TPTP Problem File: SLH0008^1.p
View Solutions
- 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_01044_038211__16282618_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1510 ( 677 unt; 237 typ; 0 def)
% Number of atoms : 2771 ( 907 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 10124 ( 163 ~; 17 |; 122 &;8805 @)
% ( 0 <=>;1017 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 31 ( 30 usr)
% Number of type conns : 757 ( 757 >; 0 *; 0 +; 0 <<)
% Number of symbols : 208 ( 207 usr; 12 con; 0-5 aty)
% Number of variables : 3451 ( 309 ^;3108 !; 34 ?;3451 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:49:33.371
%------------------------------------------------------------------------------
% Could-be-implicit typings (30)
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__Set__Oset_It__Nat__Onat_J_J_J_J,type,
produc4405103650892965957et_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc4669799618898522568at_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__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
produc2795196907576053192et_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc1300680692302638795at_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc6120947395724252946et_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__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__Nat__Onat_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
produc4215398038078006819et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
produc4412711793163744422at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc1191534996990032550at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Num__Onum_J,type,
produc1499455660341814967at_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
produc791211723392977261at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc4767002602295674295et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc7536164797210123117et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc6595053547716516154at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc85711943791777264at_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__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
product_prod_num_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Nat__Onat_J,type,
product_prod_num_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
product_prod_nat_num: $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__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (207)
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_Osecond__assumptions__axioms,type,
assump8934899134041091456axioms: nat > nat > $o ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > 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__gs,type,
clique8462013130872731469t_v_gs: set_set_set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU,type,
clique2699557479641037314nd_PLU: nat > nat > nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main,type,
clique429652266423863867U_main: nat > nat > nat > set_set_set_nat > produc4045820344675478307at_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main__graph,type,
clique711371890332037011_graph: nat > nat > nat > set_set_set_nat > produc4045820344675478307at_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main__rel,type,
clique8954521387433384062in_rel: nat > nat > nat > set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cap,type,
clique1591571987438064265eg_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cup,type,
clique1591571987439376245eg_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cap,type,
clique3314026705535538693os_cap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cup,type,
clique3314026705536850673os_cup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcap,type,
clique2586627118206219037_sqcap: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Osqcup,type,
clique2586627118207531017_sqcup: nat > nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_Ocard_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite_card_nat_nat: set_nat_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1149291290879098388et_nat: set_set_set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_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_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
minus_4365393887724441320at_nat: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
minus_4865422370612786701at_nat: produc1191534996990032550at_nat > produc1191534996990032550at_nat > produc1191534996990032550at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
minus_5802666374030397450et_nat: produc4215398038078006819et_nat > produc4215398038078006819et_nat > produc4215398038078006819et_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
minus_8086599166786498573at_nat: produc4412711793163744422at_nat > produc4412711793163744422at_nat > produc4412711793163744422at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
minus_1278274433303930802at_nat: produc1300680692302638795at_nat > produc1300680692302638795at_nat > produc1300680692302638795at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
minus_840401501723538735et_nat: produc2795196907576053192et_nat > produc2795196907576053192et_nat > produc2795196907576053192et_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
minus_5633088680627868938at_nat: produc4045820344675478307at_nat > produc4045820344675478307at_nat > produc4045820344675478307at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
minus_2715004213046008111at_nat: produc4669799618898522568at_nat > produc4669799618898522568at_nat > produc4669799618898522568at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__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__Set__Oset_It__Nat__Onat_J_J_J_J,type,
minus_5310655894399352364et_nat: produc4405103650892965957et_nat > produc4405103650892965957et_nat > produc4405103650892965957et_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
minus_3113942175840221057et_nat: set_set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
inf_inf_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in8487852146424888249at_nat: produc1300680692302638795at_nat > produc1300680692302638795at_nat > produc1300680692302638795at_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
inf_in4890968077345302454et_nat: produc2795196907576053192et_nat > produc2795196907576053192et_nat > produc2795196907576053192et_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
inf_in8110749398470801937at_nat: produc4045820344675478307at_nat > produc4045820344675478307at_nat > produc4045820344675478307at_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inf_in6765570788667771830at_nat: produc4669799618898522568at_nat > produc4669799618898522568at_nat > produc4669799618898522568at_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__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__Set__Oset_It__Nat__Onat_J_J_J_J,type,
inf_in7092662171172785971et_nat: produc4405103650892965957et_nat > produc4405103650892965957et_nat > produc4405103650892965957et_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_inf_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
inf_in5711780100303410308et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
inf_in2396666505901392698et_nat: set_set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
sup_sup_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su1050037771541313183at_nat: produc1300680692302638795at_nat > produc1300680692302638795at_nat > produc1300680692302638795at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
sup_su5047128473845832934et_nat: produc6120947395724252946et_nat > produc6120947395724252946et_nat > produc6120947395724252946et_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
sup_su1375613225412723612et_nat: produc2795196907576053192et_nat > produc2795196907576053192et_nat > produc2795196907576053192et_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__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,
sup_su6693998056113403622at_nat: produc7767816977991823634at_nat > produc7767816977991823634at_nat > produc7767816977991823634at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_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,
sup_su8202740707494180781et_nat: produc6289966885787448281et_nat > produc6289966885787448281et_nat > produc6289966885787448281et_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__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,
sup_su8797241888131514979et_nat: produc1642012749495946895et_nat > produc1642012749495946895et_nat > produc1642012749495946895et_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
sup_su2809688525313048311at_nat: produc4045820344675478307at_nat > produc4045820344675478307at_nat > produc4045820344675478307at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su3250215936735192988at_nat: produc4669799618898522568at_nat > produc4669799618898522568at_nat > produc4669799618898522568at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__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,
sup_su7542950870425426275et_nat: produc387721731789858191et_nat > produc387721731789858191et_nat > produc387721731789858191et_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__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__Set__Oset_It__Nat__Onat_J_J_J_J,type,
sup_su7058913318992137241et_nat: produc4405103650892965957et_nat > produc4405103650892965957et_nat > produc4405103650892965957et_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_sup_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
sup_su4213647025997063966et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
sup_su3906748206781935060et_nat: set_set_set_set_nat > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
ord_le2827710055974039503at_nat: produc4045820344675478307at_nat > produc4045820344675478307at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le52856854838348540et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3929206603849117072at_nat: produc85711943791777264at_nat > produc85711943791777264at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le8460144461188290721at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
ord_le9168388398137128427at_num: product_prod_nat_num > product_prod_nat_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le1758471263074804493et_nat: produc7536164797210123117et_nat > produc7536164797210123117et_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Num__Onum_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le1215176170919080154at_nat: produc6595053547716516154at_nat > produc6595053547716516154at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Num__Onum_Mt__Nat__Onat_J,type,
ord_le6590474864495760107um_nat: product_prod_num_nat > product_prod_num_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
ord_le7298718801444597813um_num: product_prod_num_num > product_prod_num_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Num__Onum_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le8212681105015131479et_nat: produc4767002602295674295et_nat > produc4767002602295674295et_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
ord_le4236890226112434445at_nat: produc791211723392977261at_nat > produc791211723392977261at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Num__Onum_J,type,
ord_le4945134163061272151at_num: produc1499455660341814967at_num > produc1499455660341814967at_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
ord_le2960050975727596227at_nat: produc4045820344675478307at_nat > produc4045820344675478307at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc7839516862119294504at_nat: nat > ( nat > nat ) > produc85711943791777264at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
product_Pair_nat_num: nat > num > product_prod_nat_num ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc6017321930131081694at_nat: nat > set_nat_nat > produc1191534996990032550at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc8033011827914384037et_nat: nat > set_set_nat > produc7536164797210123117et_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc1530211740221758555et_nat: nat > set_set_set_nat > produc4215398038078006819et_nat ).
thf(sy_c_Product__Type_OPair_001t__Num__Onum_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc5125486429189257586at_nat: num > ( nat > nat ) > produc6595053547716516154at_nat ).
thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Nat__Onat,type,
product_Pair_num_nat: num > nat > product_prod_num_nat ).
thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc5263849632999935215et_nat: num > set_set_nat > produc4767002602295674295et_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
produc4994002345568745310at_nat: set_nat_nat > nat > produc4412711793163744422at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc8650507651752646531at_nat: set_nat_nat > set_nat_nat > produc1300680692302638795at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc6308126451205306314et_nat: set_nat_nat > set_set_nat > produc6120947395724252946et_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc2168915450337631616et_nat: set_nat_nat > set_set_set_nat > produc2795196907576053192et_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
produc7293815987952286373at_nat: set_set_nat > nat > produc791211723392977261at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Num__Onum,type,
produc3851147774108065007at_num: set_set_nat > num > produc1499455660341814967at_num ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc8981877611209197642at_nat: set_set_nat > set_nat_nat > produc7767816977991823634at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc9057842353944101649et_nat: set_set_nat > set_set_nat > produc6289966885787448281et_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc7315026656311086279et_nat: set_set_nat > set_set_set_nat > produc1642012749495946895et_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Nat__Onat,type,
produc2803780273060847707at_nat: set_set_set_nat > nat > produc4045820344675478307at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc507788731782991104at_nat: set_set_set_nat > set_nat_nat > produc4669799618898522568at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc1498124630991567047et_nat: set_set_set_nat > set_set_nat > produc387721731789858191et_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc8443863378681539197et_nat: set_set_set_nat > set_set_set_nat > produc4405103650892965957et_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc4700600022864998420at_nat: produc85711943791777264at_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Num__Onum,type,
product_fst_nat_num: product_prod_nat_num > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc254584859675415498at_nat: produc1191534996990032550at_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc6232562755796831889et_nat: produc7536164797210123117et_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc5249848890643421255et_nat: produc4215398038078006819et_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Num__Onum_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc1986569589934961502at_nat: produc6595053547716516154at_nat > num ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Num__Onum_001t__Nat__Onat,type,
product_fst_num_nat: product_prod_num_nat > num ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Num__Onum_001t__Num__Onum,type,
product_fst_num_num: product_prod_num_num > num ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Num__Onum_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc3463400560882383067et_nat: produc4767002602295674295et_nat > num ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
produc8454637311967854922at_nat: produc4412711793163744422at_nat > set_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7281425617276542831at_nat: produc1300680692302638795at_nat > set_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc8269827771562123702et_nat: produc6120947395724252946et_nat > set_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc8444235278545442668et_nat: produc2795196907576053192et_nat > set_nat_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
produc5493366915834734225at_nat: produc791211723392977261at_nat > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Num__Onum,type,
produc2050698701990512859at_num: produc1499455660341814967at_num > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc1720206894711239222at_nat: produc7767816977991823634at_nat > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc945566436066627325et_nat: produc6289966885787448281et_nat > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc3064265511465314483et_nat: produc1642012749495946895et_nat > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Nat__Onat,type,
produc6523417423482510407at_nat: produc4045820344675478307at_nat > set_set_set_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc6783108559990802156at_nat: produc4669799618898522568at_nat > set_set_set_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc6470735523000571059et_nat: produc387721731789858191et_nat > set_set_set_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc1516831113742898793et_nat: produc4405103650892965957et_nat > set_set_set_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc396725235424222550at_nat: produc85711943791777264at_nat > nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Num__Onum,type,
product_snd_nat_num: product_prod_nat_num > num ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7845422228187674892at_nat: produc1191534996990032550at_nat > set_nat_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc3897700410612463315et_nat: produc7536164797210123117et_nat > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc7713928125524949129et_nat: produc4215398038078006819et_nat > set_set_set_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Num__Onum_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc6906066839348961440at_nat: produc6595053547716516154at_nat > nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Num__Onum_001t__Nat__Onat,type,
product_snd_num_nat: product_prod_num_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Num__Onum_001t__Num__Onum,type,
product_snd_num_num: product_prod_num_num > num ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Num__Onum_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc1128538215698014493et_nat: produc4767002602295674295et_nat > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
produc6822102643625338508at_nat: produc4412711793163744422at_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc5975371700627812785at_nat: produc1300680692302638795at_nat > set_nat_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc1792016581537167096et_nat: produc6120947395724252946et_nat > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc2415352471775012014et_nat: produc2795196907576053192et_nat > set_set_set_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
produc3158504570650365651at_nat: produc791211723392977261at_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Num__Onum,type,
produc8939208393660920093at_num: produc1499455660341814967at_num > num ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc4465767741541058424at_nat: produc7767816977991823634at_nat > set_nat_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc4249009452383872831et_nat: produc6289966885787448281et_nat > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc2120572880417880309et_nat: produc1642012749495946895et_nat > set_set_set_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Nat__Onat,type,
produc8987496658364038281at_nat: produc4045820344675478307at_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc754225753220371502at_nat: produc4669799618898522568at_nat > set_nat_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
produc5527042891953136885et_nat: produc387721731789858191et_nat > set_set_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
produc2905410560650206891et_nat: produc4405103650892965957et_nat > set_set_set_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_9186907679027735170et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > set_set_set_nat ).
thf(sy_c_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__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_v_X,type,
x: set_set_set_nat ).
thf(sy_v_Y,type,
y: set_set_set_nat ).
thf(sy_v_Z____,type,
z: set_set_set_nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1272)
thf(fact_0_PLU__main__graph_Ocong,axiom,
clique711371890332037011_graph = clique711371890332037011_graph ).
% PLU_main_graph.cong
thf(fact_1_X,axiom,
member2946998982187404937et_nat @ x @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ).
% X
thf(fact_2_Y,axiom,
member2946998982187404937et_nat @ y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ).
% Y
thf(fact_3_res,axiom,
( ( clique429652266423863867U_main @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ x @ y ) )
= ( produc2803780273060847707at_nat @ z @ n ) ) ).
% res
thf(fact_4_second__assumptions__axioms,axiom,
assump2881078719466019805ptions @ l @ p @ k ).
% second_assumptions_axioms
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Z_An_O_APLU__main_A_IX_A_092_060union_062_AY_J_A_061_A_IZ_M_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Z: set_set_set_nat,N: nat] :
( ( clique429652266423863867U_main @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ x @ y ) )
!= ( produc2803780273060847707at_nat @ Z @ N ) ) ).
% \<open>\<And>thesis. (\<And>Z n. PLU_main (X \<union> Y) = (Z, n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_PLU__def,axiom,
! [X: set_set_set_nat] :
( ( clique2699557479641037314nd_PLU @ l @ p @ k @ X )
= ( produc6523417423482510407at_nat @ ( clique429652266423863867U_main @ l @ p @ k @ X ) ) ) ).
% PLU_def
thf(fact_7_second__assumptions_OPLU__main_Ocong,axiom,
clique429652266423863867U_main = clique429652266423863867U_main ).
% second_assumptions.PLU_main.cong
thf(fact_8_PLU__main__rel_Ocong,axiom,
clique8954521387433384062in_rel = clique8954521387433384062in_rel ).
% PLU_main_rel.cong
thf(fact_9_fst__sup,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( produc6523417423482510407at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( produc6523417423482510407at_nat @ X2 ) @ ( produc6523417423482510407at_nat @ Y ) ) ) ).
% fst_sup
thf(fact_10_sqcup__def,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique2586627118207531017_sqcup @ l @ p @ k @ X @ Y2 )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ) ).
% sqcup_def
thf(fact_11_ACC__union,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X ) @ ( clique3210737319928189260st_ACC @ k @ Y2 ) ) ) ).
% ACC_union
thf(fact_12_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_13_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_14_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_15_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_16_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_17_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_18_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_19_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_20_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_21_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_22_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_23_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_24_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_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_sup_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_27_sup_Oidem,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_28_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_29_sup_Oidem,axiom,
! [A2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_30_sup__idem,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_31_sup__idem,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_32_sup__idem,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_33_sup__idem,axiom,
! [X2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_34_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_35_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_36_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_37_sup_Oright__idem,axiom,
! [A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) @ B2 )
= ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_38_sup__left__idem,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_39_sup__left__idem,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y ) )
= ( sup_sup_set_nat_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_40_sup__left__idem,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y ) )
= ( sup_sup_set_set_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_41_sup__left__idem,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) )
= ( sup_su2809688525313048311at_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_42_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_43_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_44_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_45_sup_Oleft__idem,axiom,
! [A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ A2 @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) )
= ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_46_sup__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: nat,C: set_set_set_nat,D: nat] :
( ( sup_su2809688525313048311at_nat @ ( produc2803780273060847707at_nat @ A2 @ B2 ) @ ( produc2803780273060847707at_nat @ C @ D ) )
= ( produc2803780273060847707at_nat @ ( sup_su4213647025997063966et_nat @ A2 @ C ) @ ( sup_sup_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_47_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat,D: set_set_nat] :
( ( sup_su8202740707494180781et_nat @ ( produc9057842353944101649et_nat @ A2 @ B2 ) @ ( produc9057842353944101649et_nat @ C @ D ) )
= ( produc9057842353944101649et_nat @ ( sup_sup_set_set_nat @ A2 @ C ) @ ( sup_sup_set_set_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_48_sup__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,C: set_set_set_nat,D: set_set_nat] :
( ( sup_su7542950870425426275et_nat @ ( produc1498124630991567047et_nat @ A2 @ B2 ) @ ( produc1498124630991567047et_nat @ C @ D ) )
= ( produc1498124630991567047et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ C ) @ ( sup_sup_set_set_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_49_sup__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_set_nat,C: set_nat_nat,D: set_set_nat] :
( ( sup_su5047128473845832934et_nat @ ( produc6308126451205306314et_nat @ A2 @ B2 ) @ ( produc6308126451205306314et_nat @ C @ D ) )
= ( produc6308126451205306314et_nat @ ( sup_sup_set_nat_nat @ A2 @ C ) @ ( sup_sup_set_set_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_50_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_set_set_nat,C: set_set_nat,D: set_set_set_nat] :
( ( sup_su8797241888131514979et_nat @ ( produc7315026656311086279et_nat @ A2 @ B2 ) @ ( produc7315026656311086279et_nat @ C @ D ) )
= ( produc7315026656311086279et_nat @ ( sup_sup_set_set_nat @ A2 @ C ) @ ( sup_su4213647025997063966et_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_51_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_nat_nat,C: set_set_nat,D: set_nat_nat] :
( ( sup_su6693998056113403622at_nat @ ( produc8981877611209197642at_nat @ A2 @ B2 ) @ ( produc8981877611209197642at_nat @ C @ D ) )
= ( produc8981877611209197642at_nat @ ( sup_sup_set_set_nat @ A2 @ C ) @ ( sup_sup_set_nat_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_52_sup__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat,D: set_set_set_nat] :
( ( sup_su7058913318992137241et_nat @ ( produc8443863378681539197et_nat @ A2 @ B2 ) @ ( produc8443863378681539197et_nat @ C @ D ) )
= ( produc8443863378681539197et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ C ) @ ( sup_su4213647025997063966et_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_53_sup__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_nat_nat,C: set_set_set_nat,D: set_nat_nat] :
( ( sup_su3250215936735192988at_nat @ ( produc507788731782991104at_nat @ A2 @ B2 ) @ ( produc507788731782991104at_nat @ C @ D ) )
= ( produc507788731782991104at_nat @ ( sup_su4213647025997063966et_nat @ A2 @ C ) @ ( sup_sup_set_nat_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_54_sup__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_set_set_nat,C: set_nat_nat,D: set_set_set_nat] :
( ( sup_su1375613225412723612et_nat @ ( produc2168915450337631616et_nat @ A2 @ B2 ) @ ( produc2168915450337631616et_nat @ C @ D ) )
= ( produc2168915450337631616et_nat @ ( sup_sup_set_nat_nat @ A2 @ C ) @ ( sup_su4213647025997063966et_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_55_sup__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat,D: set_nat_nat] :
( ( sup_su1050037771541313183at_nat @ ( produc8650507651752646531at_nat @ A2 @ B2 ) @ ( produc8650507651752646531at_nat @ C @ D ) )
= ( produc8650507651752646531at_nat @ ( sup_sup_set_nat_nat @ A2 @ C ) @ ( sup_sup_set_nat_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_56_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_57_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_58_second__assumptions_Osqcup_Ocong,axiom,
clique2586627118207531017_sqcup = clique2586627118207531017_sqcup ).
