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_605_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_606_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_607_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_608_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_609_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_610_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_611_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_612_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_613_is__singletonI,axiom,
! [X4: set_set_nat] : ( is_sin6612384548583640136et_nat @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ).
% is_singletonI
thf(fact_614_is__singletonI,axiom,
! [X4: set_nat] : ( is_singleton_set_nat @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ).
% is_singletonI
thf(fact_615_is__singletonI,axiom,
! [X4: nat] : ( is_singleton_nat @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_616_is__singletonI,axiom,
! [X4: nat > nat] : ( is_singleton_nat_nat @ ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) ).
% is_singletonI
thf(fact_617_eval__vars,axiom,
! [Phi: monoto8535755219626829232et_nat,Theta_1: set_set_set_nat > $o,Theta_2: set_set_set_nat > $o] :
( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ ( monoto7609425594988095844et_nat @ Phi ) )
=> ( ( Theta_1 @ X3 )
= ( Theta_2 @ X3 ) ) )
=> ( ( monoto4338639097878760980et_nat @ Theta_1 @ Phi )
= ( monoto4338639097878760980et_nat @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_618_eval__vars,axiom,
! [Phi: monoto5483634261523599098et_nat,Theta_1: set_set_nat > $o,Theta_2: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( monoto3765512064276419502et_nat @ Phi ) )
=> ( ( Theta_1 @ X3 )
= ( Theta_2 @ X3 ) ) )
=> ( ( monoto5763065145529399390et_nat @ Theta_1 @ Phi )
= ( monoto5763065145529399390et_nat @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_619_eval__vars,axiom,
! [Phi: monoto7244996872745832772et_nat,Theta_1: set_nat > $o,Theta_2: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( monoto8378391831928444664et_nat @ Phi ) )
=> ( ( Theta_1 @ X3 )
= ( Theta_2 @ X3 ) ) )
=> ( ( monoto7255863275561024424et_nat @ Theta_1 @ Phi )
= ( monoto7255863275561024424et_nat @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_620_eval__vars,axiom,
! [Phi: monoto8276428299528460797at_nat,Theta_1: ( nat > nat ) > $o,Theta_2: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( monoto4799612099597528497at_nat @ Phi ) )
=> ( ( Theta_1 @ X3 )
= ( Theta_2 @ X3 ) ) )
=> ( ( monoto2874428580947315297at_nat @ Theta_1 @ Phi )
= ( monoto2874428580947315297at_nat @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_621_eval__vars,axiom,
! [Phi: monotone_mformula_a,Theta_1: a > $o,Theta_2: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( monotone_vars_a @ Phi ) )
=> ( ( Theta_1 @ X3 )
= ( Theta_2 @ X3 ) ) )
=> ( ( monotone_eval_a @ Theta_1 @ Phi )
= ( monotone_eval_a @ Theta_2 @ Phi ) ) ) ).
% eval_vars
thf(fact_622_is__singletonI_H,axiom,
! [A: set_set_set_set_nat] :
( ( A != bot_bo193956671110832956et_nat )
=> ( ! [X3: set_set_set_nat,Y4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A )
=> ( ( member2946998982187404937et_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_sin2178213247671319038et_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_623_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_624_is__singletonI_H,axiom,
! [A: set_set_set_nat] :
( ( A != bot_bo7198184520161983622et_nat )
=> ( ! [X3: set_set_nat,Y4: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( ( member_set_set_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_sin6612384548583640136et_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_625_is__singletonI_H,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ( member_set_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_set_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_626_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_627_is__singletonI_H,axiom,
! [A: set_nat_nat] :
( ( A != bot_bot_set_nat_nat )
=> ( ! [X3: nat > nat,Y4: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( ( member_nat_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_nat_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_628_vars_Osimps_I4_J,axiom,
( ( monoto3765512064276419502et_nat @ monoto6214072352461320530et_nat )
= bot_bo7198184520161983622et_nat ) ).
% vars.simps(4)
thf(fact_629_vars_Osimps_I4_J,axiom,
( ( monoto8378391831928444664et_nat @ monoto2388303931541111964et_nat )
= bot_bot_set_set_nat ) ).
% vars.simps(4)
thf(fact_630_vars_Osimps_I4_J,axiom,
( ( monotone_vars_nat @ monotone_FALSE_nat )
= bot_bot_set_nat ) ).
% vars.simps(4)
thf(fact_631_vars_Osimps_I4_J,axiom,
( ( monoto4799612099597528497at_nat @ monoto3735746735589746005at_nat )
= bot_bot_set_nat_nat ) ).
% vars.simps(4)
thf(fact_632_vars_Osimps_I4_J,axiom,
( ( monotone_vars_a @ monotone_FALSE_a )
= bot_bot_set_a ) ).
% vars.simps(4)
thf(fact_633_vars_Osimps_I5_J,axiom,
( ( monoto3765512064276419502et_nat @ monoto5104785069271071961et_nat )
= bot_bo7198184520161983622et_nat ) ).
% vars.simps(5)
thf(fact_634_vars_Osimps_I5_J,axiom,
( ( monoto8378391831928444664et_nat @ monoto7549873196617247779et_nat )
= bot_bot_set_set_nat ) ).
% vars.simps(5)
thf(fact_635_vars_Osimps_I5_J,axiom,
( ( monotone_vars_nat @ monotone_TRUE_nat )
= bot_bot_set_nat ) ).
% vars.simps(5)
thf(fact_636_vars_Osimps_I5_J,axiom,
( ( monoto4799612099597528497at_nat @ monoto581402444256252508at_nat )
= bot_bot_set_nat_nat ) ).
% vars.simps(5)
thf(fact_637_vars_Osimps_I5_J,axiom,
( ( monotone_vars_a @ monotone_TRUE_a )
= bot_bot_set_a ) ).
% vars.simps(5)
thf(fact_638_vars_Osimps_I2_J,axiom,
! [Phi: monoto7244996872745832772et_nat,Psi: monoto7244996872745832772et_nat] :
( ( monoto8378391831928444664et_nat @ ( monoto3675431328128845661et_nat @ Phi @ Psi ) )
= ( sup_sup_set_set_nat @ ( monoto8378391831928444664et_nat @ Phi ) @ ( monoto8378391831928444664et_nat @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_639_vars_Osimps_I2_J,axiom,
! [Phi: monoto5483634261523599098et_nat,Psi: monoto5483634261523599098et_nat] :
( ( monoto3765512064276419502et_nat @ ( monoto78699555928797203et_nat @ Phi @ Psi ) )
= ( sup_su4213647025997063966et_nat @ ( monoto3765512064276419502et_nat @ Phi ) @ ( monoto3765512064276419502et_nat @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_640_vars_Osimps_I2_J,axiom,
! [Phi: monoto4181647612830706830la_nat,Psi: monoto4181647612830706830la_nat] :
( ( monotone_vars_nat @ ( monotone_Conj_nat @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( monotone_vars_nat @ Phi ) @ ( monotone_vars_nat @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_641_vars_Osimps_I2_J,axiom,
! [Phi: monoto8276428299528460797at_nat,Psi: monoto8276428299528460797at_nat] :
( ( monoto4799612099597528497at_nat @ ( monoto3849437543655978646at_nat @ Phi @ Psi ) )
= ( sup_sup_set_nat_nat @ ( monoto4799612099597528497at_nat @ Phi ) @ ( monoto4799612099597528497at_nat @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_642_vars_Osimps_I2_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_vars_a @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( sup_sup_set_a @ ( monotone_vars_a @ Phi ) @ ( monotone_vars_a @ Psi ) ) ) ).
% vars.simps(2)
thf(fact_643_vars_Osimps_I3_J,axiom,
! [Phi: monoto7244996872745832772et_nat,Psi: monoto7244996872745832772et_nat] :
( ( monoto8378391831928444664et_nat @ ( monoto2996447309290675281et_nat @ Phi @ Psi ) )
= ( sup_sup_set_set_nat @ ( monoto8378391831928444664et_nat @ Phi ) @ ( monoto8378391831928444664et_nat @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_644_vars_Osimps_I3_J,axiom,
! [Phi: monoto5483634261523599098et_nat,Psi: monoto5483634261523599098et_nat] :
( ( monoto3765512064276419502et_nat @ ( monoto5378078033476056327et_nat @ Phi @ Psi ) )
= ( sup_su4213647025997063966et_nat @ ( monoto3765512064276419502et_nat @ Phi ) @ ( monoto3765512064276419502et_nat @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_645_vars_Osimps_I3_J,axiom,
! [Phi: monoto4181647612830706830la_nat,Psi: monoto4181647612830706830la_nat] :
( ( monotone_vars_nat @ ( monotone_Disj_nat @ Phi @ Psi ) )
= ( sup_sup_set_nat @ ( monotone_vars_nat @ Phi ) @ ( monotone_vars_nat @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_646_vars_Osimps_I3_J,axiom,
! [Phi: monoto8276428299528460797at_nat,Psi: monoto8276428299528460797at_nat] :
( ( monoto4799612099597528497at_nat @ ( monoto6252637820927860106at_nat @ Phi @ Psi ) )
= ( sup_sup_set_nat_nat @ ( monoto4799612099597528497at_nat @ Phi ) @ ( monoto4799612099597528497at_nat @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_647_vars_Osimps_I3_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_vars_a @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( sup_sup_set_a @ ( monotone_vars_a @ Phi ) @ ( monotone_vars_a @ Psi ) ) ) ).
% vars.simps(3)
thf(fact_648_vars_Osimps_I1_J,axiom,
! [X4: set_set_nat] :
( ( monoto3765512064276419502et_nat @ ( monoto3251651810667535926et_nat @ X4 ) )
= ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ).
% vars.simps(1)
thf(fact_649_vars_Osimps_I1_J,axiom,
! [X4: set_nat] :
( ( monoto8378391831928444664et_nat @ ( monotone_Var_set_nat @ X4 ) )
= ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ).
% vars.simps(1)
thf(fact_650_vars_Osimps_I1_J,axiom,
! [X4: nat] :
( ( monotone_vars_nat @ ( monotone_Var_nat @ X4 ) )
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ).
% vars.simps(1)
thf(fact_651_vars_Osimps_I1_J,axiom,
! [X4: nat > nat] :
( ( monoto4799612099597528497at_nat @ ( monotone_Var_nat_nat @ X4 ) )
= ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) ).
% vars.simps(1)
thf(fact_652_vars_Osimps_I1_J,axiom,
! [X4: a] :
( ( monotone_vars_a @ ( monotone_Var_a @ X4 ) )
= ( insert_a @ X4 @ bot_bot_set_a ) ) ).
% vars.simps(1)
thf(fact_653_is__singleton__def,axiom,
( is_sin6612384548583640136et_nat
= ( ^ [A4: set_set_set_nat] :
? [X: set_set_nat] :
( A4
= ( insert_set_set_nat @ X @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% is_singleton_def
thf(fact_654_is__singleton__def,axiom,
( is_singleton_set_nat
= ( ^ [A4: set_set_nat] :
? [X: set_nat] :
( A4
= ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_655_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
? [X: nat] :
( A4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_656_is__singleton__def,axiom,
( is_singleton_nat_nat
= ( ^ [A4: set_nat_nat] :
? [X: nat > nat] :
( A4
= ( insert_nat_nat @ X @ bot_bot_set_nat_nat ) ) ) ) ).
% is_singleton_def
thf(fact_657_is__singletonE,axiom,
! [A: set_set_set_nat] :
( ( is_sin6612384548583640136et_nat @ A )
=> ~ ! [X3: set_set_nat] :
( A
!= ( insert_set_set_nat @ X3 @ bot_bo7198184520161983622et_nat ) ) ) ).
% is_singletonE
thf(fact_658_is__singletonE,axiom,
! [A: set_set_nat] :
( ( is_singleton_set_nat @ A )
=> ~ ! [X3: set_nat] :
( A
!= ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% is_singletonE
thf(fact_659_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_660_is__singletonE,axiom,
! [A: set_nat_nat] :
( ( is_singleton_nat_nat @ A )
=> ~ ! [X3: nat > nat] :
( A
!= ( insert_nat_nat @ X3 @ bot_bot_set_nat_nat ) ) ) ).
% is_singletonE
thf(fact_661_bot__prod__def,axiom,
( bot_bo3047382831089536473et_nat
= ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_662_bot__prod__def,axiom,
( bot_bo2908193468437106319et_nat
= ( produc1634951652945875547et_nat @ bot_bot_set_set_nat @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_663_bot__prod__def,axiom,
( bot_bo6342111893141391et_nat
= ( produc2517188444241159771et_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% bot_prod_def
thf(fact_664_bot__prod__def,axiom,
( bot_bo9070301037099493445et_nat
= ( produc8322454840080408593et_nat @ bot_bo7198184520161983622et_nat @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_665_bot__prod__def,axiom,
( bot_bo6021600403123937349et_nat
= ( produc9057842353944101649et_nat @ bot_bot_set_set_nat @ bot_bot_set_set_nat ) ) ).
% bot_prod_def
thf(fact_666_bot__prod__def,axiom,
( bot_bo7381283834780973637et_nat
= ( produc5756764756359792657et_nat @ bot_bot_set_nat @ bot_bo7198184520161983622et_nat ) ) ).
% bot_prod_def
thf(fact_667_bot__prod__def,axiom,
( bot_bo2498585933846414280at_nat
= ( produc1465592063598779540at_nat @ bot_bot_set_nat @ bot_bot_set_nat_nat ) ) ).
% bot_prod_def
thf(fact_668_bot__prod__def,axiom,
( bot_bo7186211837424989128et_nat
= ( produc1342248241601784596et_nat @ bot_bot_set_nat_nat @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_669_bot__prod__def,axiom,
( bot_bo6248027959722598907et_nat
= ( produc1498124630991567047et_nat @ bot_bo7198184520161983622et_nat @ bot_bot_set_set_nat ) ) ).
% bot_prod_def
thf(fact_670_bot__prod__def,axiom,
( bot_bo7502318977428687611et_nat
= ( produc7315026656311086279et_nat @ bot_bot_set_set_nat @ bot_bo7198184520161983622et_nat ) ) ).
% bot_prod_def
thf(fact_671_vars_Opelims,axiom,
! [X4: monoto5483634261523599098et_nat,Y2: set_set_set_nat] :
( ( ( monoto3765512064276419502et_nat @ X4 )
= Y2 )
=> ( ( accp_M7269832456818321411et_nat @ monoto3287166057840848965et_nat @ X4 )
=> ( ! [X3: set_set_nat] :
( ( X4
= ( monoto3251651810667535926et_nat @ X3 ) )
=> ( ( Y2
= ( insert_set_set_nat @ X3 @ bot_bo7198184520161983622et_nat ) )
=> ~ ( accp_M7269832456818321411et_nat @ monoto3287166057840848965et_nat @ ( monoto3251651810667535926et_nat @ X3 ) ) ) )
=> ( ! [Phi3: monoto5483634261523599098et_nat,Psi3: monoto5483634261523599098et_nat] :
( ( X4
= ( monoto78699555928797203et_nat @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_su4213647025997063966et_nat @ ( monoto3765512064276419502et_nat @ Phi3 ) @ ( monoto3765512064276419502et_nat @ Psi3 ) ) )
=> ~ ( accp_M7269832456818321411et_nat @ monoto3287166057840848965et_nat @ ( monoto78699555928797203et_nat @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monoto5483634261523599098et_nat,Psi3: monoto5483634261523599098et_nat] :
( ( X4
= ( monoto5378078033476056327et_nat @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_su4213647025997063966et_nat @ ( monoto3765512064276419502et_nat @ Phi3 ) @ ( monoto3765512064276419502et_nat @ Psi3 ) ) )
=> ~ ( accp_M7269832456818321411et_nat @ monoto3287166057840848965et_nat @ ( monoto5378078033476056327et_nat @ Phi3 @ Psi3 ) ) ) )
=> ( ( ( X4 = monoto6214072352461320530et_nat )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M7269832456818321411et_nat @ monoto3287166057840848965et_nat @ monoto6214072352461320530et_nat ) ) )
=> ~ ( ( X4 = monoto5104785069271071961et_nat )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M7269832456818321411et_nat @ monoto3287166057840848965et_nat @ monoto5104785069271071961et_nat ) ) ) ) ) ) ) ) ) ).
% vars.pelims
thf(fact_672_vars_Opelims,axiom,
! [X4: monoto7244996872745832772et_nat,Y2: set_set_nat] :
( ( ( monoto8378391831928444664et_nat @ X4 )
= Y2 )
=> ( ( accp_M6712534473328676429et_nat @ monoto1962911667133435791et_nat @ X4 )
=> ( ! [X3: set_nat] :
( ( X4
= ( monotone_Var_set_nat @ X3 ) )
=> ( ( Y2
= ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
=> ~ ( accp_M6712534473328676429et_nat @ monoto1962911667133435791et_nat @ ( monotone_Var_set_nat @ X3 ) ) ) )
=> ( ! [Phi3: monoto7244996872745832772et_nat,Psi3: monoto7244996872745832772et_nat] :
( ( X4
= ( monoto3675431328128845661et_nat @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_sup_set_set_nat @ ( monoto8378391831928444664et_nat @ Phi3 ) @ ( monoto8378391831928444664et_nat @ Psi3 ) ) )
=> ~ ( accp_M6712534473328676429et_nat @ monoto1962911667133435791et_nat @ ( monoto3675431328128845661et_nat @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monoto7244996872745832772et_nat,Psi3: monoto7244996872745832772et_nat] :
( ( X4
= ( monoto2996447309290675281et_nat @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_sup_set_set_nat @ ( monoto8378391831928444664et_nat @ Phi3 ) @ ( monoto8378391831928444664et_nat @ Psi3 ) ) )
=> ~ ( accp_M6712534473328676429et_nat @ monoto1962911667133435791et_nat @ ( monoto2996447309290675281et_nat @ Phi3 @ Psi3 ) ) ) )
=> ( ( ( X4 = monoto2388303931541111964et_nat )
=> ( ( Y2 = bot_bot_set_set_nat )
=> ~ ( accp_M6712534473328676429et_nat @ monoto1962911667133435791et_nat @ monoto2388303931541111964et_nat ) ) )
=> ~ ( ( X4 = monoto7549873196617247779et_nat )
=> ( ( Y2 = bot_bot_set_set_nat )
=> ~ ( accp_M6712534473328676429et_nat @ monoto1962911667133435791et_nat @ monoto7549873196617247779et_nat ) ) ) ) ) ) ) ) ) ).
% vars.pelims
thf(fact_673_vars_Opelims,axiom,
! [X4: monoto4181647612830706830la_nat,Y2: set_nat] :
( ( ( monotone_vars_nat @ X4 )
= Y2 )
=> ( ( accp_M5007270476854070679la_nat @ monoto8552426717926606809el_nat @ X4 )
=> ( ! [X3: nat] :
( ( X4
= ( monotone_Var_nat @ X3 ) )
=> ( ( Y2
= ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ~ ( accp_M5007270476854070679la_nat @ monoto8552426717926606809el_nat @ ( monotone_Var_nat @ X3 ) ) ) )
=> ( ! [Phi3: monoto4181647612830706830la_nat,Psi3: monoto4181647612830706830la_nat] :
( ( X4
= ( monotone_Conj_nat @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_sup_set_nat @ ( monotone_vars_nat @ Phi3 ) @ ( monotone_vars_nat @ Psi3 ) ) )
=> ~ ( accp_M5007270476854070679la_nat @ monoto8552426717926606809el_nat @ ( monotone_Conj_nat @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monoto4181647612830706830la_nat,Psi3: monoto4181647612830706830la_nat] :
( ( X4
= ( monotone_Disj_nat @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_sup_set_nat @ ( monotone_vars_nat @ Phi3 ) @ ( monotone_vars_nat @ Psi3 ) ) )
=> ~ ( accp_M5007270476854070679la_nat @ monoto8552426717926606809el_nat @ ( monotone_Disj_nat @ Phi3 @ Psi3 ) ) ) )
=> ( ( ( X4 = monotone_FALSE_nat )
=> ( ( Y2 = bot_bot_set_nat )
=> ~ ( accp_M5007270476854070679la_nat @ monoto8552426717926606809el_nat @ monotone_FALSE_nat ) ) )
=> ~ ( ( X4 = monotone_TRUE_nat )
=> ( ( Y2 = bot_bot_set_nat )
=> ~ ( accp_M5007270476854070679la_nat @ monoto8552426717926606809el_nat @ monotone_TRUE_nat ) ) ) ) ) ) ) ) ) ).
% vars.pelims
thf(fact_674_vars_Opelims,axiom,
! [X4: monoto8276428299528460797at_nat,Y2: set_nat_nat] :
( ( ( monoto4799612099597528497at_nat @ X4 )
= Y2 )
=> ( ( accp_M442112965984270726at_nat @ monoto7845514509509743560at_nat @ X4 )
=> ( ! [X3: nat > nat] :
( ( X4
= ( monotone_Var_nat_nat @ X3 ) )
=> ( ( Y2
= ( insert_nat_nat @ X3 @ bot_bot_set_nat_nat ) )
=> ~ ( accp_M442112965984270726at_nat @ monoto7845514509509743560at_nat @ ( monotone_Var_nat_nat @ X3 ) ) ) )
=> ( ! [Phi3: monoto8276428299528460797at_nat,Psi3: monoto8276428299528460797at_nat] :
( ( X4
= ( monoto3849437543655978646at_nat @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_sup_set_nat_nat @ ( monoto4799612099597528497at_nat @ Phi3 ) @ ( monoto4799612099597528497at_nat @ Psi3 ) ) )
=> ~ ( accp_M442112965984270726at_nat @ monoto7845514509509743560at_nat @ ( monoto3849437543655978646at_nat @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monoto8276428299528460797at_nat,Psi3: monoto8276428299528460797at_nat] :
( ( X4
= ( monoto6252637820927860106at_nat @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_sup_set_nat_nat @ ( monoto4799612099597528497at_nat @ Phi3 ) @ ( monoto4799612099597528497at_nat @ Psi3 ) ) )
=> ~ ( accp_M442112965984270726at_nat @ monoto7845514509509743560at_nat @ ( monoto6252637820927860106at_nat @ Phi3 @ Psi3 ) ) ) )
=> ( ( ( X4 = monoto3735746735589746005at_nat )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M442112965984270726at_nat @ monoto7845514509509743560at_nat @ monoto3735746735589746005at_nat ) ) )
=> ~ ( ( X4 = monoto581402444256252508at_nat )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M442112965984270726at_nat @ monoto7845514509509743560at_nat @ monoto581402444256252508at_nat ) ) ) ) ) ) ) ) ) ).
% vars.pelims
thf(fact_675_vars_Opelims,axiom,
! [X4: monotone_mformula_a,Y2: set_a] :
( ( ( monotone_vars_a @ X4 )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ monotone_vars_rel_a @ X4 )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( ( Y2
= ( insert_a @ X3 @ bot_bot_set_a ) )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_vars_rel_a @ ( monotone_Var_a @ X3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_sup_set_a @ ( monotone_vars_a @ Phi3 ) @ ( monotone_vars_a @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_vars_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_sup_set_a @ ( monotone_vars_a @ Phi3 ) @ ( monotone_vars_a @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_vars_rel_a @ ( monotone_Disj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( ( Y2 = bot_bot_set_a )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_vars_rel_a @ monotone_FALSE_a ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( ( Y2 = bot_bot_set_a )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_vars_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% vars.pelims
thf(fact_676_SUB_Oelims,axiom,
! [X4: monotone_mformula_a,Y2: set_Mo2626137824023173004mula_a] :
( ( ( monotone_SUB_a @ X4 )
= Y2 )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_su7438456061012554424mula_a @ ( sup_su7438456061012554424mula_a @ ( insert7703626487854729094mula_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) @ bot_bo3042613601904376864mula_a ) @ ( monotone_SUB_a @ Phi3 ) ) @ ( monotone_SUB_a @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( sup_su7438456061012554424mula_a @ ( sup_su7438456061012554424mula_a @ ( insert7703626487854729094mula_a @ ( monotone_Disj_a @ Phi3 @ Psi3 ) @ bot_bo3042613601904376864mula_a ) @ ( monotone_SUB_a @ Phi3 ) ) @ ( monotone_SUB_a @ Psi3 ) ) ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( Y2
!= ( insert7703626487854729094mula_a @ ( monotone_Var_a @ X3 ) @ bot_bo3042613601904376864mula_a ) ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( Y2
!= ( insert7703626487854729094mula_a @ monotone_FALSE_a @ bot_bo3042613601904376864mula_a ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( Y2
!= ( insert7703626487854729094mula_a @ monotone_TRUE_a @ bot_bo3042613601904376864mula_a ) ) ) ) ) ) ) ) ).
% SUB.elims
thf(fact_677_forth__assumptions_OAPR_Osimps_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ ( monotone_Var_a @ X4 ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ) ).
% forth_assumptions.APR.simps(2)
thf(fact_678_forth__assumptions_O_092_060pi_062__singleton_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_set_set_set_nat,Pi: set_set_set_nat > set_nat,X4: set_set_set_nat] :
( ( clique3407333501437444587et_nat @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member2946998982187404937et_nat @ X4 @ V3 )
=> ( member2946998982187404937et_nat @ ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ).
% forth_assumptions.\<pi>_singleton(2)
thf(fact_679_forth__assumptions_O_092_060pi_062__singleton_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_set_set_nat,Pi: set_set_nat > set_nat,X4: set_set_nat] :
( ( clique2455256169097332789et_nat @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member_set_set_nat @ X4 @ V3 )
=> ( member2946998982187404937et_nat @ ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ).
% forth_assumptions.\<pi>_singleton(2)
thf(fact_680_forth__assumptions_O_092_060pi_062__singleton_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_set_nat,Pi: set_nat > set_nat,X4: set_nat] :
( ( clique522982669833463679et_nat @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member_set_nat @ X4 @ V3 )
=> ( member2946998982187404937et_nat @ ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ).