% second_assumptions.sqcup.cong
thf(fact_59_second__assumptions_OPLU_Ocong,axiom,
clique2699557479641037314nd_PLU = clique2699557479641037314nd_PLU ).
% second_assumptions.PLU.cong
thf(fact_60_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ C @ B2 )
=> ( ord_less_set_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_61_sup_Ostrict__coboundedI2,axiom,
! [C: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ C @ B2 )
=> ( ord_less_set_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_62_sup_Ostrict__coboundedI2,axiom,
! [C: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2827710055974039503at_nat @ C @ B2 )
=> ( ord_le2827710055974039503at_nat @ C @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_63_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_64_sup_Ostrict__coboundedI2,axiom,
! [C: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ C @ B2 )
=> ( ord_le152980574450754630et_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_65_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ C @ A2 )
=> ( ord_less_set_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_66_sup_Ostrict__coboundedI1,axiom,
! [C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ C @ A2 )
=> ( ord_less_set_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_67_sup_Ostrict__coboundedI1,axiom,
! [C: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( ord_le2827710055974039503at_nat @ C @ A2 )
=> ( ord_le2827710055974039503at_nat @ C @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_68_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_69_sup_Ostrict__coboundedI1,axiom,
! [C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ C @ A2 )
=> ( ord_le152980574450754630et_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_70_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat_nat
= ( ^ [B3: set_nat_nat,A3: set_nat_nat] :
( ( A3
= ( sup_sup_set_nat_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_71_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_nat
= ( ^ [B3: set_set_nat,A3: set_set_nat] :
( ( A3
= ( sup_sup_set_set_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_72_sup_Ostrict__order__iff,axiom,
( ord_le2827710055974039503at_nat
= ( ^ [B3: produc4045820344675478307at_nat,A3: produc4045820344675478307at_nat] :
( ( A3
= ( sup_su2809688525313048311at_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_73_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_74_sup_Ostrict__order__iff,axiom,
( ord_le152980574450754630et_nat
= ( ^ [B3: set_set_set_nat,A3: set_set_set_nat] :
( ( A3
= ( sup_su4213647025997063966et_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_75_sup_Ostrict__boundedE,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ~ ( ord_less_set_nat_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_76_sup_Ostrict__boundedE,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_set_nat @ B2 @ A2 )
=> ~ ( ord_less_set_set_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_77_sup_Ostrict__boundedE,axiom,
! [B2: produc4045820344675478307at_nat,C: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2827710055974039503at_nat @ ( sup_su2809688525313048311at_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le2827710055974039503at_nat @ B2 @ A2 )
=> ~ ( ord_le2827710055974039503at_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_78_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_79_sup_Ostrict__boundedE,axiom,
! [B2: set_set_set_nat,C: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ~ ( ord_le152980574450754630et_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_80_sup_Oabsorb4,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_81_sup_Oabsorb4,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_82_sup_Oabsorb4,axiom,
! [A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( ord_le2827710055974039503at_nat @ A2 @ B2 )
=> ( ( sup_su2809688525313048311at_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_83_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_84_sup_Oabsorb4,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_85_sup_Oabsorb3,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_86_sup_Oabsorb3,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_87_sup_Oabsorb3,axiom,
! [B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2827710055974039503at_nat @ B2 @ A2 )
=> ( ( sup_su2809688525313048311at_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_88_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_89_sup_Oabsorb3,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_90_less__supI2,axiom,
! [X2: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ B2 )
=> ( ord_less_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_91_less__supI2,axiom,
! [X2: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ B2 )
=> ( ord_less_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_92_less__supI2,axiom,
! [X2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2827710055974039503at_nat @ X2 @ B2 )
=> ( ord_le2827710055974039503at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_93_less__supI2,axiom,
! [X2: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ X2 @ B2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_94_less__supI2,axiom,
! [X2: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ B2 )
=> ( ord_le152980574450754630et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_95_less__supI1,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ X2 @ A2 )
=> ( ord_less_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_96_less__supI1,axiom,
! [X2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ X2 @ A2 )
=> ( ord_less_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_97_less__supI1,axiom,
! [X2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( ord_le2827710055974039503at_nat @ X2 @ A2 )
=> ( ord_le2827710055974039503at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_98_less__supI1,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ X2 @ A2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_99_less__supI1,axiom,
! [X2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ A2 )
=> ( ord_le152980574450754630et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_100_first__assumptions_OACC__union,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K @ X ) @ ( clique3210737319928189260st_ACC @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_union
thf(fact_101_second__assumptions_Osqcup__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique2586627118207531017_sqcup @ L @ P @ K @ X @ Y2 )
= ( clique2699557479641037314nd_PLU @ L @ P @ K @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ) ) ).
% second_assumptions.sqcup_def
thf(fact_102_second__assumptions_OPLU__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique2699557479641037314nd_PLU @ L @ P @ K @ X )
= ( produc6523417423482510407at_nat @ ( clique429652266423863867U_main @ L @ P @ K @ X ) ) ) ) ).
% second_assumptions.PLU_def
thf(fact_103_sup__left__commute,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_104_sup__left__commute,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_105_sup__left__commute,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_106_sup__left__commute,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ Y @ Z2 ) )
= ( sup_su2809688525313048311at_nat @ Y @ ( sup_su2809688525313048311at_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_107_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_108_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_109_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_110_sup_Oleft__commute,axiom,
! [B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat,C: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ B2 @ ( sup_su2809688525313048311at_nat @ A2 @ C ) )
= ( sup_su2809688525313048311at_nat @ A2 @ ( sup_su2809688525313048311at_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_111_sup__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_112_sup__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [X3: set_nat_nat,Y3: set_nat_nat] : ( sup_sup_set_nat_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_113_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y3: set_set_nat] : ( sup_sup_set_set_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_114_sup__commute,axiom,
( sup_su2809688525313048311at_nat
= ( ^ [X3: produc4045820344675478307at_nat,Y3: produc4045820344675478307at_nat] : ( sup_su2809688525313048311at_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_115_mem__Collect__eq,axiom,
! [A2: set_set_set_nat,P2: set_set_set_nat > $o] :
( ( member2946998982187404937et_nat @ A2 @ ( collec7201453139178570183et_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
! [A2: nat > nat,P2: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
! [A2: set_set_nat,P2: set_set_nat > $o] :
( ( member_set_set_nat @ A2 @ ( collect_set_set_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_119_Collect__mem__eq,axiom,
! [A: set_set_set_set_nat] :
( ( collec7201453139178570183et_nat
@ ^ [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_120_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_123_Collect__cong,axiom,
! [P2: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X4: set_set_set_nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collec7201453139178570183et_nat @ P2 )
= ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_124_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_125_Collect__cong,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat_nat @ P2 )
= ( collect_nat_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_126_Collect__cong,axiom,
! [P2: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X4: set_set_nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_set_set_nat @ P2 )
= ( collect_set_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_127_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_128_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_129_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_130_sup_Ocommute,axiom,
( sup_su2809688525313048311at_nat
= ( ^ [A3: produc4045820344675478307at_nat,B3: produc4045820344675478307at_nat] : ( sup_su2809688525313048311at_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_131_sup__assoc,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ Z2 )
= ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_132_sup__assoc,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_133_sup__assoc,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_134_sup__assoc,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) @ Z2 )
= ( sup_su2809688525313048311at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_135_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_136_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_137_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_138_sup_Oassoc,axiom,
! [A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat,C: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) @ C )
= ( sup_su2809688525313048311at_nat @ A2 @ ( sup_su2809688525313048311at_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_139_inf__sup__aci_I5_J,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_140_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat_nat
= ( ^ [X3: set_nat_nat,Y3: set_nat_nat] : ( sup_sup_set_nat_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_141_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y3: set_set_nat] : ( sup_sup_set_set_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_142_inf__sup__aci_I5_J,axiom,
( sup_su2809688525313048311at_nat
= ( ^ [X3: produc4045820344675478307at_nat,Y3: produc4045820344675478307at_nat] : ( sup_su2809688525313048311at_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_143_inf__sup__aci_I6_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ Z2 )
= ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_144_inf__sup__aci_I6_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_145_inf__sup__aci_I6_J,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_146_inf__sup__aci_I6_J,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) @ Z2 )
= ( sup_su2809688525313048311at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_147_inf__sup__aci_I7_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_148_inf__sup__aci_I7_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat_nat @ Y @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_149_inf__sup__aci_I7_J,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_set_nat @ Y @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_150_inf__sup__aci_I7_J,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ Y @ Z2 ) )
= ( sup_su2809688525313048311at_nat @ Y @ ( sup_su2809688525313048311at_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_151_inf__sup__aci_I8_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_152_inf__sup__aci_I8_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y ) )
= ( sup_sup_set_nat_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_153_inf__sup__aci_I8_J,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y ) )
= ( sup_sup_set_set_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_154_inf__sup__aci_I8_J,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) )
= ( sup_su2809688525313048311at_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_Un__absorb,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_165_Un__absorb,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_166_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_167_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_168_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_169_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_170_ball__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P2: set_set_nat > $o] :
( ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( P2 @ X3 ) )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_171_ball__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P2: ( nat > nat ) > $o] :
( ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( P2 @ X3 ) )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_172_ball__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P2: set_nat > $o] :
( ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( P2 @ X3 ) )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_173_bex__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P2: set_set_nat > $o] :
( ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ( P2 @ X3 ) )
| ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_174_bex__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,P2: ( nat > nat ) > $o] :
( ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A @ B ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
& ( P2 @ X3 ) )
| ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_175_bex__Un,axiom,
! [A: set_set_nat,B: set_set_nat,P2: set_nat > $o] :
( ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A @ B ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( P2 @ X3 ) )
| ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_192_sqcap__def,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique2586627118206219037_sqcap @ l @ p @ k @ X @ Y2 )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X @ Y2 ) ) ) ).
% sqcap_def
thf(fact_193_ACC__cf__union,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_union
thf(fact_194_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_195_prod_Oinject,axiom,
! [X1: set_set_set_nat,X22: nat,Y1: set_set_set_nat,Y22: nat] :
( ( ( produc2803780273060847707at_nat @ X1 @ X22 )
= ( produc2803780273060847707at_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_196_old_Oprod_Oinject,axiom,
! [A2: set_set_set_nat,B2: nat,A5: set_set_set_nat,B5: nat] :
( ( ( produc2803780273060847707at_nat @ A2 @ B2 )
= ( produc2803780273060847707at_nat @ A5 @ B5 ) )
= ( ( A2 = A5 )
& ( B2 = B5 ) ) ) ).
% old.prod.inject
thf(fact_197_second__assumptions_Oaxioms_I1_J,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( assump5453534214990993103ptions @ L @ P @ K ) ) ).
% second_assumptions.axioms(1)
thf(fact_198_first__assumptions_Ok,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_199_first__assumptions_Okp,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ P @ K ) ) ).
% first_assumptions.kp
thf(fact_200_first__assumptions_Opl,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ L @ P ) ) ).
% first_assumptions.pl
thf(fact_201_eq__fst__iff,axiom,
! [A2: set_set_set_nat,P: produc4045820344675478307at_nat] :
( ( A2
= ( produc6523417423482510407at_nat @ P ) )
= ( ? [B3: nat] :
( P
= ( produc2803780273060847707at_nat @ A2 @ B3 ) ) ) ) ).
% eq_fst_iff
thf(fact_202_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_203_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_204_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_205_first__assumptions_OL_Ocong,axiom,
assump1710595444109740301irst_L = assump1710595444109740301irst_L ).
% first_assumptions.L.cong
thf(fact_206_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_207_second__assumptions_Osqcap_Ocong,axiom,
clique2586627118206219037_sqcap = clique2586627118206219037_sqcap ).
% second_assumptions.sqcap.cong
thf(fact_208_first__assumptions_Omp,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ P @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.mp
thf(fact_209_first__assumptions_Okm,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.km
thf(fact_210_second__assumptions_OLp,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ord_less_nat @ P @ ( assump1710595444109740301irst_L @ L @ P ) ) ) ).
% second_assumptions.Lp
thf(fact_211_second__assumptions_Osqcap__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique2586627118206219037_sqcap @ L @ P @ K @ X @ Y2 )
= ( clique2699557479641037314nd_PLU @ L @ P @ K @ ( clique7966186356931407165_odotl @ L @ K @ X @ Y2 ) ) ) ) ).
% second_assumptions.sqcap_def
thf(fact_212_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_213_Pair__inject,axiom,
! [A2: set_set_set_nat,B2: nat,A5: set_set_set_nat,B5: nat] :
( ( ( produc2803780273060847707at_nat @ A2 @ B2 )
= ( produc2803780273060847707at_nat @ A5 @ B5 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B5 ) ) ) ).
% Pair_inject
thf(fact_214_prod__cases,axiom,
! [P2: produc4045820344675478307at_nat > $o,P: produc4045820344675478307at_nat] :
( ! [A6: set_set_set_nat,B6: nat] : ( P2 @ ( produc2803780273060847707at_nat @ A6 @ B6 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_215_surj__pair,axiom,
! [P: produc4045820344675478307at_nat] :
? [X4: set_set_set_nat,Y4: nat] :
( P
= ( produc2803780273060847707at_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_216_old_Oprod_Oexhaust,axiom,
! [Y: produc4045820344675478307at_nat] :
~ ! [A6: set_set_set_nat,B6: nat] :
( Y
!= ( produc2803780273060847707at_nat @ A6 @ B6 ) ) ).
% old.prod.exhaust
thf(fact_217_fst__eqD,axiom,
! [X2: set_set_set_nat,Y: nat,A2: set_set_set_nat] :
( ( ( produc6523417423482510407at_nat @ ( produc2803780273060847707at_nat @ X2 @ Y ) )
= A2 )
=> ( X2 = A2 ) ) ).
% fst_eqD
thf(fact_218_fst__conv,axiom,
! [X1: set_set_set_nat,X22: nat] :
( ( produc6523417423482510407at_nat @ ( produc2803780273060847707at_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_219_Lm,axiom,
ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lm
thf(fact_220_deviate__neg__cup__def,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique1591571987439376245eg_cup @ l @ p @ k @ X @ Y2 )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X @ Y2 ) ) @ ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ) ) ).
% deviate_neg_cup_def
thf(fact_221_fstI,axiom,
! [X2: produc4045820344675478307at_nat,Y: set_set_set_nat,Z2: nat] :
( ( X2
= ( produc2803780273060847707at_nat @ Y @ Z2 ) )
=> ( ( produc6523417423482510407at_nat @ X2 )
= Y ) ) ).
% fstI
thf(fact_222_second__assumptions__def,axiom,
( assump2881078719466019805ptions
= ( ^ [L2: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L2 @ P3 @ K2 )
& ( assump8934899134041091456axioms @ L2 @ K2 ) ) ) ) ).
% second_assumptions_def
thf(fact_223_second__assumptions_Ointro,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( assump8934899134041091456axioms @ L @ K )
=> ( assump2881078719466019805ptions @ L @ P @ K ) ) ) ).
% second_assumptions.intro
thf(fact_224_deviate__neg__cap__def,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique1591571987438064265eg_cap @ l @ p @ k @ X @ Y2 )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X @ Y2 ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X @ Y2 ) ) ) ) ).
% deviate_neg_cap_def
thf(fact_225_joinl__join,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X @ Y2 ) @ ( clique5469973757772500719t_odot @ X @ Y2 ) ) ).