% forth_assumptions.\<pi>_singleton(2)
thf(fact_681_forth__assumptions_O_092_060pi_062__singleton_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_nat_nat,Pi: ( nat > nat ) > set_nat,X4: nat > nat] :
( ( clique5528702923696243640at_nat @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member_nat_nat @ X4 @ V3 )
=> ( member2946998982187404937et_nat @ ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ).
% forth_assumptions.\<pi>_singleton(2)
thf(fact_682_forth__assumptions_O_092_060pi_062__singleton_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member_a @ X4 @ V3 )
=> ( member2946998982187404937et_nat @ ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ).
% forth_assumptions.\<pi>_singleton(2)
thf(fact_683_forth__assumptions_Oapprox__pos_Osimps_I2_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P3 @ K @ Pi @ monotone_TRUE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(2)
thf(fact_684_forth__assumptions_Oapprox__pos_Osimps_I3_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P3 @ K @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(3)
thf(fact_685_forth__assumptions_Oapprox__pos_Osimps_I4_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,V2: a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P3 @ K @ Pi @ ( monotone_Var_a @ V2 ) )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(4)
thf(fact_686_forth__assumptions_Oapprox__pos_Ocong,axiom,
clique8538548958085942603_pos_a = clique8538548958085942603_pos_a ).
% forth_assumptions.approx_pos.cong
thf(fact_687_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_688_forth__assumptions_OAPR_Ocong,axiom,
clique3873310923663319714_APR_a = clique3873310923663319714_APR_a ).
% forth_assumptions.APR.cong
thf(fact_689_SUB_Osimps_I2_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_SUB_a @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( sup_su7438456061012554424mula_a @ ( sup_su7438456061012554424mula_a @ ( insert7703626487854729094mula_a @ ( monotone_Disj_a @ Phi @ Psi ) @ bot_bo3042613601904376864mula_a ) @ ( monotone_SUB_a @ Phi ) ) @ ( monotone_SUB_a @ Psi ) ) ) ).
% SUB.simps(2)
thf(fact_690_SUB_Osimps_I1_J,axiom,
! [Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( monotone_SUB_a @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( sup_su7438456061012554424mula_a @ ( sup_su7438456061012554424mula_a @ ( insert7703626487854729094mula_a @ ( monotone_Conj_a @ Phi @ Psi ) @ bot_bo3042613601904376864mula_a ) @ ( monotone_SUB_a @ Phi ) ) @ ( monotone_SUB_a @ Psi ) ) ) ).
% SUB.simps(1)
thf(fact_691_SUB_Osimps_I3_J,axiom,
! [X4: a] :
( ( monotone_SUB_a @ ( monotone_Var_a @ X4 ) )
= ( insert7703626487854729094mula_a @ ( monotone_Var_a @ X4 ) @ bot_bo3042613601904376864mula_a ) ) ).
% SUB.simps(3)
thf(fact_692_SUB_Osimps_I4_J,axiom,
( ( monotone_SUB_a @ monotone_FALSE_a )
= ( insert7703626487854729094mula_a @ monotone_FALSE_a @ bot_bo3042613601904376864mula_a ) ) ).
% SUB.simps(4)
thf(fact_693_SUB_Osimps_I5_J,axiom,
( ( monotone_SUB_a @ monotone_TRUE_a )
= ( insert7703626487854729094mula_a @ monotone_TRUE_a @ bot_bo3042613601904376864mula_a ) ) ).
% SUB.simps(5)
thf(fact_694_forth__assumptions_OAPR,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( member535913909593306477mula_a @ Phi @ monoto4877036962378694605mula_a )
=> ( ( member535913909593306477mula_a @ Phi @ ( clique5987991184601036204th_A_a @ V3 ) )
=> ( member2946998982187404937et_nat @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ) ).
% forth_assumptions.APR
thf(fact_695_forth__assumptions_Oempty___092_060P_062L_092_060G_062l,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( member2946998982187404937et_nat @ bot_bo7198184520161983622et_nat @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ).
% forth_assumptions.empty_\<P>L\<G>l
thf(fact_696_forth__assumptions_OAPR_Osimps_I1_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.APR.simps(1)
thf(fact_697_forth__assumptions_Oapprox__pos_Osimps_I5_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,V2: monotone_mformula_a,Va2: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P3 @ K @ Pi @ ( monotone_Disj_a @ V2 @ Va2 ) )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.approx_pos.simps(5)
thf(fact_698_SUB_Opelims,axiom,
! [X4: monotone_mformula_a,Y2: set_Mo2626137824023173004mula_a] :
( ( ( monotone_SUB_a @ X4 )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ monotone_SUB_rel_a @ X4 )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_su7438456061012554424mula_a @ ( sup_su7438456061012554424mula_a @ ( insert7703626487854729094mula_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) @ bot_bo3042613601904376864mula_a ) @ ( monotone_SUB_a @ Phi3 ) ) @ ( monotone_SUB_a @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_SUB_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( sup_su7438456061012554424mula_a @ ( sup_su7438456061012554424mula_a @ ( insert7703626487854729094mula_a @ ( monotone_Disj_a @ Phi3 @ Psi3 ) @ bot_bo3042613601904376864mula_a ) @ ( monotone_SUB_a @ Phi3 ) ) @ ( monotone_SUB_a @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_SUB_rel_a @ ( monotone_Disj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( ( Y2
= ( insert7703626487854729094mula_a @ ( monotone_Var_a @ X3 ) @ bot_bo3042613601904376864mula_a ) )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_SUB_rel_a @ ( monotone_Var_a @ X3 ) ) ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( ( Y2
= ( insert7703626487854729094mula_a @ monotone_FALSE_a @ bot_bo3042613601904376864mula_a ) )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_SUB_rel_a @ monotone_FALSE_a ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( ( Y2
= ( insert7703626487854729094mula_a @ monotone_TRUE_a @ bot_bo3042613601904376864mula_a ) )
=> ~ ( accp_M6162913489380515981mula_a @ monotone_SUB_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% SUB.pelims
thf(fact_699_forth__assumptions_Oapprox__pos_Oelims,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( ( clique8538548958085942603_pos_a @ L @ P3 @ K @ Pi @ X4 )
= Y2 )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( Y2
!= ( clique3314026705535538693os_cap @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi2 ) ) ) )
=> ( ( ( X4 = monotone_TRUE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ( ? [V: a] :
( X4
= ( monotone_Var_a @ V ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ~ ( ? [V: monotone_mformula_a,Va: monotone_mformula_a] :
( X4
= ( monotone_Disj_a @ V @ Va ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ) ).
% forth_assumptions.approx_pos.elims
thf(fact_700_forth__assumptions_OAPR_Oelims,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ X4 )
= Y2 )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( Y2
!= ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X3 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( clique2586627118207531017_sqcup @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( clique2586627118206219037_sqcap @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi3 ) ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( Y2 != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ) ).
% forth_assumptions.APR.elims
thf(fact_701_forth__assumptions_Oapprox__pos_Opelims,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( ( clique8538548958085942603_pos_a @ L @ P3 @ K @ Pi @ X4 )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ X4 )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ( Y2
= ( clique3314026705535538693os_cap @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi2 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Conj_a @ Phi2 @ Psi2 ) ) ) )
=> ( ( ( X4 = monotone_TRUE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_TRUE_a ) ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [V: a] :
( ( X4
= ( monotone_Var_a @ V ) )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Var_a @ V ) ) ) )
=> ~ ! [V: monotone_mformula_a,Va: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ V @ Va ) )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Disj_a @ V @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% forth_assumptions.approx_pos.pelims
thf(fact_702_forth__assumptions_OAPR_Opelims,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ X4 )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ X4 )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( ( Y2
= ( insert_set_set_nat @ ( insert_set_nat @ ( Pi @ X3 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Var_a @ X3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( clique2586627118207531017_sqcup @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Disj_a @ Phi3 @ Psi3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( clique2586627118206219037_sqcap @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi3 ) ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Conj_a @ Phi3 @ Psi3 ) ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( ( Y2 = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ) ).
% forth_assumptions.APR.pelims
thf(fact_703_forth__assumptions_Oapprox__pos_Osimps_I1_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi5: monotone_mformula_a,Psi5: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique8538548958085942603_pos_a @ L @ P3 @ K @ Pi @ ( monotone_Conj_a @ Phi5 @ Psi5 ) )
= ( clique3314026705535538693os_cap @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi5 ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi5 ) ) ) ) ).
% forth_assumptions.approx_pos.simps(1)
thf(fact_704_forth__assumptions_Ono__deviation_I1_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique3934260045859375359_pos_a @ L @ P3 @ K @ Pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.no_deviation(1)
thf(fact_705_second__assumptions_Osqcap_Ocong,axiom,
clique2586627118206219037_sqcap = clique2586627118206219037_sqcap ).
% second_assumptions.sqcap.cong
thf(fact_706_second__assumptions_Osqcup_Ocong,axiom,
clique2586627118207531017_sqcup = clique2586627118207531017_sqcup ).
% second_assumptions.sqcup.cong
thf(fact_707_second__assumptions_Odeviate__pos__cap_Ocong,axiom,
clique3314026705535538693os_cap = clique3314026705535538693os_cap ).
% second_assumptions.deviate_pos_cap.cong
thf(fact_708_forth__assumptions_Odeviate__pos_Ocong,axiom,
clique3934260045859375359_pos_a = clique3934260045859375359_pos_a ).
% forth_assumptions.deviate_pos.cong
thf(fact_709_forth__assumptions_OAPR_Osimps_I3_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ ( monotone_Disj_a @ Phi @ Psi ) )
= ( clique2586627118207531017_sqcup @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi ) ) ) ) ).
% forth_assumptions.APR.simps(3)
thf(fact_710_forth__assumptions_OAPR_Osimps_I4_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ ( monotone_Conj_a @ Phi @ Psi ) )
= ( clique2586627118206219037_sqcap @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi ) ) ) ) ).
% forth_assumptions.APR.simps(4)
thf(fact_711_forth__assumptions_Ono__deviation_I3_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,X4: a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ( clique3934260045859375359_pos_a @ L @ P3 @ K @ Pi @ ( monotone_Var_a @ X4 ) )
= bot_bo7198184520161983622et_nat ) ) ).
% forth_assumptions.no_deviation(3)
thf(fact_712_forth__assumptions_Odeviate__subset__Conj_I1_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ Pi )
=> ( ord_le9131159989063066194et_nat @ ( clique3934260045859375359_pos_a @ L @ P3 @ K @ Pi @ ( monotone_Conj_a @ Phi @ Psi ) ) @ ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3314026705535538693os_cap @ L @ P3 @ K @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Phi ) @ ( clique3873310923663319714_APR_a @ L @ P3 @ K @ Pi @ Psi ) ) @ ( clique3934260045859375359_pos_a @ L @ P3 @ K @ Pi @ Phi ) ) @ ( clique3934260045859375359_pos_a @ L @ P3 @ K @ Pi @ Psi ) ) ) ) ).
% forth_assumptions.deviate_subset_Conj(1)
thf(fact_713_inj__on__insert,axiom,
! [F: nat > a,A2: nat,A: set_nat] :
( ( inj_on_nat_a @ F @ ( insert_nat @ A2 @ A ) )
= ( ( inj_on_nat_a @ F @ A )
& ~ ( member_a @ ( F @ A2 ) @ ( image_nat_a @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_714_inj__on__insert,axiom,
! [F: a > set_nat,A2: a,A: set_a] :
( ( inj_on_a_set_nat @ F @ ( insert_a @ A2 @ A ) )
= ( ( inj_on_a_set_nat @ F @ A )
& ~ ( member_set_nat @ ( F @ A2 ) @ ( image_a_set_nat @ F @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_715_inj__on__insert,axiom,
! [F: set_nat > a,A2: set_nat,A: set_set_nat] :
( ( inj_on_set_nat_a @ F @ ( insert_set_nat @ A2 @ A ) )
= ( ( inj_on_set_nat_a @ F @ A )
& ~ ( member_a @ ( F @ A2 ) @ ( image_set_nat_a @ F @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_716_inj__on__insert,axiom,
! [F: nat > set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_set_nat @ F @ ( insert_nat @ A2 @ A ) )
= ( ( inj_on_nat_set_nat @ F @ A )
& ~ ( member_set_nat @ ( F @ A2 ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_717_inj__on__insert,axiom,
! [F: set_nat > set_nat,A2: set_nat,A: set_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( insert_set_nat @ A2 @ A ) )
= ( ( inj_on4604407203859583615et_nat @ F @ A )
& ~ ( member_set_nat @ ( F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_718_inj__on__insert,axiom,
! [F: nat > set_set_nat,A2: nat,A: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ ( insert_nat @ A2 @ A ) )
= ( ( inj_on8105003582846801791et_nat @ F @ A )
& ~ ( member_set_set_nat @ ( F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_719_inj__on__insert,axiom,
! [F: nat > nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat_nat @ F @ ( insert_nat @ A2 @ A ) )
= ( ( inj_on_nat_nat_nat @ F @ A )
& ~ ( member_nat_nat @ ( F @ A2 ) @ ( image_nat_nat_nat2 @ F @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_720_inj__on__insert,axiom,
! [F: set_set_nat > a,A2: set_set_nat,A: set_set_set_nat] :
( ( inj_on_set_set_nat_a @ F @ ( insert_set_set_nat @ A2 @ A ) )
= ( ( inj_on_set_set_nat_a @ F @ A )
& ~ ( member_a @ ( F @ A2 ) @ ( image_set_set_nat_a @ F @ ( minus_2447799839930672331et_nat @ A @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_721_inj__on__insert,axiom,
! [F: ( nat > nat ) > a,A2: nat > nat,A: set_nat_nat] :
( ( inj_on_nat_nat_a @ F @ ( insert_nat_nat @ A2 @ A ) )
= ( ( inj_on_nat_nat_a @ F @ A )
& ~ ( member_a @ ( F @ A2 ) @ ( image_nat_nat_a @ F @ ( minus_8121590178497047118at_nat @ A @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_722_inj__on__insert,axiom,
! [F: set_nat > set_set_nat,A2: set_nat,A: set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ ( insert_set_nat @ A2 @ A ) )
= ( ( inj_on2776966659131765557et_nat @ F @ A )
& ~ ( member_set_set_nat @ ( F @ A2 ) @ ( image_6725021117256019401et_nat @ F @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_723_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_724_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_725_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_726_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_727_Field__insert,axiom,
! [A2: set_set_nat,B2: set_set_nat,R2: set_Pr3001764113791142201et_nat] :
( ( field_set_set_nat @ ( insert3483733655121172905et_nat @ ( produc9057842353944101649et_nat @ A2 @ B2 ) @ R2 ) )
= ( sup_su4213647025997063966et_nat @ ( insert_set_set_nat @ A2 @ ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) ) @ ( field_set_set_nat @ R2 ) ) ) ).
% Field_insert
thf(fact_728_Field__insert,axiom,
! [A2: set_nat,B2: set_nat,R2: set_Pr5488025237498180813et_nat] :
( ( field_set_nat @ ( insert3810226134351308605et_nat @ ( produc4532415448927165861et_nat @ A2 @ B2 ) @ R2 ) )
= ( sup_sup_set_set_nat @ ( insert_set_nat @ A2 @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) ) @ ( field_set_nat @ R2 ) ) ) ).
% Field_insert
thf(fact_729_Field__insert,axiom,
! [A2: nat,B2: nat,R2: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R2 ) )
= ( sup_sup_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R2 ) ) ) ).
% Field_insert
thf(fact_730_Field__insert,axiom,
! [A2: nat > nat,B2: nat > nat,R2: set_Pr7682762132356531903at_nat] :
( ( field_nat_nat @ ( insert8632360194428346671at_nat @ ( produc5770335208449155351at_nat @ A2 @ B2 ) @ R2 ) )
= ( sup_sup_set_nat_nat @ ( insert_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) ) @ ( field_nat_nat @ R2 ) ) ) ).
% Field_insert
thf(fact_731_mformula_Orel__induct,axiom,
! [R: a > a > $o,X4: monotone_mformula_a,Y2: monotone_mformula_a,Q: monotone_mformula_a > monotone_mformula_a > $o] :
( ( monoto4866550245073096868la_a_a @ R @ X4 @ Y2 )
=> ( ( Q @ monotone_TRUE_a @ monotone_TRUE_a )
=> ( ( Q @ monotone_FALSE_a @ monotone_FALSE_a )
=> ( ! [A32: a,B32: a] :
( ( R @ A32 @ B32 )
=> ( Q @ ( monotone_Var_a @ A32 ) @ ( monotone_Var_a @ B32 ) ) )
=> ( ! [A41: monotone_mformula_a,A42: monotone_mformula_a,B41: monotone_mformula_a,B42: monotone_mformula_a] :
( ( Q @ A41 @ B41 )
=> ( ( Q @ A42 @ B42 )
=> ( Q @ ( monotone_Conj_a @ A41 @ A42 ) @ ( monotone_Conj_a @ B41 @ B42 ) ) ) )
=> ( ! [A51: monotone_mformula_a,A52: monotone_mformula_a,B51: monotone_mformula_a,B52: monotone_mformula_a] :
( ( Q @ A51 @ B51 )
=> ( ( Q @ A52 @ B52 )
=> ( Q @ ( monotone_Disj_a @ A51 @ A52 ) @ ( monotone_Disj_a @ B51 @ B52 ) ) ) )
=> ( Q @ X4 @ Y2 ) ) ) ) ) ) ) ).
% mformula.rel_induct
thf(fact_732_mformula_Orel__cases,axiom,
! [R: a > a > $o,A2: monotone_mformula_a,B2: monotone_mformula_a] :
( ( monoto4866550245073096868la_a_a @ R @ A2 @ B2 )
=> ( ( ( A2 = monotone_TRUE_a )
=> ( B2 != monotone_TRUE_a ) )
=> ( ( ( A2 = monotone_FALSE_a )
=> ( B2 != monotone_FALSE_a ) )
=> ( ! [X3: a] :
( ( A2
= ( monotone_Var_a @ X3 ) )
=> ! [Y4: a] :
( ( B2
= ( monotone_Var_a @ Y4 ) )
=> ~ ( R @ X3 @ Y4 ) ) )
=> ( ! [X1: monotone_mformula_a,X22: monotone_mformula_a] :
( ( A2
= ( monotone_Conj_a @ X1 @ X22 ) )
=> ! [Y1: monotone_mformula_a,Y22: monotone_mformula_a] :
( ( B2
= ( monotone_Conj_a @ Y1 @ Y22 ) )
=> ( ( monoto4866550245073096868la_a_a @ R @ X1 @ Y1 )
=> ~ ( monoto4866550245073096868la_a_a @ R @ X22 @ Y22 ) ) ) )
=> ~ ! [X1a: monotone_mformula_a,X2a: monotone_mformula_a] :
( ( A2
= ( monotone_Disj_a @ X1a @ X2a ) )
=> ! [Y1a: monotone_mformula_a,Y2a: monotone_mformula_a] :
( ( B2
= ( monotone_Disj_a @ Y1a @ Y2a ) )
=> ( ( monoto4866550245073096868la_a_a @ R @ X1a @ Y1a )
=> ~ ( monoto4866550245073096868la_a_a @ R @ X2a @ Y2a ) ) ) ) ) ) ) ) ) ).
% mformula.rel_cases
thf(fact_733_forth__assumptions_OACC__SET_I3_J,axiom,
! [L: nat,P3: nat,K: nat,V3: set_a,Pi: a > set_nat,Phi: monotone_mformula_a,Psi: monotone_mformula_a] :
( ( clique8563529963003110213ions_a @ L @ P3 @ K @ V3 @ 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_734_order__refl,axiom,
! [X4: set_set_nat] : ( ord_le6893508408891458716et_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_735_order__refl,axiom,
! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_736_order__refl,axiom,
! [X4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_737_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_738_order__refl,axiom,
! [X4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_739_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_740_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_741_dual__order_Orefl,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_742_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_743_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_744_subsetI,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A )
=> ( member2946998982187404937et_nat @ X3 @ B ) )
=> ( ord_le572741076514265352et_nat @ A @ B ) ) ).
% subsetI
thf(fact_745_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_746_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_747_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_748_subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( member_set_set_nat @ X3 @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% subsetI
thf(fact_749_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_nat_nat @ X3 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_750_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_751_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_752_subset__antisym,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_753_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_754_DiffI,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ A )
=> ( ~ ( member2946998982187404937et_nat @ C @ B )
=> ( member2946998982187404937et_nat @ C @ ( minus_3113942175840221057et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_755_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_756_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_757_DiffI,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A )
=> ( ~ ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_758_DiffI,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( ~ ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_759_Diff__iff,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( minus_3113942175840221057et_nat @ A @ B ) )
= ( ( member2946998982187404937et_nat @ C @ A )
& ~ ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_760_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ~ ( member_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_761_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_762_Diff__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
& ~ ( member_set_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_763_Diff__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
& ~ ( member_nat_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_764_Diff__idemp,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ B )
= ( minus_2447799839930672331et_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_765_Diff__idemp,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ B )
= ( minus_8121590178497047118at_nat @ A @ B ) ) ).
% Diff_idemp
thf(fact_766_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_767_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_768_subset__empty,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_769_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_770_empty__subsetI,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% empty_subsetI
thf(fact_771_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_772_empty__subsetI,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% empty_subsetI
thf(fact_773_empty__subsetI,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% empty_subsetI
thf(fact_774_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_775_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_776_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_777_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_778_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_779_le__sup__iff,axiom,
! [X4: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X4 @ Y2 ) @ Z )
= ( ( ord_le6893508408891458716et_nat @ X4 @ Z )
& ( ord_le6893508408891458716et_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_780_le__sup__iff,axiom,
! [X4: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X4 @ Y2 ) @ Z )
= ( ( ord_less_eq_set_nat @ X4 @ Z )
& ( ord_less_eq_set_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_781_le__sup__iff,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X4 @ Y2 ) @ Z )
= ( ( ord_le9131159989063066194et_nat @ X4 @ Z )
& ( ord_le9131159989063066194et_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_782_le__sup__iff,axiom,
! [X4: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y2 ) @ Z )
= ( ( ord_less_eq_nat @ X4 @ Z )
& ( ord_less_eq_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_783_le__sup__iff,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X4 @ Y2 ) @ Z )
= ( ( ord_le9059583361652607317at_nat @ X4 @ Z )
& ( ord_le9059583361652607317at_nat @ Y2 @ Z ) ) ) ).
% le_sup_iff
thf(fact_784_insert__subset,axiom,
! [X4: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ ( insert3687027775829606434et_nat @ X4 @ A ) @ B )
= ( ( member2946998982187404937et_nat @ X4 @ B )
& ( ord_le572741076514265352et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_785_insert__subset,axiom,
! [X4: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_786_insert__subset,axiom,
! [X4: set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X4 @ A ) @ B )
= ( ( member_set_nat @ X4 @ B )
& ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_787_insert__subset,axiom,
! [X4: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ A ) @ B )
= ( ( member_nat @ X4 @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_788_insert__subset,axiom,
! [X4: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ X4 @ A ) @ B )
= ( ( member_set_set_nat @ X4 @ B )
& ( ord_le9131159989063066194et_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_789_insert__subset,axiom,
! [X4: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( insert_nat_nat @ X4 @ A ) @ B )
= ( ( member_nat_nat @ X4 @ B )
& ( ord_le9059583361652607317at_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_790_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_791_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_792_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_793_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_794_Diff__empty,axiom,
! [A: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% Diff_empty
thf(fact_795_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_796_Diff__empty,axiom,
! [A: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A @ bot_bo7198184520161983622et_nat )
= A ) ).
% Diff_empty
thf(fact_797_Diff__empty,axiom,
! [A: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ bot_bot_set_nat_nat )
= A ) ).
% Diff_empty
thf(fact_798_empty__Diff,axiom,
! [A: set_set_nat] :
( ( minus_2163939370556025621et_nat @ bot_bot_set_set_nat @ A )
= bot_bot_set_set_nat ) ).
% empty_Diff
thf(fact_799_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_800_empty__Diff,axiom,
! [A: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ bot_bo7198184520161983622et_nat @ A )
= bot_bo7198184520161983622et_nat ) ).
% empty_Diff
thf(fact_801_empty__Diff,axiom,
! [A: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ bot_bot_set_nat_nat @ A )
= bot_bot_set_nat_nat ) ).
% empty_Diff
thf(fact_802_Diff__cancel,axiom,
! [A: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ A )
= bot_bot_set_set_nat ) ).
% Diff_cancel
thf(fact_803_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_804_Diff__cancel,axiom,
! [A: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A @ A )
= bot_bo7198184520161983622et_nat ) ).
% Diff_cancel
thf(fact_805_Diff__cancel,axiom,
! [A: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ A )
= bot_bot_set_nat_nat ) ).
% Diff_cancel
thf(fact_806_Field__empty,axiom,
( ( field_set_set_nat @ bot_bo8862070341723882917et_nat )
= bot_bo7198184520161983622et_nat ) ).
% Field_empty
thf(fact_807_Field__empty,axiom,
( ( field_set_nat @ bot_bo4952753636535259449et_nat )
= bot_bot_set_set_nat ) ).
% Field_empty
thf(fact_808_Field__empty,axiom,
( ( field_nat @ bot_bo2099793752762293965at_nat )
= bot_bot_set_nat ) ).
% Field_empty
thf(fact_809_Field__empty,axiom,
( ( field_nat_nat @ bot_bo1327045763863633707at_nat )
= bot_bot_set_nat_nat ) ).