% joinl_join
thf(fact_226_sup__prod__def,axiom,
( sup_su2809688525313048311at_nat
= ( ^ [X3: produc4045820344675478307at_nat,Y3: produc4045820344675478307at_nat] : ( produc2803780273060847707at_nat @ ( sup_su4213647025997063966et_nat @ ( produc6523417423482510407at_nat @ X3 ) @ ( produc6523417423482510407at_nat @ Y3 ) ) @ ( sup_sup_nat @ ( produc8987496658364038281at_nat @ X3 ) @ ( produc8987496658364038281at_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_227_sup__prod__def,axiom,
( sup_su8202740707494180781et_nat
= ( ^ [X3: produc6289966885787448281et_nat,Y3: produc6289966885787448281et_nat] : ( produc9057842353944101649et_nat @ ( sup_sup_set_set_nat @ ( produc945566436066627325et_nat @ X3 ) @ ( produc945566436066627325et_nat @ Y3 ) ) @ ( sup_sup_set_set_nat @ ( produc4249009452383872831et_nat @ X3 ) @ ( produc4249009452383872831et_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_228_sup__prod__def,axiom,
( sup_su7542950870425426275et_nat
= ( ^ [X3: produc387721731789858191et_nat,Y3: produc387721731789858191et_nat] : ( produc1498124630991567047et_nat @ ( sup_su4213647025997063966et_nat @ ( produc6470735523000571059et_nat @ X3 ) @ ( produc6470735523000571059et_nat @ Y3 ) ) @ ( sup_sup_set_set_nat @ ( produc5527042891953136885et_nat @ X3 ) @ ( produc5527042891953136885et_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_229_sup__prod__def,axiom,
( sup_su5047128473845832934et_nat
= ( ^ [X3: produc6120947395724252946et_nat,Y3: produc6120947395724252946et_nat] : ( produc6308126451205306314et_nat @ ( sup_sup_set_nat_nat @ ( produc8269827771562123702et_nat @ X3 ) @ ( produc8269827771562123702et_nat @ Y3 ) ) @ ( sup_sup_set_set_nat @ ( produc1792016581537167096et_nat @ X3 ) @ ( produc1792016581537167096et_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_230_sup__prod__def,axiom,
( sup_su8797241888131514979et_nat
= ( ^ [X3: produc1642012749495946895et_nat,Y3: produc1642012749495946895et_nat] : ( produc7315026656311086279et_nat @ ( sup_sup_set_set_nat @ ( produc3064265511465314483et_nat @ X3 ) @ ( produc3064265511465314483et_nat @ Y3 ) ) @ ( sup_su4213647025997063966et_nat @ ( produc2120572880417880309et_nat @ X3 ) @ ( produc2120572880417880309et_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_231_sup__prod__def,axiom,
( sup_su6693998056113403622at_nat
= ( ^ [X3: produc7767816977991823634at_nat,Y3: produc7767816977991823634at_nat] : ( produc8981877611209197642at_nat @ ( sup_sup_set_set_nat @ ( produc1720206894711239222at_nat @ X3 ) @ ( produc1720206894711239222at_nat @ Y3 ) ) @ ( sup_sup_set_nat_nat @ ( produc4465767741541058424at_nat @ X3 ) @ ( produc4465767741541058424at_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_232_sup__prod__def,axiom,
( sup_su7058913318992137241et_nat
= ( ^ [X3: produc4405103650892965957et_nat,Y3: produc4405103650892965957et_nat] : ( produc8443863378681539197et_nat @ ( sup_su4213647025997063966et_nat @ ( produc1516831113742898793et_nat @ X3 ) @ ( produc1516831113742898793et_nat @ Y3 ) ) @ ( sup_su4213647025997063966et_nat @ ( produc2905410560650206891et_nat @ X3 ) @ ( produc2905410560650206891et_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_233_sup__prod__def,axiom,
( sup_su3250215936735192988at_nat
= ( ^ [X3: produc4669799618898522568at_nat,Y3: produc4669799618898522568at_nat] : ( produc507788731782991104at_nat @ ( sup_su4213647025997063966et_nat @ ( produc6783108559990802156at_nat @ X3 ) @ ( produc6783108559990802156at_nat @ Y3 ) ) @ ( sup_sup_set_nat_nat @ ( produc754225753220371502at_nat @ X3 ) @ ( produc754225753220371502at_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_234_sup__prod__def,axiom,
( sup_su1375613225412723612et_nat
= ( ^ [X3: produc2795196907576053192et_nat,Y3: produc2795196907576053192et_nat] : ( produc2168915450337631616et_nat @ ( sup_sup_set_nat_nat @ ( produc8444235278545442668et_nat @ X3 ) @ ( produc8444235278545442668et_nat @ Y3 ) ) @ ( sup_su4213647025997063966et_nat @ ( produc2415352471775012014et_nat @ X3 ) @ ( produc2415352471775012014et_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_235_sup__prod__def,axiom,
( sup_su1050037771541313183at_nat
= ( ^ [X3: produc1300680692302638795at_nat,Y3: produc1300680692302638795at_nat] : ( produc8650507651752646531at_nat @ ( sup_sup_set_nat_nat @ ( produc7281425617276542831at_nat @ X3 ) @ ( produc7281425617276542831at_nat @ Y3 ) ) @ ( sup_sup_set_nat_nat @ ( produc5975371700627812785at_nat @ X3 ) @ ( produc5975371700627812785at_nat @ Y3 ) ) ) ) ) ).
% sup_prod_def
thf(fact_236_psubsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le152980574450754630et_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_237_psubsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_238_psubsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_239_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_240_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_241_subsetI,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ! [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
=> ( member2946998982187404937et_nat @ X4 @ B ) )
=> ( ord_le572741076514265352et_nat @ A @ B ) ) ).
% subsetI
thf(fact_242_subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( member_set_set_nat @ X4 @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% subsetI
thf(fact_243_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_244_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( member_set_nat @ X4 @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_245_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( member_nat_nat @ X4 @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_246_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_247_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_248_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_249_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_250_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_251_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_252_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_253_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_254_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_255_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_256_Pair__le,axiom,
! [A2: num,B2: num,C: num,D: num] :
( ( ord_le7298718801444597813um_num @ ( product_Pair_num_num @ A2 @ B2 ) @ ( product_Pair_num_num @ C @ D ) )
= ( ( ord_less_eq_num @ A2 @ C )
& ( ord_less_eq_num @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_257_Pair__le,axiom,
! [A2: num,B2: nat,C: num,D: nat] :
( ( ord_le6590474864495760107um_nat @ ( product_Pair_num_nat @ A2 @ B2 ) @ ( product_Pair_num_nat @ C @ D ) )
= ( ( ord_less_eq_num @ A2 @ C )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_258_Pair__le,axiom,
! [A2: nat,B2: num,C: nat,D: num] :
( ( ord_le9168388398137128427at_num @ ( product_Pair_nat_num @ A2 @ B2 ) @ ( product_Pair_nat_num @ C @ D ) )
= ( ( ord_less_eq_nat @ A2 @ C )
& ( ord_less_eq_num @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_259_Pair__le,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A2 @ C )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_260_Pair__le,axiom,
! [A2: set_set_nat,B2: num,C: set_set_nat,D: num] :
( ( ord_le4945134163061272151at_num @ ( produc3851147774108065007at_num @ A2 @ B2 ) @ ( produc3851147774108065007at_num @ C @ D ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ C )
& ( ord_less_eq_num @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_261_Pair__le,axiom,
! [A2: set_set_nat,B2: nat,C: set_set_nat,D: nat] :
( ( ord_le4236890226112434445at_nat @ ( produc7293815987952286373at_nat @ A2 @ B2 ) @ ( produc7293815987952286373at_nat @ C @ D ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ C )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_262_Pair__le,axiom,
! [A2: num,B2: set_set_nat,C: num,D: set_set_nat] :
( ( ord_le8212681105015131479et_nat @ ( produc5263849632999935215et_nat @ A2 @ B2 ) @ ( produc5263849632999935215et_nat @ C @ D ) )
= ( ( ord_less_eq_num @ A2 @ C )
& ( ord_le6893508408891458716et_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_263_Pair__le,axiom,
! [A2: num,B2: nat > nat,C: num,D: nat > nat] :
( ( ord_le1215176170919080154at_nat @ ( produc5125486429189257586at_nat @ A2 @ B2 ) @ ( produc5125486429189257586at_nat @ C @ D ) )
= ( ( ord_less_eq_num @ A2 @ C )
& ( ord_less_eq_nat_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_264_Pair__le,axiom,
! [A2: nat,B2: set_set_nat,C: nat,D: set_set_nat] :
( ( ord_le1758471263074804493et_nat @ ( produc8033011827914384037et_nat @ A2 @ B2 ) @ ( produc8033011827914384037et_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A2 @ C )
& ( ord_le6893508408891458716et_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_265_Pair__le,axiom,
! [A2: nat,B2: nat > nat,C: nat,D: nat > nat] :
( ( ord_le3929206603849117072at_nat @ ( produc7839516862119294504at_nat @ A2 @ B2 ) @ ( produc7839516862119294504at_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A2 @ C )
& ( ord_less_eq_nat_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_266_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_267_sup_Obounded__iff,axiom,
! [B2: produc4045820344675478307at_nat,C: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ B2 @ C ) @ A2 )
= ( ( ord_le2960050975727596227at_nat @ B2 @ A2 )
& ( ord_le2960050975727596227at_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_268_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_269_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_270_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_271_sup_Obounded__iff,axiom,
! [B2: nat > nat,C: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat_nat @ B2 @ A2 )
& ( ord_less_eq_nat_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_272_le__sup__iff,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_le9131159989063066194et_nat @ X2 @ Z2 )
& ( ord_le9131159989063066194et_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_273_le__sup__iff,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_le2960050975727596227at_nat @ X2 @ Z2 )
& ( ord_le2960050975727596227at_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_274_le__sup__iff,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Z2 )
& ( ord_le6893508408891458716et_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_275_le__sup__iff,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Z2 )
& ( ord_le9059583361652607317at_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_276_le__sup__iff,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_277_le__sup__iff,axiom,
! [X2: nat > nat,Y: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat_nat @ X2 @ Z2 )
& ( ord_less_eq_nat_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_278_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_279_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_280_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_281_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_282_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_283_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_284_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_285_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_286_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_287_prod_Ocollapse,axiom,
! [Prod: produc4045820344675478307at_nat] :
( ( produc2803780273060847707at_nat @ ( produc6523417423482510407at_nat @ Prod ) @ ( produc8987496658364038281at_nat @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_288_snd__sup,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( produc8987496658364038281at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) )
= ( sup_sup_nat @ ( produc8987496658364038281at_nat @ X2 ) @ ( produc8987496658364038281at_nat @ Y ) ) ) ).
% snd_sup
thf(fact_289_psubsetE,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ~ ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_290_psubsetE,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_291_psubsetE,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ~ ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ) ).
% psubsetE
thf(fact_292_psubset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_293_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_294_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_295_psubset__imp__ex__mem,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le52856854838348540et_nat @ A @ B )
=> ? [B6: set_set_set_nat] : ( member2946998982187404937et_nat @ B6 @ ( minus_3113942175840221057et_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_296_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_297_psubset__imp__ex__mem,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ? [B6: nat > nat] : ( member_nat_nat @ B6 @ ( minus_8121590178497047118at_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_298_psubset__imp__ex__mem,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ? [B6: set_set_nat] : ( member_set_set_nat @ B6 @ ( minus_2447799839930672331et_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_299_psubset__imp__subset,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_300_psubset__imp__subset,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_301_psubset__imp__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% psubset_imp_subset
thf(fact_302_psubset__subset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ord_le152980574450754630et_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_303_psubset__subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_less_set_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_304_psubset__subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_less_set_nat_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_305_subset__not__subset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A4 @ B4 )
& ~ ( ord_le9131159989063066194et_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_306_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ~ ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_307_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ~ ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_308_subset__psubset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ B @ C2 )
=> ( ord_le152980574450754630et_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_309_subset__psubset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C2 )
=> ( ord_less_set_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_310_subset__psubset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ B @ C2 )
=> ( ord_less_set_nat_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_311_subset__iff__psubset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_312_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_less_set_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_313_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_less_set_nat_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_314_less__eq__prod__def,axiom,
( ord_le7298718801444597813um_num
= ( ^ [X3: product_prod_num_num,Y3: product_prod_num_num] :
( ( ord_less_eq_num @ ( product_fst_num_num @ X3 ) @ ( product_fst_num_num @ Y3 ) )
& ( ord_less_eq_num @ ( product_snd_num_num @ X3 ) @ ( product_snd_num_num @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_315_less__eq__prod__def,axiom,
( ord_le6590474864495760107um_nat
= ( ^ [X3: product_prod_num_nat,Y3: product_prod_num_nat] :
( ( ord_less_eq_num @ ( product_fst_num_nat @ X3 ) @ ( product_fst_num_nat @ Y3 ) )
& ( ord_less_eq_nat @ ( product_snd_num_nat @ X3 ) @ ( product_snd_num_nat @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_316_less__eq__prod__def,axiom,
( ord_le9168388398137128427at_num
= ( ^ [X3: product_prod_nat_num,Y3: product_prod_nat_num] :
( ( ord_less_eq_nat @ ( product_fst_nat_num @ X3 ) @ ( product_fst_nat_num @ Y3 ) )
& ( ord_less_eq_num @ ( product_snd_nat_num @ X3 ) @ ( product_snd_nat_num @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_317_less__eq__prod__def,axiom,
( ord_le8460144461188290721at_nat
= ( ^ [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X3 ) @ ( product_fst_nat_nat @ Y3 ) )
& ( ord_less_eq_nat @ ( product_snd_nat_nat @ X3 ) @ ( product_snd_nat_nat @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_318_less__eq__prod__def,axiom,
( ord_le4945134163061272151at_num
= ( ^ [X3: produc1499455660341814967at_num,Y3: produc1499455660341814967at_num] :
( ( ord_le6893508408891458716et_nat @ ( produc2050698701990512859at_num @ X3 ) @ ( produc2050698701990512859at_num @ Y3 ) )
& ( ord_less_eq_num @ ( produc8939208393660920093at_num @ X3 ) @ ( produc8939208393660920093at_num @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_319_less__eq__prod__def,axiom,
( ord_le4236890226112434445at_nat
= ( ^ [X3: produc791211723392977261at_nat,Y3: produc791211723392977261at_nat] :
( ( ord_le6893508408891458716et_nat @ ( produc5493366915834734225at_nat @ X3 ) @ ( produc5493366915834734225at_nat @ Y3 ) )
& ( ord_less_eq_nat @ ( produc3158504570650365651at_nat @ X3 ) @ ( produc3158504570650365651at_nat @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_320_less__eq__prod__def,axiom,
( ord_le8212681105015131479et_nat
= ( ^ [X3: produc4767002602295674295et_nat,Y3: produc4767002602295674295et_nat] :
( ( ord_less_eq_num @ ( produc3463400560882383067et_nat @ X3 ) @ ( produc3463400560882383067et_nat @ Y3 ) )
& ( ord_le6893508408891458716et_nat @ ( produc1128538215698014493et_nat @ X3 ) @ ( produc1128538215698014493et_nat @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_321_less__eq__prod__def,axiom,
( ord_le1215176170919080154at_nat
= ( ^ [X3: produc6595053547716516154at_nat,Y3: produc6595053547716516154at_nat] :
( ( ord_less_eq_num @ ( produc1986569589934961502at_nat @ X3 ) @ ( produc1986569589934961502at_nat @ Y3 ) )
& ( ord_less_eq_nat_nat @ ( produc6906066839348961440at_nat @ X3 ) @ ( produc6906066839348961440at_nat @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_322_less__eq__prod__def,axiom,
( ord_le1758471263074804493et_nat
= ( ^ [X3: produc7536164797210123117et_nat,Y3: produc7536164797210123117et_nat] :
( ( ord_less_eq_nat @ ( produc6232562755796831889et_nat @ X3 ) @ ( produc6232562755796831889et_nat @ Y3 ) )
& ( ord_le6893508408891458716et_nat @ ( produc3897700410612463315et_nat @ X3 ) @ ( produc3897700410612463315et_nat @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_323_less__eq__prod__def,axiom,
( ord_le3929206603849117072at_nat
= ( ^ [X3: produc85711943791777264at_nat,Y3: produc85711943791777264at_nat] :
( ( ord_less_eq_nat @ ( produc4700600022864998420at_nat @ X3 ) @ ( produc4700600022864998420at_nat @ Y3 ) )
& ( ord_less_eq_nat_nat @ ( produc396725235424222550at_nat @ X3 ) @ ( produc396725235424222550at_nat @ Y3 ) ) ) ) ) ).
% less_eq_prod_def
thf(fact_324_Collect__mono__iff,axiom,
! [P2: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P2 ) @ ( collec7201453139178570183et_nat @ Q ) )
= ( ! [X3: set_set_set_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_325_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_326_Collect__mono__iff,axiom,
! [P2: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P2 ) @ ( collect_set_set_nat @ Q ) )
= ( ! [X3: set_set_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_327_Collect__mono__iff,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_328_Collect__mono__iff,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) )
= ( ! [X3: nat > nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_329_set__eq__subset,axiom,
( ( ^ [Y5: set_set_nat,Z3: set_set_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_330_set__eq__subset,axiom,
( ( ^ [Y5: set_nat_nat,Z3: set_nat_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_331_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_332_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_333_Collect__mono,axiom,
! [P2: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X4: set_set_set_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P2 ) @ ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_334_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_335_Collect__mono,axiom,
! [P2: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X4: set_set_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P2 ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_336_Collect__mono,axiom,
! [P2: set_nat > $o,Q: set_nat > $o] :
( ! [X4: set_nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P2 ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_337_Collect__mono,axiom,
! [P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X4: nat > nat] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P2 ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_338_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_339_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_340_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_341_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_342_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_343_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_344_Diff__subset,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_345_Diff__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ A ) ).