% Field_empty
thf(fact_810_Diff__insert0,axiom,
! [X4: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ~ ( member2946998982187404937et_nat @ X4 @ A )
=> ( ( minus_3113942175840221057et_nat @ A @ ( insert3687027775829606434et_nat @ X4 @ B ) )
= ( minus_3113942175840221057et_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_811_Diff__insert0,axiom,
! [X4: set_nat,A: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X4 @ A )
=> ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X4 @ B ) )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_812_Diff__insert0,axiom,
! [X4: a,A: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_813_Diff__insert0,axiom,
! [X4: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ X4 @ A )
=> ( ( minus_2447799839930672331et_nat @ A @ ( insert_set_set_nat @ X4 @ B ) )
= ( minus_2447799839930672331et_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_814_Diff__insert0,axiom,
! [X4: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ X4 @ A )
=> ( ( minus_8121590178497047118at_nat @ A @ ( insert_nat_nat @ X4 @ B ) )
= ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_815_insert__Diff1,axiom,
! [X4: set_set_set_nat,B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ B )
=> ( ( minus_3113942175840221057et_nat @ ( insert3687027775829606434et_nat @ X4 @ A ) @ B )
= ( minus_3113942175840221057et_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_816_insert__Diff1,axiom,
! [X4: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ X4 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X4 @ A ) @ B )
= ( minus_2163939370556025621et_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_817_insert__Diff1,axiom,
! [X4: a,B: set_a,A: set_a] :
( ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_818_insert__Diff1,axiom,
! [X4: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ X4 @ B )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X4 @ A ) @ B )
= ( minus_2447799839930672331et_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_819_insert__Diff1,axiom,
! [X4: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( member_nat_nat @ X4 @ B )
=> ( ( minus_8121590178497047118at_nat @ ( insert_nat_nat @ X4 @ A ) @ B )
= ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_820_Un__Diff__cancel2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ B @ A ) @ A )
= ( sup_sup_set_set_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_821_Un__Diff__cancel2,axiom,
! [B: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A ) @ A )
= ( sup_sup_set_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_822_Un__Diff__cancel2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ B @ A ) @ A )
= ( sup_su4213647025997063966et_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_823_Un__Diff__cancel2,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ B @ A ) @ A )
= ( sup_sup_set_nat_nat @ B @ A ) ) ).
% Un_Diff_cancel2
thf(fact_824_Un__Diff__cancel,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B @ A ) )
= ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_825_Un__Diff__cancel,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_826_Un__Diff__cancel,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( minus_2447799839930672331et_nat @ B @ A ) )
= ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_827_Un__Diff__cancel,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B @ A ) )
= ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_Diff_cancel
thf(fact_828_Field__Un,axiom,
! [R2: set_Pr5488025237498180813et_nat,S: set_Pr5488025237498180813et_nat] :
( ( field_set_nat @ ( sup_su4251893418135750817et_nat @ R2 @ S ) )
= ( sup_sup_set_set_nat @ ( field_set_nat @ R2 ) @ ( field_set_nat @ S ) ) ) ).
% Field_Un
thf(fact_829_Field__Un,axiom,
! [R2: set_Pr3001764113791142201et_nat,S: set_Pr3001764113791142201et_nat] :
( ( field_set_set_nat @ ( sup_su933621215571934477et_nat @ R2 @ S ) )
= ( sup_su4213647025997063966et_nat @ ( field_set_set_nat @ R2 ) @ ( field_set_set_nat @ S ) ) ) ).
% Field_Un
thf(fact_830_Field__Un,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( sup_su6327502436637775413at_nat @ R2 @ S ) )
= ( sup_sup_set_nat @ ( field_nat @ R2 ) @ ( field_nat @ S ) ) ) ).
% Field_Un
thf(fact_831_Field__Un,axiom,
! [R2: set_Pr7682762132356531903at_nat,S: set_Pr7682762132356531903at_nat] :
( ( field_nat_nat @ ( sup_su8437843545073016467at_nat @ R2 @ S ) )
= ( sup_sup_set_nat_nat @ ( field_nat_nat @ R2 ) @ ( field_nat_nat @ S ) ) ) ).
% Field_Un
thf(fact_832_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_833_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_834_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_835_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_836_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_837_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_838_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_839_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_840_Diff__eq__empty__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_841_Diff__eq__empty__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_842_Diff__eq__empty__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( minus_2447799839930672331et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
= ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_843_Diff__eq__empty__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_844_insert__Diff__single,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( insert_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
= ( insert_set_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_845_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_846_insert__Diff__single,axiom,
! [A2: set_set_nat,A: set_set_set_nat] :
( ( insert_set_set_nat @ A2 @ ( minus_2447799839930672331et_nat @ A @ ( insert_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ) )
= ( insert_set_set_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_847_insert__Diff__single,axiom,
! [A2: nat > nat,A: set_nat_nat] :
( ( insert_nat_nat @ A2 @ ( minus_8121590178497047118at_nat @ A @ ( insert_nat_nat @ A2 @ bot_bot_set_nat_nat ) ) )
= ( insert_nat_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_848_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_849_le__cases3,axiom,
! [X4: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_850_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_851_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_852_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_set_nat,Z2: set_set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [X: set_set_set_nat,Y3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y3 )
& ( ord_le9131159989063066194et_nat @ Y3 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_853_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_854_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_855_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_856_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_857_ord__eq__le__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( A2 = B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_858_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_859_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_860_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_861_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_862_ord__le__eq__trans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_863_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_864_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_865_order__antisym,axiom,
! [X4: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_866_order__antisym,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_867_order__antisym,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_868_order__antisym,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_869_order__antisym,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% order_antisym
thf(fact_870_order_Otrans,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_871_order_Otrans,axiom,
! [A2: set_nat,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_872_order_Otrans,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_873_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_874_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_875_order__trans,axiom,
! [X4: set_set_nat,Y2: set_set_nat,Z: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ Z )
=> ( ord_le6893508408891458716et_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_876_order__trans,axiom,
! [X4: set_nat,Y2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z )
=> ( ord_less_eq_set_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_877_order__trans,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat,Z: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ Z )
=> ( ord_le9131159989063066194et_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_878_order__trans,axiom,
! [X4: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_879_order__trans,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat,Z: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z )
=> ( ord_le9059583361652607317at_nat @ X4 @ Z ) ) ) ).
% order_trans
thf(fact_880_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_881_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_882_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_883_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_set_set_nat,Z2: set_set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ A3 )
& ( ord_le9131159989063066194et_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_884_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_885_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_886_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_887_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_888_dual__order_Oantisym,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_889_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_890_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_891_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_892_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_893_dual__order_Otrans,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C @ B2 )
=> ( ord_le9131159989063066194et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_894_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_895_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_896_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_897_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_898_antisym,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_899_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_900_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_901_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_902_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_903_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_set_set_nat,Z2: set_set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_904_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_905_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_906_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_907_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_908_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_909_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_910_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_911_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 )
=> ( ! [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_912_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_913_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 )
=> ( ! [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_914_order__subst1,axiom,
! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_915_order__subst1,axiom,
! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ! [X3: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_916_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_917_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_918_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_919_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 )
=> ( ! [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_920_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_921_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_922_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 )
=> ( ! [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_923_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_924_order__subst2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_925_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_926_order__eq__refl,axiom,
! [X4: set_set_nat,Y2: set_set_nat] :
( ( X4 = Y2 )
=> ( ord_le6893508408891458716et_nat @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_927_order__eq__refl,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( X4 = Y2 )
=> ( ord_less_eq_set_nat @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_928_order__eq__refl,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( X4 = Y2 )
=> ( ord_le9131159989063066194et_nat @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_929_order__eq__refl,axiom,
! [X4: nat,Y2: nat] :
( ( X4 = Y2 )
=> ( ord_less_eq_nat @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_930_order__eq__refl,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( X4 = Y2 )
=> ( ord_le9059583361652607317at_nat @ X4 @ Y2 ) ) ).
% order_eq_refl
thf(fact_931_linorder__linear,axiom,
! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X4 ) ) ).
% linorder_linear
thf(fact_932_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_933_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_934_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_935_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 )
=> ( ! [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_936_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_937_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_938_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 )
=> ( ! [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_939_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_940_ord__eq__le__subst,axiom,
! [A2: nat,F: set_set_set_nat > nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ! [X3: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_941_ord__eq__le__subst,axiom,
! [A2: set_set_set_nat,F: nat > set_set_set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_942_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_943_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_944_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_945_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 )
=> ( ! [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_946_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_947_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 )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_948_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 )
=> ( ! [X3: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_949_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 )
=> ( ! [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_950_ord__le__eq__subst,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,F: set_set_set_nat > nat,C: nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_951_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_set_set_nat,C: set_set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_952_linorder__le__cases,axiom,
! [X4: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X4 ) ) ).
% linorder_le_cases
thf(fact_953_order__antisym__conv,axiom,
! [Y2: set_set_nat,X4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X4 )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_954_order__antisym__conv,axiom,
! [Y2: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_955_order__antisym__conv,axiom,
! [Y2: set_set_set_nat,X4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X4 )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_956_order__antisym__conv,axiom,
! [Y2: nat,X4: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_957_order__antisym__conv,axiom,
! [Y2: set_nat_nat,X4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X4 )
=> ( ( ord_le9059583361652607317at_nat @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_958_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_959_DiffE,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( minus_3113942175840221057et_nat @ A @ B ) )
=> ~ ( ( member2946998982187404937et_nat @ C @ A )
=> ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_960_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_961_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_962_DiffE,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
=> ~ ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_963_DiffE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
=> ~ ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_964_DiffD1,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( minus_3113942175840221057et_nat @ A @ B ) )
=> ( member2946998982187404937et_nat @ C @ A ) ) ).
% DiffD1
thf(fact_965_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ( member_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_966_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_967_DiffD1,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
=> ( member_set_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_968_DiffD1,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
=> ( member_nat_nat @ C @ A ) ) ).
% DiffD1
thf(fact_969_DiffD2,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( minus_3113942175840221057et_nat @ A @ B ) )
=> ~ ( member2946998982187404937et_nat @ C @ B ) ) ).
% DiffD2
thf(fact_970_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B ) )
=> ~ ( member_set_nat @ C @ B ) ) ).
% DiffD2
thf(fact_971_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_972_DiffD2,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A @ B ) )
=> ~ ( member_set_set_nat @ C @ B ) ) ).
% DiffD2
thf(fact_973_DiffD2,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A @ B ) )
=> ~ ( member_nat_nat @ C @ B ) ) ).
% DiffD2
thf(fact_974_in__mono,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,X4: set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( member2946998982187404937et_nat @ X4 @ A )
=> ( member2946998982187404937et_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_975_in__mono,axiom,
! [A: set_a,B: set_a,X4: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_976_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X4: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_977_in__mono,axiom,
! [A: set_nat,B: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_978_in__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X4: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ X4 @ A )
=> ( member_set_set_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_979_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X4: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ X4 @ A )
=> ( member_nat_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_980_subsetD,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,C: set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( member2946998982187404937et_nat @ C @ A )
=> ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_981_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_982_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_983_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_984_subsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_985_subsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_986_Diff__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,D3: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ D3 @ B )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_987_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D3 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( minus_minus_set_nat @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_988_Diff__mono,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,D3: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ D3 @ B )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_989_Diff__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,D3: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ D3 @ B )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_990_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_991_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_992_equalityE,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ~ ( ord_le9131159989063066194et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_1001_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_1002_equalityD1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_1003_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_1004_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_1005_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_1006_equalityD2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ( ord_le9131159989063066194et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_1007_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_1008_subset__iff,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A4: set_set_set_set_nat,B4: set_set_set_set_nat] :
! [T: set_set_set_nat] :
( ( member2946998982187404937et_nat @ T @ A4 )
=> ( member2946998982187404937et_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1009_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1010_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A4 )
=> ( member_set_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1011_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A4 )
=> ( member_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1012_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
! [T: set_set_nat] :
( ( member_set_set_nat @ T @ A4 )
=> ( member_set_set_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1013_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [T: nat > nat] :
( ( member_nat_nat @ T @ A4 )
=> ( member_nat_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1014_Diff__subset,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1015_Diff__subset,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1016_Diff__subset,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1017_Diff__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_1018_double__diff,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ( minus_2163939370556025621et_nat @ B @ ( minus_2163939370556025621et_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1019_double__diff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1020_double__diff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ( minus_2447799839930672331et_nat @ B @ ( minus_2447799839930672331et_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1021_double__diff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ( minus_8121590178497047118at_nat @ B @ ( minus_8121590178497047118at_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_1022_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_1023_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_1024_subset__refl,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ A ) ).