% Diff_subset
thf(fact_346_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_347_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_348_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_349_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_350_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_351_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_352_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_353_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_354_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_355_subset__eq,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A4: set_set_set_set_nat,B4: set_set_set_set_nat] :
! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A4 )
=> ( member2946998982187404937et_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_356_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A4 )
=> ( member_set_set_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_357_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_358_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
=> ( member_set_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_359_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A4 )
=> ( member_nat_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_360_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_361_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_362_Diff__mono,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,D2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ D2 @ B )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_363_Diff__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,D2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_364_Diff__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,D2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ D2 @ B )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_365_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_366_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_367_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_368_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_369_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_370_in__mono,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( member2946998982187404937et_nat @ X2 @ A )
=> ( member2946998982187404937et_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_371_in__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,X2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ X2 @ A )
=> ( member_set_set_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_372_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_373_in__mono,axiom,
! [A: set_set_nat,B: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_374_in__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,X2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_375_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_376_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_377_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_378_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_379_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_380_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_381_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_382_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_383_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_384_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_385_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_386_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_387_Pair__mono,axiom,
! [X2: num,X5: num,Y: num,Y6: num] :
( ( ord_less_eq_num @ X2 @ X5 )
=> ( ( ord_less_eq_num @ Y @ Y6 )
=> ( ord_le7298718801444597813um_num @ ( product_Pair_num_num @ X2 @ Y ) @ ( product_Pair_num_num @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_388_Pair__mono,axiom,
! [X2: num,X5: num,Y: nat,Y6: nat] :
( ( ord_less_eq_num @ X2 @ X5 )
=> ( ( ord_less_eq_nat @ Y @ Y6 )
=> ( ord_le6590474864495760107um_nat @ ( product_Pair_num_nat @ X2 @ Y ) @ ( product_Pair_num_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_389_Pair__mono,axiom,
! [X2: nat,X5: nat,Y: num,Y6: num] :
( ( ord_less_eq_nat @ X2 @ X5 )
=> ( ( ord_less_eq_num @ Y @ Y6 )
=> ( ord_le9168388398137128427at_num @ ( product_Pair_nat_num @ X2 @ Y ) @ ( product_Pair_nat_num @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_390_Pair__mono,axiom,
! [X2: nat,X5: nat,Y: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ X5 )
=> ( ( ord_less_eq_nat @ Y @ Y6 )
=> ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ ( product_Pair_nat_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_391_Pair__mono,axiom,
! [X2: set_set_nat,X5: set_set_nat,Y: num,Y6: num] :
( ( ord_le6893508408891458716et_nat @ X2 @ X5 )
=> ( ( ord_less_eq_num @ Y @ Y6 )
=> ( ord_le4945134163061272151at_num @ ( produc3851147774108065007at_num @ X2 @ Y ) @ ( produc3851147774108065007at_num @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_392_Pair__mono,axiom,
! [X2: set_set_nat,X5: set_set_nat,Y: nat,Y6: nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ X5 )
=> ( ( ord_less_eq_nat @ Y @ Y6 )
=> ( ord_le4236890226112434445at_nat @ ( produc7293815987952286373at_nat @ X2 @ Y ) @ ( produc7293815987952286373at_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_393_Pair__mono,axiom,
! [X2: num,X5: num,Y: set_set_nat,Y6: set_set_nat] :
( ( ord_less_eq_num @ X2 @ X5 )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Y6 )
=> ( ord_le8212681105015131479et_nat @ ( produc5263849632999935215et_nat @ X2 @ Y ) @ ( produc5263849632999935215et_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_394_Pair__mono,axiom,
! [X2: num,X5: num,Y: nat > nat,Y6: nat > nat] :
( ( ord_less_eq_num @ X2 @ X5 )
=> ( ( ord_less_eq_nat_nat @ Y @ Y6 )
=> ( ord_le1215176170919080154at_nat @ ( produc5125486429189257586at_nat @ X2 @ Y ) @ ( produc5125486429189257586at_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_395_Pair__mono,axiom,
! [X2: nat,X5: nat,Y: set_set_nat,Y6: set_set_nat] :
( ( ord_less_eq_nat @ X2 @ X5 )
=> ( ( ord_le6893508408891458716et_nat @ Y @ Y6 )
=> ( ord_le1758471263074804493et_nat @ ( produc8033011827914384037et_nat @ X2 @ Y ) @ ( produc8033011827914384037et_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_396_Pair__mono,axiom,
! [X2: nat,X5: nat,Y: nat > nat,Y6: nat > nat] :
( ( ord_less_eq_nat @ X2 @ X5 )
=> ( ( ord_less_eq_nat_nat @ Y @ Y6 )
=> ( ord_le3929206603849117072at_nat @ ( produc7839516862119294504at_nat @ X2 @ Y ) @ ( produc7839516862119294504at_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_397_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_398_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_399_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_400_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_401_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_402_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_403_fst__mono,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ X2 @ Y )
=> ( ord_le9131159989063066194et_nat @ ( produc6523417423482510407at_nat @ X2 ) @ ( produc6523417423482510407at_nat @ Y ) ) ) ).
% fst_mono
thf(fact_404_sndI,axiom,
! [X2: produc4045820344675478307at_nat,Y: set_set_set_nat,Z2: nat] :
( ( X2
= ( produc2803780273060847707at_nat @ Y @ Z2 ) )
=> ( ( produc8987496658364038281at_nat @ X2 )
= Z2 ) ) ).
% sndI
thf(fact_405_second__assumptions_Odeviate__neg__cup_Ocong,axiom,
clique1591571987439376245eg_cup = clique1591571987439376245eg_cup ).
% second_assumptions.deviate_neg_cup.cong
thf(fact_406_second__assumptions_Odeviate__neg__cap_Ocong,axiom,
clique1591571987438064265eg_cap = clique1591571987438064265eg_cap ).
% second_assumptions.deviate_neg_cap.cong
thf(fact_407_snd__eqD,axiom,
! [X2: set_set_set_nat,Y: nat,A2: nat] :
( ( ( produc8987496658364038281at_nat @ ( produc2803780273060847707at_nat @ X2 @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_408_snd__conv,axiom,
! [X1: set_set_set_nat,X22: nat] :
( ( produc8987496658364038281at_nat @ ( produc2803780273060847707at_nat @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_409_eq__snd__iff,axiom,
! [B2: nat,P: produc4045820344675478307at_nat] :
( ( B2
= ( produc8987496658364038281at_nat @ P ) )
= ( ? [A3: set_set_set_nat] :
( P
= ( produc2803780273060847707at_nat @ A3 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_410_prod__eq__iff,axiom,
( ( ^ [Y5: produc4045820344675478307at_nat,Z3: produc4045820344675478307at_nat] : ( Y5 = Z3 ) )
= ( ^ [S: produc4045820344675478307at_nat,T: produc4045820344675478307at_nat] :
( ( ( produc6523417423482510407at_nat @ S )
= ( produc6523417423482510407at_nat @ T ) )
& ( ( produc8987496658364038281at_nat @ S )
= ( produc8987496658364038281at_nat @ T ) ) ) ) ) ).
% prod_eq_iff
thf(fact_411_prod__eqI,axiom,
! [P: produc4045820344675478307at_nat,Q2: produc4045820344675478307at_nat] :
( ( ( produc6523417423482510407at_nat @ P )
= ( produc6523417423482510407at_nat @ Q2 ) )
=> ( ( ( produc8987496658364038281at_nat @ P )
= ( produc8987496658364038281at_nat @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_412_prod_Oexpand,axiom,
! [Prod: produc4045820344675478307at_nat,Prod2: produc4045820344675478307at_nat] :
( ( ( ( produc6523417423482510407at_nat @ Prod )
= ( produc6523417423482510407at_nat @ Prod2 ) )
& ( ( produc8987496658364038281at_nat @ Prod )
= ( produc8987496658364038281at_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_413_sup_OcoboundedI2,axiom,
! [C: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ B2 )
=> ( ord_le9131159989063066194et_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_414_sup_OcoboundedI2,axiom,
! [C: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ C @ B2 )
=> ( ord_le2960050975727596227at_nat @ C @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_415_sup_OcoboundedI2,axiom,
! [C: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ B2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_416_sup_OcoboundedI2,axiom,
! [C: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ B2 )
=> ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_417_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_418_sup_OcoboundedI2,axiom,
! [C: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ C @ B2 )
=> ( ord_less_eq_nat_nat @ C @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_419_sup_OcoboundedI1,axiom,
! [C: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ A2 )
=> ( ord_le9131159989063066194et_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_420_sup_OcoboundedI1,axiom,
! [C: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ C @ A2 )
=> ( ord_le2960050975727596227at_nat @ C @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_421_sup_OcoboundedI1,axiom,
! [C: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_422_sup_OcoboundedI1,axiom,
! [C: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_423_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_424_sup_OcoboundedI1,axiom,
! [C: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ C @ A2 )
=> ( ord_less_eq_nat_nat @ C @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_425_sup_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_426_sup_Oabsorb__iff2,axiom,
( ord_le2960050975727596227at_nat
= ( ^ [A3: produc4045820344675478307at_nat,B3: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_427_sup_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_428_sup_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_429_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_430_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [A3: nat > nat,B3: nat > nat] :
( ( sup_sup_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_431_sup_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B3: set_set_set_nat,A3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_432_sup_Oabsorb__iff1,axiom,
( ord_le2960050975727596227at_nat
= ( ^ [B3: produc4045820344675478307at_nat,A3: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_433_sup_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B3: set_set_nat,A3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_434_sup_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B3: set_nat_nat,A3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_435_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_436_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [B3: nat > nat,A3: nat > nat] :
( ( sup_sup_nat_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_437_sup_Ocobounded2,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_438_sup_Ocobounded2,axiom,
! [B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ B2 @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_439_sup_Ocobounded2,axiom,
! [B2: set_set_nat,A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_440_sup_Ocobounded2,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_441_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_442_sup_Ocobounded2,axiom,
! [B2: nat > nat,A2: nat > nat] : ( ord_less_eq_nat_nat @ B2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_443_sup_Ocobounded1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_444_sup_Ocobounded1,axiom,
! [A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ A2 @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_445_sup_Ocobounded1,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_446_sup_Ocobounded1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_447_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_448_sup_Ocobounded1,axiom,
! [A2: nat > nat,B2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_449_sup_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B3: set_set_set_nat,A3: set_set_set_nat] :
( A3
= ( sup_su4213647025997063966et_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_450_sup_Oorder__iff,axiom,
( ord_le2960050975727596227at_nat
= ( ^ [B3: produc4045820344675478307at_nat,A3: produc4045820344675478307at_nat] :
( A3
= ( sup_su2809688525313048311at_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_451_sup_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B3: set_set_nat,A3: set_set_nat] :
( A3
= ( sup_sup_set_set_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_452_sup_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B3: set_nat_nat,A3: set_nat_nat] :
( A3
= ( sup_sup_set_nat_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_453_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_454_sup_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [B3: nat > nat,A3: nat > nat] :
( A3
= ( sup_sup_nat_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_455_sup_OboundedI,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_456_sup_OboundedI,axiom,
! [B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat,C: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ B2 @ A2 )
=> ( ( ord_le2960050975727596227at_nat @ C @ A2 )
=> ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_457_sup_OboundedI,axiom,
! [B2: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_458_sup_OboundedI,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_459_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_460_sup_OboundedI,axiom,
! [B2: nat > nat,A2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ C @ A2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_461_sup_OboundedE,axiom,
! [B2: set_set_set_nat,C: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ~ ( ord_le9131159989063066194et_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_462_sup_OboundedE,axiom,
! [B2: produc4045820344675478307at_nat,C: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le2960050975727596227at_nat @ B2 @ A2 )
=> ~ ( ord_le2960050975727596227at_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_463_sup_OboundedE,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_464_sup_OboundedE,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_465_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_466_sup_OboundedE,axiom,
! [B2: nat > nat,C: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_467_sup__absorb2,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_468_sup__absorb2,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ X2 @ Y )
=> ( ( sup_su2809688525313048311at_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_469_sup__absorb2,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( sup_sup_set_set_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_470_sup__absorb2,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_471_sup__absorb2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_472_sup__absorb2,axiom,
! [X2: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y )
=> ( ( sup_sup_nat_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_473_sup__absorb1,axiom,
! [Y: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X2 )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_474_sup__absorb1,axiom,
! [Y: produc4045820344675478307at_nat,X2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ Y @ X2 )
=> ( ( sup_su2809688525313048311at_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_475_sup__absorb1,axiom,
! [Y: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( ( sup_sup_set_set_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_476_sup__absorb1,axiom,
! [Y: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X2 )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_477_sup__absorb1,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_478_sup__absorb1,axiom,
! [Y: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X2 )
=> ( ( sup_sup_nat_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_479_sup_Oabsorb2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_480_sup_Oabsorb2,axiom,
! [A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ A2 @ B2 )
=> ( ( sup_su2809688525313048311at_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_481_sup_Oabsorb2,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_482_sup_Oabsorb2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_483_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_484_sup_Oabsorb2,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( sup_sup_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_485_sup_Oabsorb1,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_486_sup_Oabsorb1,axiom,
! [B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ B2 @ A2 )
=> ( ( sup_su2809688525313048311at_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_487_sup_Oabsorb1,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( sup_sup_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_488_sup_Oabsorb1,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_489_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_490_sup_Oabsorb1,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( sup_sup_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_491_sup__unique,axiom,
! [F: set_set_set_nat > set_set_set_nat > set_set_set_nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ! [X4: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat,Z4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y4 @ X4 )
=> ( ( ord_le9131159989063066194et_nat @ Z4 @ X4 )
=> ( ord_le9131159989063066194et_nat @ ( F @ Y4 @ Z4 ) @ X4 ) ) )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_492_sup__unique,axiom,
! [F: produc4045820344675478307at_nat > produc4045820344675478307at_nat > produc4045820344675478307at_nat,X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ! [X4: produc4045820344675478307at_nat,Y4: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: produc4045820344675478307at_nat,Y4: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: produc4045820344675478307at_nat,Y4: produc4045820344675478307at_nat,Z4: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ Y4 @ X4 )
=> ( ( ord_le2960050975727596227at_nat @ Z4 @ X4 )
=> ( ord_le2960050975727596227at_nat @ ( F @ Y4 @ Z4 ) @ X4 ) ) )
=> ( ( sup_su2809688525313048311at_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_493_sup__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X2: set_set_nat,Y: set_set_nat] :
( ! [X4: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_nat,Y4: set_set_nat,Z4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y4 @ X4 )
=> ( ( ord_le6893508408891458716et_nat @ Z4 @ X4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ Y4 @ Z4 ) @ X4 ) ) )
=> ( ( sup_sup_set_set_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_494_sup__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X2: set_nat_nat,Y: set_nat_nat] :
( ! [X4: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat,Z4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y4 @ X4 )
=> ( ( ord_le9059583361652607317at_nat @ Z4 @ X4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ Y4 @ Z4 ) @ X4 ) ) )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_495_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_nat @ Z4 @ X4 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z4 ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_496_sup__unique,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,X2: nat > nat,Y: nat > nat] :
( ! [X4: nat > nat,Y4: nat > nat] : ( ord_less_eq_nat_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat > nat,Y4: nat > nat] : ( ord_less_eq_nat_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat > nat,Y4: nat > nat,Z4: nat > nat] :
( ( ord_less_eq_nat_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_nat_nat @ Z4 @ X4 )
=> ( ord_less_eq_nat_nat @ ( F @ Y4 @ Z4 ) @ X4 ) ) )
=> ( ( sup_sup_nat_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_497_sup_OorderI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_498_sup_OorderI,axiom,
! [A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( A2
= ( sup_su2809688525313048311at_nat @ A2 @ B2 ) )
=> ( ord_le2960050975727596227at_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_499_sup_OorderI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_500_sup_OorderI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2
= ( sup_sup_set_nat_nat @ A2 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_501_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_502_sup_OorderI,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( A2
= ( sup_sup_nat_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_503_sup_OorderE,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( A2
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_504_sup_OorderE,axiom,
! [B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ B2 @ A2 )
=> ( A2
= ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_505_sup_OorderE,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_506_sup_OorderE,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_507_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_508_sup_OorderE,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_509_le__iff__sup,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_510_le__iff__sup,axiom,
( ord_le2960050975727596227at_nat
= ( ^ [X3: produc4045820344675478307at_nat,Y3: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_511_le__iff__sup,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y3: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_512_le__iff__sup,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_513_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( sup_sup_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_514_le__iff__sup,axiom,
( ord_less_eq_nat_nat
= ( ^ [X3: nat > nat,Y3: nat > nat] :
( ( sup_sup_nat_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_515_sup__least,axiom,
! [Y: set_set_set_nat,X2: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X2 )
=> ( ( ord_le9131159989063066194et_nat @ Z2 @ X2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_516_sup__least,axiom,
! [Y: produc4045820344675478307at_nat,X2: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ Y @ X2 )
=> ( ( ord_le2960050975727596227at_nat @ Z2 @ X2 )
=> ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_517_sup__least,axiom,
! [Y: set_set_nat,X2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ Z2 @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_518_sup__least,axiom,
! [Y: set_nat_nat,X2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ Z2 @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_519_sup__least,axiom,
! [Y: nat,X2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_520_sup__least,axiom,
! [Y: nat > nat,X2: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_521_sup__mono,axiom,
! [A2: set_set_set_nat,C: set_set_set_nat,B2: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ ( sup_su4213647025997063966et_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_522_sup__mono,axiom,
! [A2: produc4045820344675478307at_nat,C: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat,D: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ A2 @ C )
=> ( ( ord_le2960050975727596227at_nat @ B2 @ D )
=> ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) @ ( sup_su2809688525313048311at_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_523_sup__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_524_sup__mono,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ ( sup_sup_set_nat_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_525_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_526_sup__mono,axiom,
! [A2: nat > nat,C: nat > nat,B2: nat > nat,D: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C )
=> ( ( ord_less_eq_nat_nat @ B2 @ D )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ ( sup_sup_nat_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_527_sup_Omono,axiom,
! [C: set_set_set_nat,A2: set_set_set_nat,D: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ D @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ C @ D ) @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_528_sup_Omono,axiom,
! [C: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat,D: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ C @ A2 )
=> ( ( ord_le2960050975727596227at_nat @ D @ B2 )
=> ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ C @ D ) @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_529_sup_Omono,axiom,
! [C: set_set_nat,A2: set_set_nat,D: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ D @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ C @ D ) @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_530_sup_Omono,axiom,
! [C: set_nat_nat,A2: set_nat_nat,D: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ D @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ C @ D ) @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_531_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_532_sup_Omono,axiom,
! [C: nat > nat,A2: nat > nat,D: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ C @ A2 )
=> ( ( ord_less_eq_nat_nat @ D @ B2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ C @ D ) @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_533_le__supI2,axiom,
! [X2: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_534_le__supI2,axiom,
! [X2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ X2 @ B2 )
=> ( ord_le2960050975727596227at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_535_le__supI2,axiom,
! [X2: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_536_le__supI2,axiom,
! [X2: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_537_le__supI2,axiom,
! [X2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_538_le__supI2,axiom,
! [X2: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ B2 )
=> ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_539_le__supI1,axiom,
! [X2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_540_le__supI1,axiom,
! [X2: produc4045820344675478307at_nat,A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ X2 @ A2 )
=> ( ord_le2960050975727596227at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_541_le__supI1,axiom,
! [X2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_542_le__supI1,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_543_le__supI1,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_544_le__supI1,axiom,
! [X2: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ A2 )
=> ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_545_sup__ge2,axiom,
! [Y: set_set_set_nat,X2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_546_sup__ge2,axiom,
! [Y: produc4045820344675478307at_nat,X2: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ Y @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_547_sup__ge2,axiom,
! [Y: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( sup_sup_set_set_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_548_sup__ge2,axiom,
! [Y: set_nat_nat,X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y @ ( sup_sup_set_nat_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_549_sup__ge2,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_550_sup__ge2,axiom,
! [Y: nat > nat,X2: nat > nat] : ( ord_less_eq_nat_nat @ Y @ ( sup_sup_nat_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_551_sup__ge1,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_552_sup__ge1,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_553_sup__ge1,axiom,
! [X2: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_554_sup__ge1,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_555_sup__ge1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_556_sup__ge1,axiom,
! [X2: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_557_le__supI,axiom,
! [A2: set_set_set_nat,X2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ X2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ X2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_558_le__supI,axiom,
! [A2: produc4045820344675478307at_nat,X2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ A2 @ X2 )
=> ( ( ord_le2960050975727596227at_nat @ B2 @ X2 )
=> ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_559_le__supI,axiom,
! [A2: set_set_nat,X2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_560_le__supI,axiom,
! [A2: set_nat_nat,X2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_561_le__supI,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_562_le__supI,axiom,
! [A2: nat > nat,X2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ X2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_563_le__supE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ X2 )
=> ~ ( ord_le9131159989063066194et_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_564_le__supE,axiom,
! [A2: produc4045820344675478307at_nat,B2: produc4045820344675478307at_nat,X2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le2960050975727596227at_nat @ A2 @ X2 )
=> ~ ( ord_le2960050975727596227at_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_565_le__supE,axiom,
! [A2: set_set_nat,B2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ X2 )
=> ~ ( ord_le6893508408891458716et_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_566_le__supE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ X2 )
=> ~ ( ord_le9059583361652607317at_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_567_le__supE,axiom,
! [A2: nat,B2: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_568_le__supE,axiom,
! [A2: nat > nat,B2: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_nat_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_569_inf__sup__ord_I3_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_570_inf__sup__ord_I3_J,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_571_inf__sup__ord_I3_J,axiom,
! [X2: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_572_inf__sup__ord_I3_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_573_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_574_inf__sup__ord_I3_J,axiom,
! [X2: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_575_inf__sup__ord_I4_J,axiom,
! [Y: set_set_set_nat,X2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_576_inf__sup__ord_I4_J,axiom,
! [Y: produc4045820344675478307at_nat,X2: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ Y @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_577_inf__sup__ord_I4_J,axiom,
! [Y: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y @ ( sup_sup_set_set_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_578_inf__sup__ord_I4_J,axiom,
! [Y: set_nat_nat,X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y @ ( sup_sup_set_nat_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_579_inf__sup__ord_I4_J,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_580_inf__sup__ord_I4_J,axiom,
! [Y: nat > nat,X2: nat > nat] : ( ord_less_eq_nat_nat @ Y @ ( sup_sup_nat_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_581_Un__Diff,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ C2 ) @ ( minus_2163939370556025621et_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_582_Un__Diff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ C2 ) @ ( minus_8121590178497047118at_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_583_Un__Diff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C2 )
= ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ A @ C2 ) @ ( minus_2447799839930672331et_nat @ B @ C2 ) ) ) ).