% subset_refl
thf(fact_1025_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_1026_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1027_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1028_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1029_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1030_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_1031_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_1032_subset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ord_le9131159989063066194et_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_1033_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_1034_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_1035_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_1036_set__eq__subset,axiom,
( ( ^ [Y5: set_set_set_nat,Z2: set_set_set_nat] : ( Y5 = Z2 ) )
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A4 @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_1037_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_1038_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_1039_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_1040_Collect__mono__iff,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) )
= ( ! [X: set_set_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_1041_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_1042_image__diff__subset,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,B: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_5842784325960735177et_nat @ F @ A ) @ ( image_5842784325960735177et_nat @ F @ B ) ) @ ( image_5842784325960735177et_nat @ F @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1043_image__diff__subset,axiom,
! [F: ( nat > nat ) > set_nat,A: set_nat_nat,B: set_nat_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_7432509271690132940et_nat @ F @ A ) @ ( image_7432509271690132940et_nat @ F @ B ) ) @ ( image_7432509271690132940et_nat @ F @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1044_image__diff__subset,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat,B: set_set_set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_1454916318497077779at_nat @ F @ A ) @ ( image_1454916318497077779at_nat @ F @ B ) ) @ ( image_1454916318497077779at_nat @ F @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1045_image__diff__subset,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,B: set_nat_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat_nat @ F @ A ) @ ( image_nat_nat_nat @ F @ B ) ) @ ( image_nat_nat_nat @ F @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1046_image__diff__subset,axiom,
! [F: set_set_nat > set_set_nat,A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_7884819252390400639et_nat @ F @ A ) @ ( image_7884819252390400639et_nat @ F @ B ) ) @ ( image_7884819252390400639et_nat @ F @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1047_image__diff__subset,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,B: set_nat_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_9186907679027735170et_nat @ F @ A ) @ ( image_9186907679027735170et_nat @ F @ B ) ) @ ( image_9186907679027735170et_nat @ F @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1048_image__diff__subset,axiom,
! [F: set_set_nat > nat > nat,A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_8441894408526374658at_nat @ F @ A ) @ ( image_8441894408526374658at_nat @ F @ B ) ) @ ( image_8441894408526374658at_nat @ F @ ( minus_2447799839930672331et_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1049_image__diff__subset,axiom,
! [F: ( nat > nat ) > nat > nat,A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B ) ) @ ( image_3205354838064109189at_nat @ F @ ( minus_8121590178497047118at_nat @ A @ B ) ) ) ).
% image_diff_subset
thf(fact_1050_diff__shunt__var,axiom,
! [X4: set_set_nat,Y2: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ X4 @ Y2 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ X4 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1051_diff__shunt__var,axiom,
! [X4: set_nat,Y2: set_nat] :
( ( ( minus_minus_set_nat @ X4 @ Y2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X4 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1052_diff__shunt__var,axiom,
! [X4: set_set_set_nat,Y2: set_set_set_nat] :
( ( ( minus_2447799839930672331et_nat @ X4 @ Y2 )
= bot_bo7198184520161983622et_nat )
= ( ord_le9131159989063066194et_nat @ X4 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1053_diff__shunt__var,axiom,
! [X4: set_nat_nat,Y2: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ X4 @ Y2 )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ X4 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_1054_Diff__subset__conv,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ C2 )
= ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1055_Diff__subset__conv,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
= ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1056_Diff__subset__conv,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ C2 )
= ( ord_le9131159989063066194et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1057_Diff__subset__conv,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ C2 )
= ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1058_Diff__partition,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( sup_sup_set_set_nat @ A @ ( minus_2163939370556025621et_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1059_Diff__partition,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1060_Diff__partition,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( sup_su4213647025997063966et_nat @ A @ ( minus_2447799839930672331et_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1061_Diff__partition,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( sup_sup_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B @ A ) )
= B ) ) ).
% Diff_partition
thf(fact_1062_subset__Diff__insert,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,X4: set_set_set_nat,C2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ ( minus_3113942175840221057et_nat @ B @ ( insert3687027775829606434et_nat @ X4 @ C2 ) ) )
= ( ( ord_le572741076514265352et_nat @ A @ ( minus_3113942175840221057et_nat @ B @ C2 ) )
& ~ ( member2946998982187404937et_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1063_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X4: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X4 @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1064_subset__Diff__insert,axiom,
! [A: set_set_nat,B: set_set_nat,X4: set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B @ ( insert_set_nat @ X4 @ C2 ) ) )
= ( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B @ C2 ) )
& ~ ( member_set_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1065_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X4: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X4 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1066_subset__Diff__insert,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X4: set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( minus_2447799839930672331et_nat @ B @ ( insert_set_set_nat @ X4 @ C2 ) ) )
= ( ( ord_le9131159989063066194et_nat @ A @ ( minus_2447799839930672331et_nat @ B @ C2 ) )
& ~ ( member_set_set_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1067_subset__Diff__insert,axiom,
! [A: set_nat_nat,B: set_nat_nat,X4: nat > nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( minus_8121590178497047118at_nat @ B @ ( insert_nat_nat @ X4 @ C2 ) ) )
= ( ( ord_le9059583361652607317at_nat @ A @ ( minus_8121590178497047118at_nat @ B @ C2 ) )
& ~ ( member_nat_nat @ X4 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1068_subset__insert__iff,axiom,
! [A: set_set_set_set_nat,X4: set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ ( insert3687027775829606434et_nat @ X4 @ B ) )
= ( ( ( member2946998982187404937et_nat @ X4 @ A )
=> ( ord_le572741076514265352et_nat @ ( minus_3113942175840221057et_nat @ A @ ( insert3687027775829606434et_nat @ X4 @ bot_bo193956671110832956et_nat ) ) @ B ) )
& ( ~ ( member2946998982187404937et_nat @ X4 @ A )
=> ( ord_le572741076514265352et_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1069_subset__insert__iff,axiom,
! [A: set_a,X4: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X4 @ B ) )
= ( ( ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X4 @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1070_subset__insert__iff,axiom,
! [A: set_set_nat,X4: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X4 @ B ) )
= ( ( ( member_set_nat @ X4 @ A )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ B ) )
& ( ~ ( member_set_nat @ X4 @ A )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1071_subset__insert__iff,axiom,
! [A: set_nat,X4: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B ) )
= ( ( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1072_subset__insert__iff,axiom,
! [A: set_set_set_nat,X4: set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ ( insert_set_set_nat @ X4 @ B ) )
= ( ( ( member_set_set_nat @ X4 @ A )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) @ B ) )
& ( ~ ( member_set_set_nat @ X4 @ A )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1073_subset__insert__iff,axiom,
! [A: set_nat_nat,X4: nat > nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ ( insert_nat_nat @ X4 @ B ) )
= ( ( ( member_nat_nat @ X4 @ A )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) @ B ) )
& ( ~ ( member_nat_nat @ X4 @ A )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1074_Diff__single__insert,axiom,
! [A: set_set_nat,X4: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) @ B )
=> ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1075_Diff__single__insert,axiom,
! [A: set_nat,X4: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1076_Diff__single__insert,axiom,
! [A: set_set_set_nat,X4: set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) @ B )
=> ( ord_le9131159989063066194et_nat @ A @ ( insert_set_set_nat @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1077_Diff__single__insert,axiom,
! [A: set_nat_nat,X4: nat > nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) @ B )
=> ( ord_le9059583361652607317at_nat @ A @ ( insert_nat_nat @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_1078_inj__on__image__set__diff,axiom,
! [F: a > set_nat,C2: set_a,A: set_a,B: set_a] :
( ( inj_on_a_set_nat @ F @ C2 )
=> ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B ) @ C2 )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ( image_a_set_nat @ F @ ( minus_minus_set_a @ A @ B ) )
= ( minus_2163939370556025621et_nat @ ( image_a_set_nat @ F @ A ) @ ( image_a_set_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1079_inj__on__image__set__diff,axiom,
! [F: set_nat > set_set_nat,C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( inj_on2776966659131765557et_nat @ F @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ( image_6725021117256019401et_nat @ F @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( minus_2447799839930672331et_nat @ ( image_6725021117256019401et_nat @ F @ A ) @ ( image_6725021117256019401et_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1080_inj__on__image__set__diff,axiom,
! [F: set_nat > nat > nat,C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( inj_on4369475957891034808at_nat @ F @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ( image_8569768528772619084at_nat @ F @ ( minus_2163939370556025621et_nat @ A @ B ) )
= ( minus_8121590178497047118at_nat @ ( image_8569768528772619084at_nat @ F @ A ) @ ( image_8569768528772619084at_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1081_inj__on__image__set__diff,axiom,
! [F: nat > set_set_nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on8105003582846801791et_nat @ F @ C2 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( image_2194112158459175443et_nat @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_2447799839930672331et_nat @ ( image_2194112158459175443et_nat @ F @ A ) @ ( image_2194112158459175443et_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1082_inj__on__image__set__diff,axiom,
! [F: nat > nat > nat,C2: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat_nat @ F @ C2 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B ) @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ( image_nat_nat_nat2 @ F @ ( minus_minus_set_nat @ A @ B ) )
= ( minus_8121590178497047118at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ ( image_nat_nat_nat2 @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1083_inj__on__image__set__diff,axiom,
! [F: set_set_nat > set_nat,C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( inj_on1894729867836481333et_nat @ F @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ( image_5842784325960735177et_nat @ F @ ( minus_2447799839930672331et_nat @ A @ B ) )
= ( minus_2163939370556025621et_nat @ ( image_5842784325960735177et_nat @ F @ A ) @ ( image_5842784325960735177et_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1084_inj__on__image__set__diff,axiom,
! [F: set_set_nat > set_set_nat,C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( inj_on2040386338155636715et_nat @ F @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ( image_7884819252390400639et_nat @ F @ ( minus_2447799839930672331et_nat @ A @ B ) )
= ( minus_2447799839930672331et_nat @ ( image_7884819252390400639et_nat @ F @ A ) @ ( image_7884819252390400639et_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1085_inj__on__image__set__diff,axiom,
! [F: set_set_nat > nat > nat,C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( inj_on3419524245016971886at_nat @ F @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ( image_8441894408526374658at_nat @ F @ ( minus_2447799839930672331et_nat @ A @ B ) )
= ( minus_8121590178497047118at_nat @ ( image_8441894408526374658at_nat @ F @ A ) @ ( image_8441894408526374658at_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1086_inj__on__image__set__diff,axiom,
! [F: ( nat > nat ) > set_set_nat,C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( inj_on4164537515518332398et_nat @ F @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ( image_9186907679027735170et_nat @ F @ ( minus_8121590178497047118at_nat @ A @ B ) )
= ( minus_2447799839930672331et_nat @ ( image_9186907679027735170et_nat @ F @ A ) @ ( image_9186907679027735170et_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1087_inj__on__image__set__diff,axiom,
! [F: ( nat > nat ) > nat > nat,C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( inj_on2461717442902640625at_nat @ F @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ( image_3205354838064109189at_nat @ F @ ( minus_8121590178497047118at_nat @ A @ B ) )
= ( minus_8121590178497047118at_nat @ ( image_3205354838064109189at_nat @ F @ A ) @ ( image_3205354838064109189at_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1088_mformula_Orel__intros_I4_J,axiom,
! [R: a > a > $o,X41: monotone_mformula_a,Y41: monotone_mformula_a,X42: monotone_mformula_a,Y42: monotone_mformula_a] :
( ( monoto4866550245073096868la_a_a @ R @ X41 @ Y41 )
=> ( ( monoto4866550245073096868la_a_a @ R @ X42 @ Y42 )
=> ( monoto4866550245073096868la_a_a @ R @ ( monotone_Conj_a @ X41 @ X42 ) @ ( monotone_Conj_a @ Y41 @ Y42 ) ) ) ) ).
% mformula.rel_intros(4)
thf(fact_1089_mformula_Orel__intros_I5_J,axiom,
! [R: a > a > $o,X51: monotone_mformula_a,Y51: monotone_mformula_a,X52: monotone_mformula_a,Y52: monotone_mformula_a] :
( ( monoto4866550245073096868la_a_a @ R @ X51 @ Y51 )
=> ( ( monoto4866550245073096868la_a_a @ R @ X52 @ Y52 )
=> ( monoto4866550245073096868la_a_a @ R @ ( monotone_Disj_a @ X51 @ X52 ) @ ( monotone_Disj_a @ Y51 @ Y52 ) ) ) ) ).
% mformula.rel_intros(5)
thf(fact_1090_mformula_Orel__inject_I4_J,axiom,
! [R: a > a > $o,X41: monotone_mformula_a,X42: monotone_mformula_a,Y41: monotone_mformula_a,Y42: monotone_mformula_a] :
( ( monoto4866550245073096868la_a_a @ R @ ( monotone_Conj_a @ X41 @ X42 ) @ ( monotone_Conj_a @ Y41 @ Y42 ) )
= ( ( monoto4866550245073096868la_a_a @ R @ X41 @ Y41 )
& ( monoto4866550245073096868la_a_a @ R @ X42 @ Y42 ) ) ) ).
% mformula.rel_inject(4)
thf(fact_1091_mformula_Orel__inject_I5_J,axiom,
! [R: a > a > $o,X51: monotone_mformula_a,X52: monotone_mformula_a,Y51: monotone_mformula_a,Y52: monotone_mformula_a] :
( ( monoto4866550245073096868la_a_a @ R @ ( monotone_Disj_a @ X51 @ X52 ) @ ( monotone_Disj_a @ Y51 @ Y52 ) )
= ( ( monoto4866550245073096868la_a_a @ R @ X51 @ Y51 )
& ( monoto4866550245073096868la_a_a @ R @ X52 @ Y52 ) ) ) ).