% Un_Diff
thf(fact_584_subset__Un__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_585_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_586_subset__Un__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_587_subset__UnE,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ~ ! [A7: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A7 @ A )
=> ! [B7: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B7 @ B )
=> ( C2
!= ( sup_su4213647025997063966et_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_588_subset__UnE,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ~ ! [A7: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A7 @ A )
=> ! [B7: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B7 @ B )
=> ( C2
!= ( sup_sup_set_set_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_589_subset__UnE,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ~ ! [A7: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A7 @ A )
=> ! [B7: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B7 @ B )
=> ( C2
!= ( sup_sup_set_nat_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_590_Un__absorb2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( sup_su4213647025997063966et_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_591_Un__absorb2,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( sup_sup_set_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_592_Un__absorb2,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( sup_sup_set_nat_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_593_Un__absorb1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( sup_su4213647025997063966et_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_594_Un__absorb1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( sup_sup_set_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_595_Un__absorb1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( sup_sup_set_nat_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_596_Un__upper2,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ B @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_597_Un__upper2,axiom,
! [B: set_set_nat,A: set_set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_598_Un__upper2,axiom,
! [B: set_nat_nat,A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ B @ ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_599_Un__upper1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_600_Un__upper1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( sup_sup_set_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_601_Un__upper1,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ ( sup_sup_set_nat_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_602_Un__least,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_603_Un__least,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_604_Un__least,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_605_Un__mono,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_606_Un__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_607_Un__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_608_first__assumptions_Ojoinl__join,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K @ X @ Y2 ) @ ( clique5469973757772500719t_odot @ X @ Y2 ) ) ) ).
% first_assumptions.joinl_join
thf(fact_609_second__assumptions_Odeviate__neg__cap__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique1591571987438064265eg_cap @ L @ P @ K @ X @ Y2 )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118206219037_sqcap @ L @ P @ K @ X @ Y2 ) ) @ ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X @ Y2 ) ) ) ) ) ).
% second_assumptions.deviate_neg_cap_def
thf(fact_610_surjective__pairing,axiom,
! [T2: produc4045820344675478307at_nat] :
( T2
= ( produc2803780273060847707at_nat @ ( produc6523417423482510407at_nat @ T2 ) @ ( produc8987496658364038281at_nat @ T2 ) ) ) ).
% surjective_pairing
thf(fact_611_prod_Oexhaust__sel,axiom,
! [Prod: produc4045820344675478307at_nat] :
( Prod
= ( produc2803780273060847707at_nat @ ( produc6523417423482510407at_nat @ Prod ) @ ( produc8987496658364038281at_nat @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_612_second__assumptions_Oaxioms_I2_J,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( assump8934899134041091456axioms @ L @ K ) ) ).
% second_assumptions.axioms(2)
thf(fact_613_second__assumptions_Odeviate__neg__cup__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique1591571987439376245eg_cup @ L @ P @ K @ X @ Y2 )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X @ Y2 ) ) @ ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ) ) ) ).
% second_assumptions.deviate_neg_cup_def
thf(fact_614_second__assumptions_OLm,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ord_less_eq_nat @ ( assump1710595444109740334irst_m @ K ) @ ( assump1710595444109740301irst_L @ L @ P ) ) ) ).
% second_assumptions.Lm
thf(fact_615_fst__diff,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( produc6523417423482510407at_nat @ ( minus_5633088680627868938at_nat @ X2 @ Y ) )
= ( minus_2447799839930672331et_nat @ ( produc6523417423482510407at_nat @ X2 ) @ ( produc6523417423482510407at_nat @ Y ) ) ) ).
% fst_diff
thf(fact_616_diff__Pair,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat,D: set_nat_nat] :
( ( minus_1278274433303930802at_nat @ ( produc8650507651752646531at_nat @ A2 @ B2 ) @ ( produc8650507651752646531at_nat @ C @ D ) )
= ( produc8650507651752646531at_nat @ ( minus_8121590178497047118at_nat @ A2 @ C ) @ ( minus_8121590178497047118at_nat @ B2 @ D ) ) ) ).
% diff_Pair
thf(fact_617_diff__Pair,axiom,
! [A2: set_nat_nat,B2: nat,C: set_nat_nat,D: nat] :
( ( minus_8086599166786498573at_nat @ ( produc4994002345568745310at_nat @ A2 @ B2 ) @ ( produc4994002345568745310at_nat @ C @ D ) )
= ( produc4994002345568745310at_nat @ ( minus_8121590178497047118at_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ D ) ) ) ).
% diff_Pair
thf(fact_618_diff__Pair,axiom,
! [A2: set_nat_nat,B2: set_set_set_nat,C: set_nat_nat,D: set_set_set_nat] :
( ( minus_840401501723538735et_nat @ ( produc2168915450337631616et_nat @ A2 @ B2 ) @ ( produc2168915450337631616et_nat @ C @ D ) )
= ( produc2168915450337631616et_nat @ ( minus_8121590178497047118at_nat @ A2 @ C ) @ ( minus_2447799839930672331et_nat @ B2 @ D ) ) ) ).
% diff_Pair
thf(fact_619_diff__Pair,axiom,
! [A2: nat,B2: set_nat_nat,C: nat,D: set_nat_nat] :
( ( minus_4865422370612786701at_nat @ ( produc6017321930131081694at_nat @ A2 @ B2 ) @ ( produc6017321930131081694at_nat @ C @ D ) )
= ( produc6017321930131081694at_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_8121590178497047118at_nat @ B2 @ D ) ) ) ).
% diff_Pair
thf(fact_620_diff__Pair,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( minus_4365393887724441320at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ C @ D ) )
= ( product_Pair_nat_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ D ) ) ) ).
% diff_Pair
thf(fact_621_diff__Pair,axiom,
! [A2: nat,B2: set_set_set_nat,C: nat,D: set_set_set_nat] :
( ( minus_5802666374030397450et_nat @ ( produc1530211740221758555et_nat @ A2 @ B2 ) @ ( produc1530211740221758555et_nat @ C @ D ) )
= ( produc1530211740221758555et_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_2447799839930672331et_nat @ B2 @ D ) ) ) ).
% diff_Pair
thf(fact_622_diff__Pair,axiom,
! [A2: set_set_set_nat,B2: set_nat_nat,C: set_set_set_nat,D: set_nat_nat] :
( ( minus_2715004213046008111at_nat @ ( produc507788731782991104at_nat @ A2 @ B2 ) @ ( produc507788731782991104at_nat @ C @ D ) )
= ( produc507788731782991104at_nat @ ( minus_2447799839930672331et_nat @ A2 @ C ) @ ( minus_8121590178497047118at_nat @ B2 @ D ) ) ) ).
% diff_Pair
thf(fact_623_diff__Pair,axiom,
! [A2: set_set_set_nat,B2: nat,C: set_set_set_nat,D: nat] :
( ( minus_5633088680627868938at_nat @ ( produc2803780273060847707at_nat @ A2 @ B2 ) @ ( produc2803780273060847707at_nat @ C @ D ) )
= ( produc2803780273060847707at_nat @ ( minus_2447799839930672331et_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ D ) ) ) ).
% diff_Pair
thf(fact_624_diff__Pair,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat,D: set_set_set_nat] :
( ( minus_5310655894399352364et_nat @ ( produc8443863378681539197et_nat @ A2 @ B2 ) @ ( produc8443863378681539197et_nat @ C @ D ) )
= ( produc8443863378681539197et_nat @ ( minus_2447799839930672331et_nat @ A2 @ C ) @ ( minus_2447799839930672331et_nat @ B2 @ D ) ) ) ).
% diff_Pair
thf(fact_625_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_626_minus__prod__def,axiom,
( minus_1278274433303930802at_nat
= ( ^ [X3: produc1300680692302638795at_nat,Y3: produc1300680692302638795at_nat] : ( produc8650507651752646531at_nat @ ( minus_8121590178497047118at_nat @ ( produc7281425617276542831at_nat @ X3 ) @ ( produc7281425617276542831at_nat @ Y3 ) ) @ ( minus_8121590178497047118at_nat @ ( produc5975371700627812785at_nat @ X3 ) @ ( produc5975371700627812785at_nat @ Y3 ) ) ) ) ) ).
% minus_prod_def
thf(fact_627_minus__prod__def,axiom,
( minus_8086599166786498573at_nat
= ( ^ [X3: produc4412711793163744422at_nat,Y3: produc4412711793163744422at_nat] : ( produc4994002345568745310at_nat @ ( minus_8121590178497047118at_nat @ ( produc8454637311967854922at_nat @ X3 ) @ ( produc8454637311967854922at_nat @ Y3 ) ) @ ( minus_minus_nat @ ( produc6822102643625338508at_nat @ X3 ) @ ( produc6822102643625338508at_nat @ Y3 ) ) ) ) ) ).
% minus_prod_def
thf(fact_628_minus__prod__def,axiom,
( minus_840401501723538735et_nat
= ( ^ [X3: produc2795196907576053192et_nat,Y3: produc2795196907576053192et_nat] : ( produc2168915450337631616et_nat @ ( minus_8121590178497047118at_nat @ ( produc8444235278545442668et_nat @ X3 ) @ ( produc8444235278545442668et_nat @ Y3 ) ) @ ( minus_2447799839930672331et_nat @ ( produc2415352471775012014et_nat @ X3 ) @ ( produc2415352471775012014et_nat @ Y3 ) ) ) ) ) ).
% minus_prod_def
thf(fact_629_minus__prod__def,axiom,
( minus_4865422370612786701at_nat
= ( ^ [X3: produc1191534996990032550at_nat,Y3: produc1191534996990032550at_nat] : ( produc6017321930131081694at_nat @ ( minus_minus_nat @ ( produc254584859675415498at_nat @ X3 ) @ ( produc254584859675415498at_nat @ Y3 ) ) @ ( minus_8121590178497047118at_nat @ ( produc7845422228187674892at_nat @ X3 ) @ ( produc7845422228187674892at_nat @ Y3 ) ) ) ) ) ).
% minus_prod_def
thf(fact_630_minus__prod__def,axiom,
( minus_4365393887724441320at_nat
= ( ^ [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( product_Pair_nat_nat @ ( minus_minus_nat @ ( product_fst_nat_nat @ X3 ) @ ( product_fst_nat_nat @ Y3 ) ) @ ( minus_minus_nat @ ( product_snd_nat_nat @ X3 ) @ ( product_snd_nat_nat @ Y3 ) ) ) ) ) ).
% minus_prod_def
thf(fact_631_minus__prod__def,axiom,
( minus_5802666374030397450et_nat
= ( ^ [X3: produc4215398038078006819et_nat,Y3: produc4215398038078006819et_nat] : ( produc1530211740221758555et_nat @ ( minus_minus_nat @ ( produc5249848890643421255et_nat @ X3 ) @ ( produc5249848890643421255et_nat @ Y3 ) ) @ ( minus_2447799839930672331et_nat @ ( produc7713928125524949129et_nat @ X3 ) @ ( produc7713928125524949129et_nat @ Y3 ) ) ) ) ) ).
% minus_prod_def
thf(fact_632_minus__prod__def,axiom,
( minus_2715004213046008111at_nat
= ( ^ [X3: produc4669799618898522568at_nat,Y3: produc4669799618898522568at_nat] : ( produc507788731782991104at_nat @ ( minus_2447799839930672331et_nat @ ( produc6783108559990802156at_nat @ X3 ) @ ( produc6783108559990802156at_nat @ Y3 ) ) @ ( minus_8121590178497047118at_nat @ ( produc754225753220371502at_nat @ X3 ) @ ( produc754225753220371502at_nat @ Y3 ) ) ) ) ) ).
% minus_prod_def
thf(fact_633_minus__prod__def,axiom,
( minus_5633088680627868938at_nat
= ( ^ [X3: produc4045820344675478307at_nat,Y3: produc4045820344675478307at_nat] : ( produc2803780273060847707at_nat @ ( minus_2447799839930672331et_nat @ ( produc6523417423482510407at_nat @ X3 ) @ ( produc6523417423482510407at_nat @ Y3 ) ) @ ( minus_minus_nat @ ( produc8987496658364038281at_nat @ X3 ) @ ( produc8987496658364038281at_nat @ Y3 ) ) ) ) ) ).
% minus_prod_def
thf(fact_634_minus__prod__def,axiom,
( minus_5310655894399352364et_nat
= ( ^ [X3: produc4405103650892965957et_nat,Y3: produc4405103650892965957et_nat] : ( produc8443863378681539197et_nat @ ( minus_2447799839930672331et_nat @ ( produc1516831113742898793et_nat @ X3 ) @ ( produc1516831113742898793et_nat @ Y3 ) ) @ ( minus_2447799839930672331et_nat @ ( produc2905410560650206891et_nat @ X3 ) @ ( produc2905410560650206891et_nat @ Y3 ) ) ) ) ) ).
% minus_prod_def
thf(fact_635_ACC__odot,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X ) @ ( clique3210737319928189260st_ACC @ k @ Y2 ) ) ) ).
% ACC_odot
thf(fact_636_ACC__cf__mono,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_mono
thf(fact_637_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P2: set_set_set_nat > nat > $o,X2: set_set_set_nat,Y: nat,A2: produc4045820344675478307at_nat] :
( ( P2 @ X2 @ Y )
=> ( ( A2
= ( produc2803780273060847707at_nat @ X2 @ Y ) )
=> ( P2 @ ( produc6523417423482510407at_nat @ A2 ) @ ( produc8987496658364038281at_nat @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_638_inf_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_639_inf_Oidem,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_640_inf__idem,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_641_inf__idem,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_642_inf_Oleft__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_643_inf_Oleft__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_644_inf__left__idem,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) )
= ( inf_in5711780100303410308et_nat @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_645_inf__left__idem,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ X2 @ Y ) )
= ( inf_inf_set_nat_nat @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_646_inf_Oright__idem,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ B2 )
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_647_inf_Oright__idem,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_648_inf__right__idem,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ Y )
= ( inf_in5711780100303410308et_nat @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_649_inf__right__idem,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ Y )
= ( inf_inf_set_nat_nat @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_650_IntI,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ A )
=> ( ( member2946998982187404937et_nat @ C @ B )
=> ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_651_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_652_IntI,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A )
=> ( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_653_IntI,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_654_Int__iff,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) )
= ( ( member2946998982187404937et_nat @ C @ A )
& ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_655_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_656_Int__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
& ( member_set_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_657_Int__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
& ( member_nat_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_658_le__inf__iff,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) )
= ( ( ord_le9131159989063066194et_nat @ X2 @ Y )
& ( ord_le9131159989063066194et_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_659_le__inf__iff,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y @ Z2 ) )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Y )
& ( ord_le6893508408891458716et_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_660_le__inf__iff,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z2 ) )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Y )
& ( ord_le9059583361652607317at_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_661_le__inf__iff,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_662_le__inf__iff,axiom,
! [X2: nat > nat,Y: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat_nat @ X2 @ Y )
& ( ord_less_eq_nat_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_663_inf_Obounded__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) )
= ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_664_inf_Obounded__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_665_inf_Obounded__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_666_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_667_inf_Obounded__iff,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat_nat @ A2 @ B2 )
& ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_668_inf__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: nat,C: set_set_set_nat,D: nat] :
( ( inf_in8110749398470801937at_nat @ ( produc2803780273060847707at_nat @ A2 @ B2 ) @ ( produc2803780273060847707at_nat @ C @ D ) )
= ( produc2803780273060847707at_nat @ ( inf_in5711780100303410308et_nat @ A2 @ C ) @ ( inf_inf_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_669_inf__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat,D: set_set_set_nat] :
( ( inf_in7092662171172785971et_nat @ ( produc8443863378681539197et_nat @ A2 @ B2 ) @ ( produc8443863378681539197et_nat @ C @ D ) )
= ( produc8443863378681539197et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ C ) @ ( inf_in5711780100303410308et_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_670_inf__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_nat_nat,C: set_set_set_nat,D: set_nat_nat] :
( ( inf_in6765570788667771830at_nat @ ( produc507788731782991104at_nat @ A2 @ B2 ) @ ( produc507788731782991104at_nat @ C @ D ) )
= ( produc507788731782991104at_nat @ ( inf_in5711780100303410308et_nat @ A2 @ C ) @ ( inf_inf_set_nat_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_671_inf__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_set_set_nat,C: set_nat_nat,D: set_set_set_nat] :
( ( inf_in4890968077345302454et_nat @ ( produc2168915450337631616et_nat @ A2 @ B2 ) @ ( produc2168915450337631616et_nat @ C @ D ) )
= ( produc2168915450337631616et_nat @ ( inf_inf_set_nat_nat @ A2 @ C ) @ ( inf_in5711780100303410308et_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_672_inf__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat,D: set_nat_nat] :
( ( inf_in8487852146424888249at_nat @ ( produc8650507651752646531at_nat @ A2 @ B2 ) @ ( produc8650507651752646531at_nat @ C @ D ) )
= ( produc8650507651752646531at_nat @ ( inf_inf_set_nat_nat @ A2 @ C ) @ ( inf_inf_set_nat_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_673_Int__subset__iff,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( ord_le9131159989063066194et_nat @ C2 @ A )
& ( ord_le9131159989063066194et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_674_Int__subset__iff,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( ord_le6893508408891458716et_nat @ C2 @ A )
& ( ord_le6893508408891458716et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_675_Int__subset__iff,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C2 @ A )
& ( ord_le9059583361652607317at_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_676_inf__sup__absorb,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_677_inf__sup__absorb,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_678_inf__sup__absorb,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_679_inf__sup__absorb,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( inf_in8110749398470801937at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_680_sup__inf__absorb,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_681_sup__inf__absorb,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_682_sup__inf__absorb,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_683_sup__inf__absorb,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ X2 @ ( inf_in8110749398470801937at_nat @ X2 @ Y ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_684_Int__Un__eq_I4_J,axiom,
! [T3: set_set_set_nat,S2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ T3 @ ( inf_in5711780100303410308et_nat @ S2 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_685_Int__Un__eq_I4_J,axiom,
! [T3: set_nat_nat,S2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T3 @ ( inf_inf_set_nat_nat @ S2 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_686_Int__Un__eq_I4_J,axiom,
! [T3: set_set_nat,S2: set_set_nat] :
( ( sup_sup_set_set_nat @ T3 @ ( inf_inf_set_set_nat @ S2 @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_687_Int__Un__eq_I3_J,axiom,
! [S2: set_set_set_nat,T3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ S2 @ ( inf_in5711780100303410308et_nat @ S2 @ T3 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_688_Int__Un__eq_I3_J,axiom,
! [S2: set_nat_nat,T3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S2 @ ( inf_inf_set_nat_nat @ S2 @ T3 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_689_Int__Un__eq_I3_J,axiom,
! [S2: set_set_nat,T3: set_set_nat] :
( ( sup_sup_set_set_nat @ S2 @ ( inf_inf_set_set_nat @ S2 @ T3 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_690_Int__Un__eq_I2_J,axiom,
! [S2: set_set_set_nat,T3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_691_Int__Un__eq_I2_J,axiom,
! [S2: set_nat_nat,T3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_692_Int__Un__eq_I2_J,axiom,
! [S2: set_set_nat,T3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_693_Int__Un__eq_I1_J,axiom,
! [S2: set_set_set_nat,T3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_694_Int__Un__eq_I1_J,axiom,
! [S2: set_nat_nat,T3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_695_Int__Un__eq_I1_J,axiom,
! [S2: set_set_nat,T3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_696_Un__Int__eq_I4_J,axiom,
! [T3: set_set_set_nat,S2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ T3 @ ( sup_su4213647025997063966et_nat @ S2 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_697_Un__Int__eq_I4_J,axiom,
! [T3: set_nat_nat,S2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T3 @ ( sup_sup_set_nat_nat @ S2 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_698_Un__Int__eq_I4_J,axiom,
! [T3: set_set_nat,S2: set_set_nat] :
( ( inf_inf_set_set_nat @ T3 @ ( sup_sup_set_set_nat @ S2 @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_699_Un__Int__eq_I3_J,axiom,
! [S2: set_set_set_nat,T3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ S2 @ ( sup_su4213647025997063966et_nat @ S2 @ T3 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_700_Un__Int__eq_I3_J,axiom,
! [S2: set_nat_nat,T3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S2 @ ( sup_sup_set_nat_nat @ S2 @ T3 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_701_Un__Int__eq_I3_J,axiom,
! [S2: set_set_nat,T3: set_set_nat] :
( ( inf_inf_set_set_nat @ S2 @ ( sup_sup_set_set_nat @ S2 @ T3 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_702_Un__Int__eq_I2_J,axiom,
! [S2: set_set_set_nat,T3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_703_Un__Int__eq_I2_J,axiom,
! [S2: set_nat_nat,T3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_704_Un__Int__eq_I2_J,axiom,
! [S2: set_set_nat,T3: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S2 @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_705_Un__Int__eq_I1_J,axiom,
! [S2: set_set_set_nat,T3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_706_Un__Int__eq_I1_J,axiom,
! [S2: set_nat_nat,T3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_707_Un__Int__eq_I1_J,axiom,
! [S2: set_set_nat,T3: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S2 @ T3 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_708_fst__inf,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat] :
( ( produc6523417423482510407at_nat @ ( inf_in8110749398470801937at_nat @ X2 @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( produc6523417423482510407at_nat @ X2 ) @ ( produc6523417423482510407at_nat @ Y ) ) ) ).
% fst_inf
thf(fact_709_inf__sup__aci_I4_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) )
= ( inf_in5711780100303410308et_nat @ X2 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_710_inf__sup__aci_I4_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ X2 @ Y ) )
= ( inf_inf_set_nat_nat @ X2 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_711_inf__sup__aci_I3_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ Y @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_712_inf__sup__aci_I3_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat_nat @ Y @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_713_inf__sup__aci_I2_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ Z2 )
= ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_714_inf__sup__aci_I2_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ Z2 )
= ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_715_inf__sup__aci_I1_J,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_716_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat_nat
= ( ^ [X3: set_nat_nat,Y3: set_nat_nat] : ( inf_inf_set_nat_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_717_inf_Oassoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_718_inf_Oassoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_719_inf__assoc,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ Z2 )
= ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_720_inf__assoc,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ Z2 )
= ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_721_inf_Ocommute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_722_inf_Ocommute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] : ( inf_inf_set_nat_nat @ B3 @ A3 ) ) ) ).
% inf.commute
thf(fact_723_inf__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_724_inf__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [X3: set_nat_nat,Y3: set_nat_nat] : ( inf_inf_set_nat_nat @ Y3 @ X3 ) ) ) ).
% inf_commute
thf(fact_725_inf_Oleft__commute,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ B2 @ ( inf_in5711780100303410308et_nat @ A2 @ C ) )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_726_inf_Oleft__commute,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ B2 @ ( inf_inf_set_nat_nat @ A2 @ C ) )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_727_inf__left__commute,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ Y @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_728_inf__left__commute,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat_nat @ Y @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_729_IntE,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) )
=> ~ ( ( member2946998982187404937et_nat @ C @ A )
=> ~ ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% IntE
thf(fact_730_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_731_IntE,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ~ ( ( member_set_set_nat @ C @ A )
=> ~ ( member_set_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_732_IntE,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ~ ( ( member_nat_nat @ C @ A )
=> ~ ( member_nat_nat @ C @ B ) ) ) ).
% IntE
thf(fact_733_IntD1,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) )
=> ( member2946998982187404937et_nat @ C @ A ) ) ).
% IntD1
thf(fact_734_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_735_IntD1,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ( member_set_set_nat @ C @ A ) ) ).
% IntD1
thf(fact_736_IntD1,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C @ A ) ) ).
% IntD1
thf(fact_737_IntD2,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) )
=> ( member2946998982187404937et_nat @ C @ B ) ) ).
% IntD2
thf(fact_738_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_739_IntD2,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ( member_set_set_nat @ C @ B ) ) ).
% IntD2
thf(fact_740_IntD2,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C @ B ) ) ).
% IntD2
thf(fact_741_Int__assoc,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_742_Int__assoc,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_743_Int__absorb,axiom,
! [A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_744_Int__absorb,axiom,
! [A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_745_Int__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_746_Int__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( inf_inf_set_nat_nat @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_747_Int__left__absorb,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( inf_in5711780100303410308et_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_748_Int__left__absorb,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( inf_inf_set_nat_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_749_Int__left__commute,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( inf_in5711780100303410308et_nat @ B @ ( inf_in5711780100303410308et_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_750_Int__left__commute,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ B @ ( inf_inf_set_nat_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_751_psubset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ B @ C2 )
=> ( ord_le152980574450754630et_nat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_752_psubsetD,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,C: set_set_set_nat] :
( ( ord_le52856854838348540et_nat @ A @ B )
=> ( ( member2946998982187404937et_nat @ C @ A )
=> ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_753_psubsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_754_psubsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_755_psubsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_756_inf__sup__ord_I2_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_757_inf__sup__ord_I2_J,axiom,
! [X2: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_758_inf__sup__ord_I2_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_759_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_760_inf__sup__ord_I2_J,axiom,
! [X2: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_761_inf__sup__ord_I1_J,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_762_inf__sup__ord_I1_J,axiom,
! [X2: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_763_inf__sup__ord_I1_J,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_764_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_765_inf__sup__ord_I1_J,axiom,
! [X2: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_766_inf__le1,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_767_inf__le1,axiom,
! [X2: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_768_inf__le1,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_769_inf__le1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_770_inf__le1,axiom,
! [X2: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_771_inf__le2,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_772_inf__le2,axiom,
! [X2: set_set_nat,Y: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_773_inf__le2,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_774_inf__le2,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_775_inf__le2,axiom,
! [X2: nat > nat,Y: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_776_le__infE,axiom,
! [X2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ X2 @ A2 )
=> ~ ( ord_le9131159989063066194et_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_777_le__infE,axiom,
! [X2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ X2 @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_778_le__infE,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ X2 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_779_le__infE,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_780_le__infE,axiom,
! [X2: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_781_le__infI,axiom,
! [X2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_782_le__infI,axiom,
! [X2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_783_le__infI,axiom,
! [X2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_784_le__infI,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_785_le__infI,axiom,
! [X2: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ B2 )
=> ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_786_inf__mono,axiom,
! [A2: set_set_set_nat,C: set_set_set_nat,B2: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ ( inf_in5711780100303410308et_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_787_inf__mono,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_788_inf__mono,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ ( inf_inf_set_nat_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_789_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_790_inf__mono,axiom,
! [A2: nat > nat,C: nat > nat,B2: nat > nat,D: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C )
=> ( ( ord_less_eq_nat_nat @ B2 @ D )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ ( inf_inf_nat_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_791_le__infI1,axiom,
! [A2: set_set_set_nat,X2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ X2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_792_le__infI1,axiom,
! [A2: set_set_nat,X2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_793_le__infI1,axiom,
! [A2: set_nat_nat,X2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_794_le__infI1,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_795_le__infI1,axiom,
! [A2: nat > nat,X2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ X2 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_796_le__infI2,axiom,
! [B2: set_set_set_nat,X2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ X2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_797_le__infI2,axiom,
! [B2: set_set_nat,X2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_798_le__infI2,axiom,
! [B2: set_nat_nat,X2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_799_le__infI2,axiom,
! [B2: nat,X2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_800_le__infI2,axiom,
! [B2: nat > nat,X2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ X2 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_801_inf_OorderE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( A2
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_802_inf_OorderE,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_803_inf_OorderE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_804_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_805_inf_OorderE,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_806_inf_OorderI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( A2
= ( inf_in5711780100303410308et_nat @ A2 @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_807_inf_OorderI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( A2
= ( inf_inf_set_set_nat @ A2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_808_inf_OorderI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( A2
= ( inf_inf_set_nat_nat @ A2 @ B2 ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_809_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_810_inf_OorderI,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( A2
= ( inf_inf_nat_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_811_inf__unique,axiom,
! [F: set_set_set_nat > set_set_set_nat > set_set_set_nat,X2: set_set_set_nat,Y: set_set_set_nat] :
( ! [X4: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F @ X4 @ Y4 ) @ X4 )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( F @ X4 @ Y4 ) @ Y4 )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat,Z4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y4 )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Z4 )
=> ( ord_le9131159989063066194et_nat @ X4 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_812_inf__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X2: set_set_nat,Y: set_set_nat] :
( ! [X4: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X4 @ Y4 ) @ X4 )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( F @ X4 @ Y4 ) @ Y4 )
=> ( ! [X4: set_set_nat,Y4: set_set_nat,Z4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ( ord_le6893508408891458716et_nat @ X4 @ Z4 )
=> ( ord_le6893508408891458716et_nat @ X4 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_set_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_813_inf__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X2: set_nat_nat,Y: set_nat_nat] :
( ! [X4: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X4 @ Y4 ) @ X4 )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( F @ X4 @ Y4 ) @ Y4 )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat,Z4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y4 )
=> ( ( ord_le9059583361652607317at_nat @ X4 @ Z4 )
=> ( ord_le9059583361652607317at_nat @ X4 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_814_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y4 ) @ X4 )
=> ( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y4 ) @ Y4 )
=> ( ! [X4: nat,Y4: nat,Z4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat @ X4 @ Z4 )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_815_inf__unique,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,X2: nat > nat,Y: nat > nat] :
( ! [X4: nat > nat,Y4: nat > nat] : ( ord_less_eq_nat_nat @ ( F @ X4 @ Y4 ) @ X4 )
=> ( ! [X4: nat > nat,Y4: nat > nat] : ( ord_less_eq_nat_nat @ ( F @ X4 @ Y4 ) @ Y4 )
=> ( ! [X4: nat > nat,Y4: nat > nat,Z4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat_nat @ X4 @ Z4 )
=> ( ord_less_eq_nat_nat @ X4 @ ( F @ Y4 @ Z4 ) ) ) )
=> ( ( inf_inf_nat_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_816_le__iff__inf,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X3: set_set_set_nat,Y3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_817_le__iff__inf,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y3: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_818_le__iff__inf,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_819_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_820_le__iff__inf,axiom,
( ord_less_eq_nat_nat
= ( ^ [X3: nat > nat,Y3: nat > nat] :
( ( inf_inf_nat_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_821_inf_Oabsorb1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_822_inf_Oabsorb1,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_823_inf_Oabsorb1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_824_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_825_inf_Oabsorb1,axiom,
! [A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( inf_inf_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_826_inf_Oabsorb2,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_827_inf_Oabsorb2,axiom,
! [B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_828_inf_Oabsorb2,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_829_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_830_inf_Oabsorb2,axiom,
! [B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( inf_inf_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_831_inf__absorb1,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_832_inf__absorb1,axiom,
! [X2: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( inf_inf_set_set_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_833_inf__absorb1,axiom,
! [X2: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_834_inf__absorb1,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( inf_inf_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_835_inf__absorb1,axiom,
! [X2: nat > nat,Y: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y )
=> ( ( inf_inf_nat_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_836_inf__absorb2,axiom,
! [Y: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y @ X2 )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_837_inf__absorb2,axiom,
! [Y: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( ( inf_inf_set_set_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_838_inf__absorb2,axiom,
! [Y: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X2 )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_839_inf__absorb2,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_840_inf__absorb2,axiom,
! [Y: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X2 )
=> ( ( inf_inf_nat_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_841_inf_OboundedE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ~ ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_842_inf_OboundedE,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_843_inf_OboundedE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_844_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_845_inf_OboundedE,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_846_inf_OboundedI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_847_inf_OboundedI,axiom,
! [A2: set_set_nat,B2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_848_inf_OboundedI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_849_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_850_inf_OboundedI,axiom,