% mformula.rel_inject(5)
thf(fact_1092_mformula_Orel__intros_I3_J,axiom,
! [R: a > a > $o,X32: a,Y32: a] :
( ( R @ X32 @ Y32 )
=> ( monoto4866550245073096868la_a_a @ R @ ( monotone_Var_a @ X32 ) @ ( monotone_Var_a @ Y32 ) ) ) ).
% mformula.rel_intros(3)
thf(fact_1093_mformula_Orel__inject_I3_J,axiom,
! [R: a > a > $o,X32: a,Y32: a] :
( ( monoto4866550245073096868la_a_a @ R @ ( monotone_Var_a @ X32 ) @ ( monotone_Var_a @ Y32 ) )
= ( R @ X32 @ Y32 ) ) ).
% mformula.rel_inject(3)
thf(fact_1094_mformula_Octr__transfer_I2_J,axiom,
! [R: a > a > $o] : ( monoto4866550245073096868la_a_a @ R @ monotone_FALSE_a @ monotone_FALSE_a ) ).
% mformula.ctr_transfer(2)
thf(fact_1095_mformula_Octr__transfer_I1_J,axiom,
! [R: a > a > $o] : ( monoto4866550245073096868la_a_a @ R @ monotone_TRUE_a @ monotone_TRUE_a ) ).
% mformula.ctr_transfer(1)
thf(fact_1096_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1097_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1098_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1099_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1100_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1101_bot_Oextremum__unique,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1102_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1103_bot_Oextremum__unique,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_unique
thf(fact_1104_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1105_bot_Oextremum__unique,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_1106_bot_Oextremum,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% bot.extremum
thf(fact_1107_first__assumptions_OACC__empty,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.ACC_empty
thf(fact_1108_first__assumptions_OACC__union,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K @ X2 ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_union
thf(fact_1109_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_1110_v__gs__mono,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_mono
thf(fact_1111_second__assumptions_Odeviate__pos__cup_Ocong,axiom,
clique3314026705536850673os_cup = clique3314026705536850673os_cup ).
% second_assumptions.deviate_pos_cup.cong
thf(fact_1112_first__assumptions_Ov__mono,axiom,
! [L: nat,P3: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_1113_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_1114_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X7: set_set_set_nat,G3: set_set_nat] :
? [X: set_set_nat] :
( ( member_set_set_nat @ X @ X7 )
& ( ord_le6893508408891458716et_nat @ X @ G3 ) ) ) ) ).
% accepts_def
thf(fact_1115_acceptsI,axiom,
! [D3: set_set_nat,G: set_set_nat,X2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D3 @ G )
=> ( ( member_set_set_nat @ D3 @ X2 )
=> ( clique3686358387679108662ccepts @ X2 @ G ) ) ) ).
% acceptsI
thf(fact_1116_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3686358387679108662ccepts @ X2 @ G )
= ( ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ X2 )
& ( ord_le6893508408891458716et_nat @ X @ G ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_1117_first__assumptions_OacceptsI,axiom,
! [L: nat,P3: nat,K: nat,D3: set_set_nat,G: set_set_nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le6893508408891458716et_nat @ D3 @ G )
=> ( ( member_set_set_nat @ D3 @ X2 )
=> ( clique3686358387679108662ccepts @ X2 @ G ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_1118_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_1119_first__assumptions_Ojoinl__join,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K @ X2 @ Y ) @ ( clique5469973757772500719t_odot @ X2 @ Y ) ) ) ).
% first_assumptions.joinl_join
thf(fact_1120_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_1121_first__assumptions_OACC__odot,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X2 @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K @ X2 ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_odot
thf(fact_1122_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_1123_ACC__union,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X2 ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ).
% ACC_union
thf(fact_1124_ACC__odot,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X2 @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X2 ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ).
% ACC_odot
thf(fact_1125_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_1126_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_1127_ACC__SET_I2_J,axiom,
( ( clique3210737319928189260st_ACC @ k @ ( clique6509092761774629891_SET_a @ pi @ monotone_FALSE_a ) )
= bot_bo7198184520161983622et_nat ) ).
% ACC_SET(2)
thf(fact_1128_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_1129_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_1130_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_1131_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_1132_second__assumptions_Odeviate__pos__cup__def,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( clique3314026705536850673os_cup @ L @ P3 @ K @ X2 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P3 @ K @ X2 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_pos_cup_def
thf(fact_1133_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_1134_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_1135_second__assumptions_Osqcup,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ L @ P3 @ K @ X2 @ Y ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ) ).
% second_assumptions.sqcup
thf(fact_1136_second__assumptions_Osqcap,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118206219037_sqcap @ L @ P3 @ K @ X2 @ Y ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ) ).
% second_assumptions.sqcap
thf(fact_1137_second__assumptions_Ov__sameprod__subset,axiom,
! [L: nat,P3: nat,K: nat,Vs: set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ) ).
% second_assumptions.v_sameprod_subset
thf(fact_1138_first__assumptions_Oempty__CLIQUE,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.empty_CLIQUE
thf(fact_1139_second__assumptions_Odeviate__pos__cup,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( ( clique3314026705536850673os_cup @ L @ P3 @ K @ X2 @ Y )
= bot_bo7198184520161983622et_nat ) ) ) ) ).
% second_assumptions.deviate_pos_cup
thf(fact_1140_first__assumptions_OPOS__sub__CLIQUE,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_sub_CLIQUE
thf(fact_1141_second__assumptions_Osqcup__sub,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P3 @ K @ X2 @ Y ) ) ) ) ) ) ).
% second_assumptions.sqcup_sub
thf(fact_1142_second__assumptions_Odeviate__pos__cap__def,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( clique3314026705535538693os_cap @ L @ P3 @ K @ X2 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X2 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118206219037_sqcap @ L @ P3 @ K @ X2 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_pos_cap_def
thf(fact_1143_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_1144_first__assumptions_OCLIQUE__NEG,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K ) @ ( clique3210737375870294875st_NEG @ K ) )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.CLIQUE_NEG
thf(fact_1145_APR_Opelims,axiom,
! [X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ X4 )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ X4 )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( ( Y2
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X3 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ ( monotone_Var_a @ X3 ) ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( 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] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( ( Y2
= ( 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 ) ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( ( Y2 = undefi6751788150640612746et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique5870032674357670943_rel_a @ monotone_TRUE_a ) ) ) ) ) ) ) ) ) ).
% APR.pelims
thf(fact_1146_forth__assumptions__axioms,axiom,
clique8563529963003110213ions_a @ l @ p @ k @ v @ pi ).
% forth_assumptions_axioms
thf(fact_1147_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_1148_second__assumptions__axioms,axiom,
assump2881078719466019805ptions @ l @ p @ k ).
% second_assumptions_axioms
thf(fact_1149_joinl__join,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X2 @ Y ) @ ( clique5469973757772500719t_odot @ X2 @ Y ) ) ).
% joinl_join
thf(fact_1150_sqcup,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X2 @ Y ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% sqcup
thf(fact_1151_sqcap,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X2 @ Y ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% sqcap
thf(fact_1152_deviate__pos__cup,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( clique3314026705536850673os_cup @ l @ p @ k @ X2 @ Y )
= bot_bo7198184520161983622et_nat ) ) ) ).
% deviate_pos_cup
thf(fact_1153_L0,axiom,
ord_less_eq_nat @ assumptions_and_L0 @ l ).
% L0
thf(fact_1154_L0_H,axiom,
ord_less_eq_nat @ assumptions_and_L02 @ l ).
% L0'
thf(fact_1155_approx__pos_Osimps_I5_J,axiom,
! [V2: monotone_mformula_a,Va2: monotone_mformula_a] :
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ V2 @ Va2 ) )
= bot_bo7198184520161983622et_nat ) ).
% approx_pos.simps(5)
thf(fact_1156_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_1157_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_1158_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_1159_APR_Osimps_I1_J,axiom,
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% APR.simps(1)
thf(fact_1160_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_1161_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_1162_approx__pos_Osimps_I1_J,axiom,
! [Phi5: monotone_mformula_a,Psi5: monotone_mformula_a] :
( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi5 @ Psi5 ) )
= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi5 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi5 ) ) ) ).
% approx_pos.simps(1)
thf(fact_1163_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_1164_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_1165_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_1166_sqcup__sub,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X2 @ Y ) ) ) ) ) ).
% sqcup_sub
thf(fact_1167_deviate__pos__cap__def,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705535538693os_cap @ l @ p @ k @ X2 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X2 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X2 @ Y ) ) ) ) ).
% deviate_pos_cap_def
thf(fact_1168_APR_Osimps_I2_J,axiom,
! [X4: a] :
( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X4 ) )
= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) ).
% APR.simps(2)
thf(fact_1169_deviate__pos__cup__def,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705536850673os_cup @ l @ p @ k @ X2 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X2 @ Y ) ) ) ) ).
% deviate_pos_cup_def
thf(fact_1170_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_1171_approx__pos_Oelims,axiom,
! [X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X4 )
= Y2 )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( Y2
!= ( clique3314026705535538693os_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) )
=> ( ( ( X4 = monotone_TRUE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ( ? [V: a] :
( X4
= ( monotone_Var_a @ V ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ~ ( ? [V: monotone_mformula_a,Va: monotone_mformula_a] :
( X4
= ( monotone_Disj_a @ V @ Va ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) ) ) ) ) ) ) ).
% approx_pos.elims
thf(fact_1172_approx__pos_Opelims,axiom,
! [X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ X4 )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ X4 )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ( Y2
= ( 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 ) ) ) )
=> ( ( ( X4 = monotone_TRUE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_TRUE_a ) ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ monotone_FALSE_a ) ) )
=> ( ! [V: a] :
( ( X4
= ( monotone_Var_a @ V ) )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Var_a @ V ) ) ) )
=> ~ ! [V: monotone_mformula_a,Va: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ V @ Va ) )
=> ( ( Y2 = bot_bo7198184520161983622et_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique4465983624924118198_rel_a @ ( monotone_Disj_a @ V @ Va ) ) ) ) ) ) ) ) ) ) ).
% approx_pos.pelims
thf(fact_1173_APR_Oelims,axiom,
! [X4: monotone_mformula_a,Y2: set_set_set_nat] :
( ( ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ X4 )
= Y2 )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( Y2 != bot_bo7198184520161983622et_nat ) )
=> ( ! [X3: a] :
( ( X4
= ( monotone_Var_a @ X3 ) )
=> ( Y2
!= ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X3 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) ) )
=> ( ! [Phi3: monotone_mformula_a,Psi3: monotone_mformula_a] :
( ( X4
= ( monotone_Disj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( 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] :
( ( X4
= ( monotone_Conj_a @ Phi3 @ Psi3 ) )
=> ( Y2
!= ( clique2586627118206219037_sqcap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi3 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi3 ) ) ) )
=> ~ ( ( X4 = monotone_TRUE_a )
=> ( Y2 != undefi6751788150640612746et_nat ) ) ) ) ) ) ) ).
% APR.elims
thf(fact_1174_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_1175_no__deviation_I1_J,axiom,
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ monotone_FALSE_a )
= bot_bo7198184520161983622et_nat ) ).
% no_deviation(1)
thf(fact_1176_no__deviation_I3_J,axiom,
! [X4: a] :
( ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X4 ) )
= bot_bo7198184520161983622et_nat ) ).
% no_deviation(3)
thf(fact_1177__092_060pi_062__singleton_I2_J,axiom,
! [X4: a] :
( ( member_a @ X4 @ v )
=> ( member2946998982187404937et_nat @ ( insert_set_set_nat @ ( insert_set_nat @ ( pi @ X4 ) @ bot_bot_set_set_nat ) @ bot_bo7198184520161983622et_nat ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ).
% \<pi>_singleton(2)
thf(fact_1178_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_1179_first__assumptions_OPOS__CLIQUE,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_CLIQUE
thf(fact_1180_sqcap__def,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118206219037_sqcap @ l @ p @ k @ X2 @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X2 @ Y ) ) ) ).
% sqcap_def
thf(fact_1181_PLU__joinl,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X2 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% PLU_joinl
thf(fact_1182_sqcup__def,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118207531017_sqcup @ l @ p @ k @ X2 @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ) ).
% sqcup_def
thf(fact_1183_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_1184_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_1185_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_1186_PLU__union,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% PLU_union
thf(fact_1187_second__assumptions_OPLU_Ocong,axiom,
clique2699557479641037314nd_PLU = clique2699557479641037314nd_PLU ).
% second_assumptions.PLU.cong
thf(fact_1188_second__assumptions_OPLU__union,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P3 @ K @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ) ).
% second_assumptions.PLU_union
thf(fact_1189_second__assumptions_Osqcup__def,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( clique2586627118207531017_sqcup @ L @ P3 @ K @ X2 @ Y )
= ( clique2699557479641037314nd_PLU @ L @ P3 @ K @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ) ) ).
% second_assumptions.sqcup_def
thf(fact_1190_second__assumptions_OPLU__joinl,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( member2946998982187404937et_nat @ X2 @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P3 @ K @ ( clique7966186356931407165_odotl @ L @ K @ X2 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ L @ P3 @ K ) ) ) ) ) ).
% second_assumptions.PLU_joinl
thf(fact_1191_second__assumptions_Osqcap__def,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( clique2586627118206219037_sqcap @ L @ P3 @ K @ X2 @ Y )
= ( clique2699557479641037314nd_PLU @ L @ P3 @ K @ ( clique7966186356931407165_odotl @ L @ K @ X2 @ Y ) ) ) ) ).
% second_assumptions.sqcap_def
thf(fact_1192_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_1193_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_1194_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_1195_approx__neg_Osimps_I2_J,axiom,
! [Phi5: monotone_mformula_a,Psi5: monotone_mformula_a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Disj_a @ Phi5 @ Psi5 ) )
= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi5 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi5 ) ) ) ).
% approx_neg.simps(2)
thf(fact_1196_approx__neg_Osimps_I1_J,axiom,
! [Phi5: monotone_mformula_a,Psi5: monotone_mformula_a] :
( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ ( monotone_Conj_a @ Phi5 @ Psi5 ) )
= ( clique1591571987438064265eg_cap @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi5 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi5 ) ) ) ).
% approx_neg.simps(1)
thf(fact_1197_approx__neg_Oelims,axiom,
! [X4: monotone_mformula_a,Y2: set_nat_nat] :
( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X4 )
= Y2 )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( Y2
!= ( 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] :
( ( X4
= ( monotone_Disj_a @ Phi2 @ Psi2 ) )
=> ( Y2
!= ( clique1591571987439376245eg_cup @ l @ p @ k @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Phi2 ) @ ( clique3873310923663319714_APR_a @ l @ p @ k @ pi @ Psi2 ) ) ) )
=> ( ( ( X4 = monotone_TRUE_a )
=> ( Y2 != bot_bot_set_nat_nat ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( Y2 != bot_bot_set_nat_nat ) )
=> ~ ( ? [V: a] :
( X4
= ( monotone_Var_a @ V ) )
=> ( Y2 != bot_bot_set_nat_nat ) ) ) ) ) ) ) ).
% approx_neg.elims
thf(fact_1198_second__assumptions_Odeviate__neg__cup_Ocong,axiom,
clique1591571987439376245eg_cup = clique1591571987439376245eg_cup ).
% second_assumptions.deviate_neg_cup.cong
thf(fact_1199_second__assumptions_Odeviate__neg__cap_Ocong,axiom,
clique1591571987438064265eg_cap = clique1591571987438064265eg_cap ).
% second_assumptions.deviate_neg_cap.cong
thf(fact_1200_approx__neg_Opelims,axiom,
! [X4: monotone_mformula_a,Y2: set_nat_nat] :
( ( ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ X4 )
= Y2 )
=> ( ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ X4 )
=> ( ! [Phi2: monotone_mformula_a,Psi2: monotone_mformula_a] :
( ( X4
= ( monotone_Conj_a @ Phi2 @ Psi2 ) )
=> ( ( Y2
= ( 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] :
( ( X4
= ( monotone_Disj_a @ Phi2 @ Psi2 ) )
=> ( ( Y2
= ( 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 ) ) ) )
=> ( ( ( X4 = monotone_TRUE_a )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_TRUE_a ) ) )
=> ( ( ( X4 = monotone_FALSE_a )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ monotone_FALSE_a ) ) )
=> ~ ! [V: a] :
( ( X4
= ( monotone_Var_a @ V ) )
=> ( ( Y2 = bot_bot_set_nat_nat )
=> ~ ( accp_M6162913489380515981mula_a @ clique6353239774569474354_rel_a @ ( monotone_Var_a @ V ) ) ) ) ) ) ) ) ) ) ).
% approx_neg.pelims
thf(fact_1201_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_1202_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_1203_no__deviation_I4_J,axiom,
! [X4: a] :
( ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ ( monotone_Var_a @ X4 ) )
= bot_bot_set_nat_nat ) ).
% no_deviation(4)
thf(fact_1204_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_1205_third__assumptions__axioms,axiom,
assump2119784843035796504ptions @ l @ p @ k ).
% third_assumptions_axioms
thf(fact_1206_odotl__def,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k @ X2 @ Y )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X2 @ Y ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).
% odotl_def
thf(fact_1207_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_1208_first__assumptions_Oodotl__def,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique7966186356931407165_odotl @ L @ K @ X2 @ Y )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X2 @ Y ) @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ).
% first_assumptions.odotl_def
thf(fact_1209_finite__approx__pos,axiom,
! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique8538548958085942603_pos_a @ l @ p @ k @ pi @ Phi ) ) ).
% finite_approx_pos
thf(fact_1210_deviate__neg__cap__def,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987438064265eg_cap @ l @ p @ k @ X2 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X2 @ Y ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X2 @ Y ) ) ) ) ).
% deviate_neg_cap_def
thf(fact_1211_ACC__cf__odot,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X2 @ Y ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X2 ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_odot
thf(fact_1212_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_1213_ACC__cf__union,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X2 ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_union
thf(fact_1214_ACC__cf__mono,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X2 ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_mono
thf(fact_1215_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_1216_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_1217_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_1218_deviate__finite_I1_J,axiom,
! [Phi: monotone_mformula_a] : ( finite6739761609112101331et_nat @ ( clique3934260045859375359_pos_a @ l @ p @ k @ pi @ Phi ) ) ).
% deviate_finite(1)
thf(fact_1219_deviate__neg__cup__def,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987439376245eg_cup @ l @ p @ k @ X2 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X2 @ Y ) ) @ ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ) ) ).
% deviate_neg_cup_def
thf(fact_1220_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_1221_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_1222_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_1223_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_1224_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_1225_first__assumptions_OACC__cf__odot,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X2 @ Y ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X2 ) @ ( clique951075384711337423ACC_cf @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_odot
thf(fact_1226_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_1227_first__assumptions_OACC__cf__empty,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ) ).
% first_assumptions.ACC_cf_empty
thf(fact_1228_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X2 ) @ ( clique951075384711337423ACC_cf @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_1229_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X2 ) @ ( clique951075384711337423ACC_cf @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_1230_second__assumptions_Odeviate__neg__cup__def,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( clique1591571987439376245eg_cup @ L @ P3 @ K @ X2 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118207531017_sqcup @ L @ P3 @ K @ X2 @ Y ) ) @ ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_neg_cup_def
thf(fact_1231_second__assumptions_Odeviate__neg__cap__def,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( clique1591571987438064265eg_cap @ L @ P3 @ K @ X2 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118206219037_sqcap @ L @ P3 @ K @ X2 @ Y ) ) @ ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X2 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_neg_cap_def
thf(fact_1232_first__assumptions_Ofinite__POS__NEG,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737375870294875st_NEG @ K ) ) ) ) ).
% first_assumptions.finite_POS_NEG
thf(fact_1233_ACC__cf___092_060F_062,axiom,
! [X2: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X2 ) @ ( clique2971579238625216137irst_F @ k ) ) ).
% ACC_cf_\<F>
thf(fact_1234_finite__approx__neg,axiom,
! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique6623365555141101007_neg_a @ l @ p @ k @ pi @ Phi ) ) ).
% finite_approx_neg
thf(fact_1235_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_1236_finite__v__gs__Gl,axiom,
! [X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) ) ).
% finite_v_gs_Gl
thf(fact_1237_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_1238_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_1239_deviate__finite_I2_J,axiom,
! [Phi: monotone_mformula_a] : ( finite2115694454571419734at_nat @ ( clique2019076642914533763_neg_a @ l @ p @ k @ pi @ Phi ) ) ).
% deviate_finite(2)
thf(fact_1240_finite__ACC,axiom,
! [X2: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X2 ) ) ).
% finite_ACC
thf(fact_1241_first__assumptions_Ofinite___092_060F_062,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.finite_\<F>
thf(fact_1242_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K @ X2 ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_1243_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_1244_first__assumptions_Ofinite__v__gs__Gl,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) ) ) ).
% first_assumptions.finite_v_gs_Gl
thf(fact_1245_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X2 ) @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_1246_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_1247_ACC__cf__I,axiom,
! [F3: nat > nat,X2: set_set_set_nat] :
( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ k ) )
=> ( ( clique3686358387679108662ccepts @ X2 @ ( clique5033774636164728462irst_C @ k @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ k @ X2 ) ) ) ) ).
% ACC_cf_I
thf(fact_1248_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1249_first__assumptions_ONEG__def,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( clique3210737375870294875st_NEG @ K )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K ) @ ( clique2971579238625216137irst_F @ K ) ) ) ) ).
% first_assumptions.NEG_def
thf(fact_1250_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P3: nat,K: nat,F3: nat > nat,X2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ K ) )
=> ( ( clique3686358387679108662ccepts @ X2 @ ( clique5033774636164728462irst_C @ K @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ K @ X2 ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_1251_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_1252_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_1253_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_1254_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_1255_Lm,axiom,
ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lm
thf(fact_1256_M0,axiom,
ord_less_eq_nat @ assumptions_and_M0 @ ( assump1710595444109740334irst_m @ k ) ).
% M0
thf(fact_1257_M0_H,axiom,
ord_less_eq_nat @ assumptions_and_M02 @ ( assump1710595444109740334irst_m @ k ) ).
% M0'
thf(fact_1258_plucking__step_I3_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X2 ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ X2 ) ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ) ) ).
% plucking_step(3)
thf(fact_1259_first__assumptions_Omp,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ P3 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.mp
thf(fact_1260_plucking__step_I2_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X2 ) )
=> ( ord_le9131159989063066194et_nat @ Y @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ) ) ).
% plucking_step(2)
thf(fact_1261_plucking__step_I5_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X2 ) )
=> ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ).
% plucking_step(5)
thf(fact_1262_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_1263_second__assumptions_Oplucking__step_I2_J,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P3 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P3 @ X2 ) )
=> ( ord_le9131159989063066194et_nat @ Y @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ) ) ).
% second_assumptions.plucking_step(2)
thf(fact_1264_second__assumptions_Oplucking__step_I5_J,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P3 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P3 @ X2 ) )
=> ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% second_assumptions.plucking_step(5)
thf(fact_1265_second__assumptions_Oplucking__step_I3_J,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P3 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P3 @ X2 ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ X2 ) ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ) ) ).
% second_assumptions.plucking_step(3)
thf(fact_1266_first__assumptions_Ok,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_1267_first__assumptions_Okp,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ P3 @ K ) ) ).
% first_assumptions.kp
thf(fact_1268_first__assumptions_Opl,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ L @ P3 ) ) ).
% first_assumptions.pl
thf(fact_1269_second__assumptions_Oaxioms_I1_J,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( assump5453534214990993103ptions @ L @ P3 @ K ) ) ).
% second_assumptions.axioms(1)
thf(fact_1270_first__assumptions_Okml,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_1271_first__assumptions_Okm,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.km
thf(fact_1272_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 )
=> ( ! [X8: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k ) @ X8 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X8 @ ( clique7840962075309931874st_G_l @ l @ k ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X8 ) ) ) )
=> ( P @ ( clique4095374090462327202g_step @ p @ X8 ) ) )
=> ( P @ X8 ) ) )
=> ( P @ A0 ) ) ) ).
% PLU_main.pinduct
thf(fact_1273_plucking__step_I1_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) @ p ) @ one_one_nat ) ) ) ) ) ).
% plucking_step(1)
thf(fact_1274_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1275_first__assumptions_Olm,axiom,
! [L: nat,P3: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P3 @ K )
=> ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.lm
thf(fact_1276_second__assumptions_Oplucking__step_I1_J,axiom,
! [L: nat,P3: nat,K: nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P3 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P3 @ X2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X2 ) ) @ P3 ) @ one_one_nat ) ) ) ) ) ) ).
% second_assumptions.plucking_step(1)
thf(fact_1277_second__assumptions_OPLU__main_Opinduct,axiom,
! [L: nat,P3: nat,K: nat,A0: set_set_set_nat,P: set_set_set_nat > $o] :
( ( assump2881078719466019805ptions @ L @ P3 @ K )
=> ( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P3 @ K ) @ A0 )
=> ( ! [X8: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P3 @ K ) @ X8 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X8 @ ( clique7840962075309931874st_G_l @ L @ K ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P3 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X8 ) ) ) )
=> ( P @ ( clique4095374090462327202g_step @ P3 @ X8 ) ) )
=> ( P @ X8 ) ) )
=> ( P @ A0 ) ) ) ) ).
% second_assumptions.PLU_main.pinduct
% Conjectures (1)
thf(conj_0,conjecture,
! [X3: a] :
( ( member_a @ X3 @ v )
=> ( ( member_set_nat @ ( pi @ X3 ) @ d )
=> ( theta @ X3 ) ) ) ).
%------------------------------------------------------------------------------