! [A2: nat > nat,B2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_nat @ A2 @ C )
=> ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_851_inf__greatest,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Z2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_852_inf__greatest,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_853_inf__greatest,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_854_inf__greatest,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_855_inf__greatest,axiom,
! [X2: nat > nat,Y: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_856_inf_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( A3
= ( inf_in5711780100303410308et_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_857_inf_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( A3
= ( inf_inf_set_set_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_858_inf_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( A3
= ( inf_inf_set_nat_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_859_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_860_inf_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [A3: nat > nat,B3: nat > nat] :
( A3
= ( inf_inf_nat_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_861_inf_Ocobounded1,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_862_inf_Ocobounded1,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_863_inf_Ocobounded1,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_864_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_865_inf_Ocobounded1,axiom,
! [A2: nat > nat,B2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_866_inf_Ocobounded2,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_867_inf_Ocobounded2,axiom,
! [A2: set_set_nat,B2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_868_inf_Ocobounded2,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_869_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_870_inf_Ocobounded2,axiom,
! [A2: nat > nat,B2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_871_inf_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_872_inf_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( inf_inf_set_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_873_inf_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_874_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_875_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [A3: nat > nat,B3: nat > nat] :
( ( inf_inf_nat_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_876_inf_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B3: set_set_set_nat,A3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_877_inf_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B3: set_set_nat,A3: set_set_nat] :
( ( inf_inf_set_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_878_inf_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B3: set_nat_nat,A3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_879_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_880_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [B3: nat > nat,A3: nat > nat] :
( ( inf_inf_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_881_inf_OcoboundedI1,axiom,
! [A2: set_set_set_nat,C: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_882_inf_OcoboundedI1,axiom,
! [A2: set_set_nat,C: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_883_inf_OcoboundedI1,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_884_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_885_inf_OcoboundedI1,axiom,
! [A2: nat > nat,C: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_886_inf_OcoboundedI2,axiom,
! [B2: set_set_set_nat,C: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ C )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_887_inf_OcoboundedI2,axiom,
! [B2: set_set_nat,C: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ C )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_888_inf_OcoboundedI2,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ C )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_889_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_890_inf_OcoboundedI2,axiom,
! [B2: nat > nat,C: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_891_inf_Ostrict__coboundedI2,axiom,
! [B2: set_nat_nat,C: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ C )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_892_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_893_inf_Ostrict__coboundedI2,axiom,
! [B2: set_set_set_nat,C: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ C )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_894_inf_Ostrict__coboundedI1,axiom,
! [A2: set_nat_nat,C: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ C )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_895_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_896_inf_Ostrict__coboundedI1,axiom,
! [A2: set_set_set_nat,C: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ C )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_897_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( A3
= ( inf_inf_set_nat_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_898_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_899_inf_Ostrict__order__iff,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( A3
= ( inf_in5711780100303410308et_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_900_inf_Ostrict__boundedE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C ) )
=> ~ ( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ~ ( ord_less_set_nat_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_901_inf_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_902_inf_Ostrict__boundedE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C ) )
=> ~ ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ~ ( ord_le152980574450754630et_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_903_inf_Oabsorb4,axiom,
! [B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_904_inf_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_905_inf_Oabsorb4,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_906_inf_Oabsorb3,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_907_inf_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_908_inf_Oabsorb3,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_909_less__infI2,axiom,
! [B2: set_nat_nat,X2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ X2 )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_910_less__infI2,axiom,
! [B2: nat,X2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_911_less__infI2,axiom,
! [B2: set_set_set_nat,X2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ X2 )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_912_less__infI1,axiom,
! [A2: set_nat_nat,X2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ X2 )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_913_less__infI1,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_914_less__infI1,axiom,
! [A2: set_set_set_nat,X2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ X2 )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_915_distrib__imp1,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ! [X4: set_set_set_nat,Y4: set_set_set_nat,Z4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X4 @ ( sup_su4213647025997063966et_nat @ Y4 @ Z4 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X4 @ Y4 ) @ ( inf_in5711780100303410308et_nat @ X4 @ Z4 ) ) )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_916_distrib__imp1,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ! [X4: set_nat_nat,Y4: set_nat_nat,Z4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X4 @ ( sup_sup_set_nat_nat @ Y4 @ Z4 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X4 @ Y4 ) @ ( inf_inf_set_nat_nat @ X4 @ Z4 ) ) )
=> ( ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_917_distrib__imp1,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ! [X4: set_set_nat,Y4: set_set_nat,Z4: set_set_nat] :
( ( inf_inf_set_set_nat @ X4 @ ( sup_sup_set_set_nat @ Y4 @ Z4 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X4 @ Y4 ) @ ( inf_inf_set_set_nat @ X4 @ Z4 ) ) )
=> ( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_918_distrib__imp1,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ! [X4: produc4045820344675478307at_nat,Y4: produc4045820344675478307at_nat,Z4: produc4045820344675478307at_nat] :
( ( inf_in8110749398470801937at_nat @ X4 @ ( sup_su2809688525313048311at_nat @ Y4 @ Z4 ) )
= ( sup_su2809688525313048311at_nat @ ( inf_in8110749398470801937at_nat @ X4 @ Y4 ) @ ( inf_in8110749398470801937at_nat @ X4 @ Z4 ) ) )
=> ( ( sup_su2809688525313048311at_nat @ X2 @ ( inf_in8110749398470801937at_nat @ Y @ Z2 ) )
= ( inf_in8110749398470801937at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) @ ( sup_su2809688525313048311at_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_919_distrib__imp2,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ! [X4: set_set_set_nat,Y4: set_set_set_nat,Z4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X4 @ ( inf_in5711780100303410308et_nat @ Y4 @ Z4 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X4 @ Y4 ) @ ( sup_su4213647025997063966et_nat @ X4 @ Z4 ) ) )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_920_distrib__imp2,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ! [X4: set_nat_nat,Y4: set_nat_nat,Z4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X4 @ ( inf_inf_set_nat_nat @ Y4 @ Z4 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X4 @ Y4 ) @ ( sup_sup_set_nat_nat @ X4 @ Z4 ) ) )
=> ( ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_921_distrib__imp2,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ! [X4: set_set_nat,Y4: set_set_nat,Z4: set_set_nat] :
( ( sup_sup_set_set_nat @ X4 @ ( inf_inf_set_set_nat @ Y4 @ Z4 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X4 @ Y4 ) @ ( sup_sup_set_set_nat @ X4 @ Z4 ) ) )
=> ( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y ) @ ( inf_inf_set_set_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_922_distrib__imp2,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ! [X4: produc4045820344675478307at_nat,Y4: produc4045820344675478307at_nat,Z4: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ X4 @ ( inf_in8110749398470801937at_nat @ Y4 @ Z4 ) )
= ( inf_in8110749398470801937at_nat @ ( sup_su2809688525313048311at_nat @ X4 @ Y4 ) @ ( sup_su2809688525313048311at_nat @ X4 @ Z4 ) ) )
=> ( ( inf_in8110749398470801937at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ Y @ Z2 ) )
= ( sup_su2809688525313048311at_nat @ ( inf_in8110749398470801937at_nat @ X2 @ Y ) @ ( inf_in8110749398470801937at_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_923_inf__sup__distrib1,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_924_inf__sup__distrib1,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_925_inf__sup__distrib1,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y ) @ ( inf_inf_set_set_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_926_inf__sup__distrib1,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ( inf_in8110749398470801937at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ Y @ Z2 ) )
= ( sup_su2809688525313048311at_nat @ ( inf_in8110749398470801937at_nat @ X2 @ Y ) @ ( inf_in8110749398470801937at_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_927_inf__sup__distrib2,axiom,
! [Y: set_set_set_nat,Z2: set_set_set_nat,X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y @ Z2 ) @ X2 )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y @ X2 ) @ ( inf_in5711780100303410308et_nat @ Z2 @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_928_inf__sup__distrib2,axiom,
! [Y: set_nat_nat,Z2: set_nat_nat,X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y @ Z2 ) @ X2 )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y @ X2 ) @ ( inf_inf_set_nat_nat @ Z2 @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_929_inf__sup__distrib2,axiom,
! [Y: set_set_nat,Z2: set_set_nat,X2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y @ Z2 ) @ X2 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y @ X2 ) @ ( inf_inf_set_set_nat @ Z2 @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_930_inf__sup__distrib2,axiom,
! [Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat,X2: produc4045820344675478307at_nat] :
( ( inf_in8110749398470801937at_nat @ ( sup_su2809688525313048311at_nat @ Y @ Z2 ) @ X2 )
= ( sup_su2809688525313048311at_nat @ ( inf_in8110749398470801937at_nat @ Y @ X2 ) @ ( inf_in8110749398470801937at_nat @ Z2 @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_931_sup__inf__distrib1,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_932_sup__inf__distrib1,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_933_sup__inf__distrib1,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_934_sup__inf__distrib1,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ X2 @ ( inf_in8110749398470801937at_nat @ Y @ Z2 ) )
= ( inf_in8110749398470801937at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) @ ( sup_su2809688525313048311at_nat @ X2 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_935_sup__inf__distrib2,axiom,
! [Y: set_set_set_nat,Z2: set_set_set_nat,X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) @ X2 )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y @ X2 ) @ ( sup_su4213647025997063966et_nat @ Z2 @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_936_sup__inf__distrib2,axiom,
! [Y: set_nat_nat,Z2: set_nat_nat,X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y @ Z2 ) @ X2 )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y @ X2 ) @ ( sup_sup_set_nat_nat @ Z2 @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_937_sup__inf__distrib2,axiom,
! [Y: set_set_nat,Z2: set_set_nat,X2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y @ Z2 ) @ X2 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y @ X2 ) @ ( sup_sup_set_set_nat @ Z2 @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_938_sup__inf__distrib2,axiom,
! [Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat,X2: produc4045820344675478307at_nat] :
( ( sup_su2809688525313048311at_nat @ ( inf_in8110749398470801937at_nat @ Y @ Z2 ) @ X2 )
= ( inf_in8110749398470801937at_nat @ ( sup_su2809688525313048311at_nat @ Y @ X2 ) @ ( sup_su2809688525313048311at_nat @ Z2 @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_939_Int__mono,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_940_Int__mono,axiom,
! [A: set_set_nat,C2: set_set_nat,B: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_941_Int__mono,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_942_Int__lower1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_943_Int__lower1,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_944_Int__lower1,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_945_Int__lower2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_946_Int__lower2,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_947_Int__lower2,axiom,
! [A: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_948_Int__absorb1,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( ( inf_in5711780100303410308et_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_949_Int__absorb1,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ A )
=> ( ( inf_inf_set_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_950_Int__absorb1,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B @ A )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_951_Int__absorb2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( inf_in5711780100303410308et_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_952_Int__absorb2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( inf_inf_set_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_953_Int__absorb2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_954_Int__greatest,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A )
=> ( ( ord_le9131159989063066194et_nat @ C2 @ B )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_955_Int__greatest,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ B )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_956_Int__greatest,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ B )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_957_Int__Collect__mono,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,P2: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ! [X4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ A )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le572741076514265352et_nat @ ( inf_in2396666505901392698et_nat @ A @ ( collec7201453139178570183et_nat @ P2 ) ) @ ( inf_in2396666505901392698et_nat @ B @ ( collec7201453139178570183et_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_958_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_959_Int__Collect__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,P2: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ ( collect_set_set_nat @ P2 ) ) @ ( inf_in5711780100303410308et_nat @ B @ ( collect_set_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_960_Int__Collect__mono,axiom,
! [A: set_set_nat,B: set_set_nat,P2: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P2 ) ) @ ( inf_inf_set_set_nat @ B @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_961_Int__Collect__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,P2: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ ( collect_nat_nat @ P2 ) ) @ ( inf_inf_set_nat_nat @ B @ ( collect_nat_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_962_Int__Diff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( minus_8121590178497047118at_nat @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_963_Int__Diff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A @ ( minus_2447799839930672331et_nat @ B @ C2 ) ) ) ).
% Int_Diff
thf(fact_964_Diff__Int2,axiom,
! [A: set_nat_nat,C2: set_nat_nat,B: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ ( inf_inf_set_nat_nat @ A @ C2 ) @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( minus_8121590178497047118at_nat @ ( inf_inf_set_nat_nat @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_965_Diff__Int2,axiom,
! [A: set_set_set_nat,C2: set_set_set_nat,B: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ A @ C2 ) @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ A @ C2 ) @ B ) ) ).
% Diff_Int2
thf(fact_966_Diff__Diff__Int,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ ( minus_8121590178497047118at_nat @ A @ B ) )
= ( inf_inf_set_nat_nat @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_967_Diff__Diff__Int,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A @ ( minus_2447799839930672331et_nat @ A @ B ) )
= ( inf_in5711780100303410308et_nat @ A @ B ) ) ).
% Diff_Diff_Int
thf(fact_968_Diff__Int__distrib,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ C2 @ ( minus_8121590178497047118at_nat @ A @ B ) )
= ( minus_8121590178497047118at_nat @ ( inf_inf_set_nat_nat @ C2 @ A ) @ ( inf_inf_set_nat_nat @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_969_Diff__Int__distrib,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ C2 @ ( minus_2447799839930672331et_nat @ A @ B ) )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ C2 @ A ) @ ( inf_in5711780100303410308et_nat @ C2 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_970_Diff__Int__distrib2,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ C2 )
= ( minus_8121590178497047118at_nat @ ( inf_inf_set_nat_nat @ A @ C2 ) @ ( inf_inf_set_nat_nat @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_971_Diff__Int__distrib2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ C2 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ A @ C2 ) @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_972_Un__Int__crazy,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ B @ C2 ) ) @ ( inf_in5711780100303410308et_nat @ C2 @ A ) )
= ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) @ ( sup_su4213647025997063966et_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_973_Un__Int__crazy,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ B @ C2 ) ) @ ( inf_inf_set_nat_nat @ C2 @ A ) )
= ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ B @ C2 ) ) @ ( sup_sup_set_nat_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_974_Un__Int__crazy,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ B @ C2 ) ) @ ( inf_inf_set_set_nat @ C2 @ A ) )
= ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ B @ C2 ) ) @ ( sup_sup_set_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_975_Int__Un__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_976_Int__Un__distrib,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_977_Int__Un__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_978_Un__Int__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_979_Un__Int__distrib,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_980_Un__Int__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_981_Int__Un__distrib2,axiom,
! [B: set_set_set_nat,C2: set_set_set_nat,A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ B @ C2 ) @ A )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ B @ A ) @ ( inf_in5711780100303410308et_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_982_Int__Un__distrib2,axiom,
! [B: set_nat_nat,C2: set_nat_nat,A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ B @ C2 ) @ A )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ B @ A ) @ ( inf_inf_set_nat_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_983_Int__Un__distrib2,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ C2 ) @ A )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ A ) @ ( inf_inf_set_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_984_Un__Int__distrib2,axiom,
! [B: set_set_set_nat,C2: set_set_set_nat,A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ B @ C2 ) @ A )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ B @ A ) @ ( sup_su4213647025997063966et_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_985_Un__Int__distrib2,axiom,
! [B: set_nat_nat,C2: set_nat_nat,A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ B @ C2 ) @ A )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ B @ A ) @ ( sup_sup_set_nat_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_986_Un__Int__distrib2,axiom,
! [B: set_set_nat,C2: set_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ C2 ) @ A )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ A ) @ ( sup_sup_set_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_987_inf__prod__def,axiom,
( inf_in8110749398470801937at_nat
= ( ^ [X3: produc4045820344675478307at_nat,Y3: produc4045820344675478307at_nat] : ( produc2803780273060847707at_nat @ ( inf_in5711780100303410308et_nat @ ( produc6523417423482510407at_nat @ X3 ) @ ( produc6523417423482510407at_nat @ Y3 ) ) @ ( inf_inf_nat @ ( produc8987496658364038281at_nat @ X3 ) @ ( produc8987496658364038281at_nat @ Y3 ) ) ) ) ) ).
% inf_prod_def
thf(fact_988_inf__prod__def,axiom,
( inf_in7092662171172785971et_nat
= ( ^ [X3: produc4405103650892965957et_nat,Y3: produc4405103650892965957et_nat] : ( produc8443863378681539197et_nat @ ( inf_in5711780100303410308et_nat @ ( produc1516831113742898793et_nat @ X3 ) @ ( produc1516831113742898793et_nat @ Y3 ) ) @ ( inf_in5711780100303410308et_nat @ ( produc2905410560650206891et_nat @ X3 ) @ ( produc2905410560650206891et_nat @ Y3 ) ) ) ) ) ).
% inf_prod_def
thf(fact_989_inf__prod__def,axiom,
( inf_in6765570788667771830at_nat
= ( ^ [X3: produc4669799618898522568at_nat,Y3: produc4669799618898522568at_nat] : ( produc507788731782991104at_nat @ ( inf_in5711780100303410308et_nat @ ( produc6783108559990802156at_nat @ X3 ) @ ( produc6783108559990802156at_nat @ Y3 ) ) @ ( inf_inf_set_nat_nat @ ( produc754225753220371502at_nat @ X3 ) @ ( produc754225753220371502at_nat @ Y3 ) ) ) ) ) ).
% inf_prod_def
thf(fact_990_inf__prod__def,axiom,
( inf_in4890968077345302454et_nat
= ( ^ [X3: produc2795196907576053192et_nat,Y3: produc2795196907576053192et_nat] : ( produc2168915450337631616et_nat @ ( inf_inf_set_nat_nat @ ( produc8444235278545442668et_nat @ X3 ) @ ( produc8444235278545442668et_nat @ Y3 ) ) @ ( inf_in5711780100303410308et_nat @ ( produc2415352471775012014et_nat @ X3 ) @ ( produc2415352471775012014et_nat @ Y3 ) ) ) ) ) ).
% inf_prod_def
thf(fact_991_inf__prod__def,axiom,
( inf_in8487852146424888249at_nat
= ( ^ [X3: produc1300680692302638795at_nat,Y3: produc1300680692302638795at_nat] : ( produc8650507651752646531at_nat @ ( inf_inf_set_nat_nat @ ( produc7281425617276542831at_nat @ X3 ) @ ( produc7281425617276542831at_nat @ Y3 ) ) @ ( inf_inf_set_nat_nat @ ( produc5975371700627812785at_nat @ X3 ) @ ( produc5975371700627812785at_nat @ Y3 ) ) ) ) ) ).
% inf_prod_def
thf(fact_992_distrib__inf__le,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) @ ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_993_distrib__inf__le,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ ( inf_in8110749398470801937at_nat @ X2 @ Y ) @ ( inf_in8110749398470801937at_nat @ X2 @ Z2 ) ) @ ( inf_in8110749398470801937at_nat @ X2 @ ( sup_su2809688525313048311at_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_994_distrib__inf__le,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y ) @ ( inf_inf_set_set_nat @ X2 @ Z2 ) ) @ ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_995_distrib__inf__le,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y ) @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) @ ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_996_distrib__inf__le,axiom,
! [X2: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y ) @ ( inf_inf_nat @ X2 @ Z2 ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_997_distrib__inf__le,axiom,
! [X2: nat > nat,Y: nat > nat,Z2: nat > nat] : ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y ) @ ( inf_inf_nat_nat @ X2 @ Z2 ) ) @ ( inf_inf_nat_nat @ X2 @ ( sup_sup_nat_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_998_distrib__sup__le,axiom,
! [X2: set_set_set_nat,Y: set_set_set_nat,Z2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y @ Z2 ) ) @ ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_999_distrib__sup__le,axiom,
! [X2: produc4045820344675478307at_nat,Y: produc4045820344675478307at_nat,Z2: produc4045820344675478307at_nat] : ( ord_le2960050975727596227at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ ( inf_in8110749398470801937at_nat @ Y @ Z2 ) ) @ ( inf_in8110749398470801937at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ Y ) @ ( sup_su2809688525313048311at_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1000_distrib__sup__le,axiom,
! [X2: set_set_nat,Y: set_set_nat,Z2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y @ Z2 ) ) @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y ) @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1001_distrib__sup__le,axiom,
! [X2: set_nat_nat,Y: set_nat_nat,Z2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y @ Z2 ) ) @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y ) @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1002_distrib__sup__le,axiom,
! [X2: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y ) @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1003_distrib__sup__le,axiom,
! [X2: nat > nat,Y: nat > nat,Z2: nat > nat] : ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y @ Z2 ) ) @ ( inf_inf_nat_nat @ ( sup_sup_nat_nat @ X2 @ Y ) @ ( sup_sup_nat_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1004_Un__Int__assoc__eq,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) )
= ( ord_le9131159989063066194et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1005_Un__Int__assoc__eq,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) )
= ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1006_Un__Int__assoc__eq,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) )
= ( ord_le9059583361652607317at_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_1007_Un__Diff__Int,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( inf_inf_set_set_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1008_Un__Diff__Int,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1009_Un__Diff__Int,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= A ) ).
% Un_Diff_Int
thf(fact_1010_Int__Diff__Un,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1011_Int__Diff__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1012_Int__Diff__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ A @ B ) )
= A ) ).
% Int_Diff_Un
thf(fact_1013_Diff__Int,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( inf_inf_set_set_nat @ B @ C2 ) )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1014_Diff__Int,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ ( inf_inf_set_nat_nat @ B @ C2 ) )
= ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1015_Diff__Int,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C2 ) )
= ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ A @ C2 ) ) ) ).
% Diff_Int
thf(fact_1016_Diff__Un,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) )
= ( inf_inf_set_set_nat @ ( minus_2163939370556025621et_nat @ A @ B ) @ ( minus_2163939370556025621et_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1017_Diff__Un,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) )
= ( inf_inf_set_nat_nat @ ( minus_8121590178497047118at_nat @ A @ B ) @ ( minus_8121590178497047118at_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1018_Diff__Un,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) )
= ( inf_in5711780100303410308et_nat @ ( minus_2447799839930672331et_nat @ A @ B ) @ ( minus_2447799839930672331et_nat @ A @ C2 ) ) ) ).
% Diff_Un
thf(fact_1019_first__assumptions_Okml,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_1020_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ Y2 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_1021_first__assumptions_OACC__odot,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K @ X ) @ ( clique3210737319928189260st_ACC @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_odot
thf(fact_1022_ACC__cf___092_060F_062,axiom,
! [X: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X ) @ ( clique2971579238625216137irst_F @ k ) ) ).
% ACC_cf_\<F>
thf(fact_1023_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X6: set_set_set_nat,G: set_set_nat] :
? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X6 )
& ( ord_le6893508408891458716et_nat @ X3 @ G ) ) ) ) ).
% accepts_def
thf(fact_1024_odotl__def,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k @ X @ Y2 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X @ Y2 ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).
% odotl_def
thf(fact_1025_deviate__pos__cap__def,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3314026705535538693os_cap @ l @ p @ k @ X @ Y2 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X @ Y2 ) ) ) ) ).
% deviate_pos_cap_def
thf(fact_1026_deviate__pos__cup__def,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique3314026705536850673os_cup @ l @ p @ k @ X @ Y2 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X @ Y2 ) ) ) ) ).
% deviate_pos_cup_def
thf(fact_1027_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1028_exI__realizer,axiom,
! [P2: nat > set_set_set_nat > $o,Y: nat,X2: set_set_set_nat] :
( ( P2 @ Y @ X2 )
=> ( P2 @ ( produc8987496658364038281at_nat @ ( produc2803780273060847707at_nat @ X2 @ Y ) ) @ ( produc6523417423482510407at_nat @ ( produc2803780273060847707at_nat @ X2 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_1029_ACC__cf__odot,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X @ Y2 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X ) @ ( clique951075384711337423ACC_cf @ k @ Y2 ) ) ) ).
% ACC_cf_odot
thf(fact_1030_acceptsI,axiom,
! [D2: set_set_nat,G2: set_set_nat,X: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D2 @ G2 )
=> ( ( member_set_set_nat @ D2 @ X )
=> ( clique3686358387679108662ccepts @ X @ G2 ) ) ) ).
% acceptsI
thf(fact_1031_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_1032_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_1033_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_1034_second__assumptions_Odeviate__pos__cup_Ocong,axiom,
clique3314026705536850673os_cup = clique3314026705536850673os_cup ).
% second_assumptions.deviate_pos_cup.cong
thf(fact_1035_second__assumptions_Odeviate__pos__cap_Ocong,axiom,
clique3314026705535538693os_cap = clique3314026705535538693os_cap ).
% second_assumptions.deviate_pos_cap.cong
thf(fact_1036_first__assumptions_OacceptsI,axiom,
! [L: nat,P: nat,K: nat,D2: set_set_nat,G2: set_set_nat,X: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ G2 )
=> ( ( member_set_set_nat @ D2 @ X )
=> ( clique3686358387679108662ccepts @ X @ G2 ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_1037_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique3686358387679108662ccepts @ X @ G2 )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X )
& ( ord_le6893508408891458716et_nat @ X3 @ G2 ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_1038_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1039_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1040_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_1041_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_1042_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_1043_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B2: nat] :
( ( P2 @ K )
=> ( ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y7: nat] :
( ( P2 @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1044_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_1045_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1046_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_1047_less__not__refl3,axiom,
! [S3: nat,T2: nat] :
( ( ord_less_nat @ S3 @ T2 )
=> ( S3 != T2 ) ) ).
% less_not_refl3
thf(fact_1048_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1049_nat__less__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P2 @ M2 ) )
=> ( P2 @ N ) )
=> ( P2 @ N2 ) ) ).
% nat_less_induct
thf(fact_1050_infinite__descent,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ~ ( P2 @ N )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N2 ) ) ).
% infinite_descent
thf(fact_1051_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1052_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1053_first__assumptions_OACC__cf__odot,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X @ Y2 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X ) @ ( clique951075384711337423ACC_cf @ K @ Y2 ) ) ) ) ).
% first_assumptions.ACC_cf_odot
thf(fact_1054_second__assumptions_Odeviate__pos__cup__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique3314026705536850673os_cup @ L @ P @ K @ X @ Y2 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X @ Y2 ) ) ) ) ) ).
% second_assumptions.deviate_pos_cup_def
thf(fact_1055_second__assumptions_Odeviate__pos__cap__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique3314026705535538693os_cap @ L @ P @ K @ X @ Y2 )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X @ Y2 ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118206219037_sqcap @ L @ P @ K @ X @ Y2 ) ) ) ) ) ).
% second_assumptions.deviate_pos_cap_def
thf(fact_1056_first__assumptions_Oodotl__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique7966186356931407165_odotl @ L @ K @ X @ Y2 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X @ Y2 ) @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ).
% first_assumptions.odotl_def
thf(fact_1057_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X ) @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_1058_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1059_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1060_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1061_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1062_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1063_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1064_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1065_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1066_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1067_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1068_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_1069_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1070_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1071_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1072_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1073_exE__realizer_H,axiom,
! [P2: nat > set_set_set_nat > $o,P: produc4045820344675478307at_nat] :
( ( P2 @ ( produc8987496658364038281at_nat @ P ) @ ( produc6523417423482510407at_nat @ P ) )
=> ~ ! [X4: set_set_set_nat,Y4: nat] :
~ ( P2 @ Y4 @ X4 ) ) ).
% exE_realizer'
thf(fact_1074_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1075_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1076_conjI__realizer,axiom,
! [P2: set_set_set_nat > $o,P: set_set_set_nat,Q: nat > $o,Q2: nat] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( produc6523417423482510407at_nat @ ( produc2803780273060847707at_nat @ P @ Q2 ) ) )
& ( Q @ ( produc8987496658364038281at_nat @ ( produc2803780273060847707at_nat @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_1077_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_1078_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_1079_ACC__cf__I,axiom,
! [F2: nat > nat,X: set_set_set_nat] :
( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ k ) )
=> ( ( clique3686358387679108662ccepts @ X @ ( clique5033774636164728462irst_C @ k @ F2 ) )
=> ( member_nat_nat @ F2 @ ( clique951075384711337423ACC_cf @ k @ X ) ) ) ) ).
% ACC_cf_I
thf(fact_1080_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P: nat,K: nat,F2: nat > nat,X: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ K ) )
=> ( ( clique3686358387679108662ccepts @ X @ ( clique5033774636164728462irst_C @ K @ F2 ) )
=> ( member_nat_nat @ F2 @ ( clique951075384711337423ACC_cf @ K @ X ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_1081_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_1082_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_1083_order__refl,axiom,
! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% order_refl
thf(fact_1084_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_1085_order__refl,axiom,
! [X2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_1086_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1087_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1088_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1089_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1090_dual__order_Orefl,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_1091_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1092_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_1093_order__antisym__conv,axiom,
! [Y: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_1094_order__antisym__conv,axiom,
! [Y: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_1095_order__antisym__conv,axiom,
! [Y: num,X2: num] :
( ( ord_less_eq_num @ Y @ X2 )
=> ( ( ord_less_eq_num @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_1096_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_1097_order__antisym__conv,axiom,
! [Y: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_1098_linorder__le__cases,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_1099_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_1100_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1101_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1102_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1103_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1104_ord__le__eq__subst,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > num,C: num] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1105_ord__le__eq__subst,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > nat,C: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1106_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1107_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > nat > nat,C: nat > nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1108_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_set_nat,C: set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1109_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1110_ord__eq__le__subst,axiom,
! [A2: set_set_nat,F: ( nat > nat ) > set_set_nat,B2: nat > nat,C: nat > nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ! [X4: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1111_ord__eq__le__subst,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat_nat,B2: nat > nat,C: nat > nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ! [X4: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1112_ord__eq__le__subst,axiom,
! [A2: num,F: ( nat > nat ) > num,B2: nat > nat,C: nat > nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ! [X4: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1113_ord__eq__le__subst,axiom,
! [A2: nat,F: ( nat > nat ) > nat,B2: nat > nat,C: nat > nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ! [X4: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1114_ord__eq__le__subst,axiom,
! [A2: nat > nat,F: ( nat > nat ) > nat > nat,B2: nat > nat,C: nat > nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat_nat @ B2 @ C )
=> ( ! [X4: nat > nat,Y4: nat > nat] :
( ( ord_less_eq_nat_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1115_first__assumptions_OPOS__CLIQUE,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_CLIQUE
thf(fact_1116_first__assumptions_OPOS__sub__CLIQUE,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_sub_CLIQUE
thf(fact_1117_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_1118_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_1119_v__gs__mono,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ).
% v_gs_mono
thf(fact_1120_nat__descend__induct,axiom,
! [N2: nat,P2: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P2 @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P2 @ I3 ) )
=> ( P2 @ K3 ) ) )
=> ( P2 @ M ) ) ) ).
% nat_descend_induct
thf(fact_1121_v__gs__union,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ).
% v_gs_union
thf(fact_1122_finite__ACC,axiom,
! [X: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X ) ) ).
% finite_ACC
thf(fact_1123_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K @ X ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_1124_first__assumptions_Ofinite___092_060F_062,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.finite_\<F>
thf(fact_1125_first__assumptions_Ov__gs__union,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X @ Y2 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ).
% first_assumptions.v_gs_union
thf(fact_1126_first__assumptions_Oempty__CLIQUE,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.empty_CLIQUE
thf(fact_1127_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X ) @ ( clique8462013130872731469t_v_gs @ Y2 ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_1128_finite__v__gs__Gl,axiom,
! [X: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X ) ) ) ).
% finite_v_gs_Gl
thf(fact_1129_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_1130_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_1131_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_1132_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_1133_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_1134_first__assumptions_OACC__cf__empty,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ) ).
% first_assumptions.ACC_cf_empty
thf(fact_1135_first__assumptions_OACC__empty,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.ACC_empty
thf(fact_1136_first__assumptions_Ov__gs__empty,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ) ).
% first_assumptions.v_gs_empty
thf(fact_1137_first__assumptions_Ofinite__v__gs__Gl,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X ) ) ) ) ).
% first_assumptions.finite_v_gs_Gl
thf(fact_1138_plucking__step_I3_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ p @ X ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ X ) ) @ ( clique3210737319928189260st_ACC @ k @ Y2 ) ) ) ) ) ).
% plucking_step(3)
thf(fact_1139_plucking__step_I2_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ p @ X ) )
=> ( ord_le9131159989063066194et_nat @ Y2 @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ) ) ).
% plucking_step(2)
thf(fact_1140_plucking__step_I5_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ p @ X ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) ) ) ) ).
% plucking_step(5)
thf(fact_1141_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_1142_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_1143_second__assumptions_Oplucking__step_I2_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ P @ X ) )
=> ( ord_le9131159989063066194et_nat @ Y2 @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ) ) ).
% second_assumptions.plucking_step(2)
thf(fact_1144_second__assumptions_Oplucking__step_I5_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ P @ X ) )
=> ( Y2 != bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% second_assumptions.plucking_step(5)
thf(fact_1145_second__assumptions_Oplucking__step_I3_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ P @ X ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ X ) ) @ ( clique3210737319928189260st_ACC @ K @ Y2 ) ) ) ) ) ) ).
% second_assumptions.plucking_step(3)
thf(fact_1146_first__assumptions_Ofinite__POS__NEG,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737375870294875st_NEG @ K ) ) ) ) ).
% first_assumptions.finite_POS_NEG
thf(fact_1147_first__assumptions_OCLIQUE__NEG,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K ) @ ( clique3210737375870294875st_NEG @ K ) )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.CLIQUE_NEG
thf(fact_1148_PLU__main_Opinduct,axiom,
! [A0: set_set_set_nat,P2: set_set_set_nat > $o] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k ) @ A0 )
=> ( ! [X7: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k ) @ X7 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X7 @ ( clique7840962075309931874st_G_l @ l @ k ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X7 ) ) ) )
=> ( P2 @ ( clique4095374090462327202g_step @ p @ X7 ) ) )
=> ( P2 @ X7 ) ) )
=> ( P2 @ A0 ) ) ) ).
% PLU_main.pinduct
thf(fact_1149_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_1150_plucking__step_I1_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ p @ X ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y2 ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) @ p ) @ one_one_nat ) ) ) ) ) ).
% plucking_step(1)
thf(fact_1151_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1152_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1153_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1154_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1155_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1156_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1157_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1158_first__assumptions_Olm,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.lm
thf(fact_1159_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1160_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1161_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1162_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1163_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1164_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N: nat] :
( L
= ( plus_plus_nat @ K @ N ) ) ) ).
% le_Suc_ex
thf(fact_1165_add__leD2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1166_add__leD1,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% add_leD1
thf(fact_1167_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_1168_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_1169_add__leE,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1170_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1171_diff__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_cancel2
thf(fact_1172_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_1173_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_1174_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1175_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1176_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1177_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1178_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1179_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1180_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1181_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1182_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1183_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1184_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1185_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1186_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1187_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1188_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1189_add__diff__inverse__nat,axiom,
! [M: nat,N2: nat] :
( ~ ( ord_less_nat @ M @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1190_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1191_first__assumptions_ONEG__def,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique3210737375870294875st_NEG @ K )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K ) @ ( clique2971579238625216137irst_F @ K ) ) ) ) ).
% first_assumptions.NEG_def
thf(fact_1192_second__assumptions_Oplucking__step_I1_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ P @ X ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y2 ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) @ P ) @ one_one_nat ) ) ) ) ) ) ).
% second_assumptions.plucking_step(1)
thf(fact_1193_second__assumptions_OPLU__main_Opinduct,axiom,
! [L: nat,P: nat,K: nat,A0: set_set_set_nat,P2: set_set_set_nat > $o] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P @ K ) @ A0 )
=> ( ! [X7: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P @ K ) @ X7 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X7 @ ( clique7840962075309931874st_G_l @ L @ K ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X7 ) ) ) )
=> ( P2 @ ( clique4095374090462327202g_step @ P @ X7 ) ) )
=> ( P2 @ X7 ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% second_assumptions.PLU_main.pinduct
thf(fact_1194_PLU__main__n,axiom,
! [X: set_set_set_nat,Z5: set_set_set_nat,N2: nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ( clique429652266423863867U_main @ l @ p @ k @ X )
= ( produc2803780273060847707at_nat @ Z5 @ N2 ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ p @ one_one_nat ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) ) ) ) ).
% PLU_main_n
thf(fact_1195_nat__1__eq__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N2 ) )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1196_nat__mult__eq__1__iff,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1197_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1198_add__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1199_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1200_add__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1201_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1202_diff__mult__distrib,axiom,
! [M: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1203_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1204_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1205_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1206_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1207_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1208_second__assumptions_OPLU__main__n,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Z5: set_set_set_nat,N2: nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ( clique429652266423863867U_main @ L @ P @ K @ X )
= ( produc2803780273060847707at_nat @ Z5 @ N2 ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ P @ one_one_nat ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) ) ) ) ) ).
% second_assumptions.PLU_main_n
thf(fact_1209_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1210_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1211_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1212_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1213_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1214_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1215_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1216_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1217_bounded__Max__nat,axiom,
! [P2: nat > $o,X2: nat,M5: nat] :
( ( P2 @ X2 )
=> ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P2 @ M4 )
=> ~ ! [X8: nat] :
( ( P2 @ X8 )
=> ( ord_less_eq_nat @ X8 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1218_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1219_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_nat @ X4 @ N2 ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1220_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1221_pointwise__minimal__pointwise__maximal_I1_J,axiom,
! [S3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat_nat )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S3 )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S3 )
=> ( ( ord_less_eq_nat_nat @ X4 @ Xa )
| ( ord_less_eq_nat_nat @ Xa @ X4 ) ) ) )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S3 )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S3 )
=> ( ord_less_eq_nat_nat @ X4 @ Xa2 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(1)
thf(fact_1222_pointwise__minimal__pointwise__maximal_I2_J,axiom,
! [S3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat_nat )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S3 )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S3 )
=> ( ( ord_less_eq_nat_nat @ X4 @ Xa )
| ( ord_less_eq_nat_nat @ Xa @ X4 ) ) ) )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S3 )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S3 )
=> ( ord_less_eq_nat_nat @ Xa2 @ X4 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(2)
thf(fact_1223_card__POS,axiom,
( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ k ) )
= ( binomial @ ( assump1710595444109740334irst_m @ k ) @ k ) ) ).
% card_POS
thf(fact_1224_first__assumptions_Ocard__POS,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ K ) )
= ( binomial @ ( assump1710595444109740334irst_m @ K ) @ K ) ) ) ).
% first_assumptions.card_POS
thf(fact_1225_binomial__absorb__comp,axiom,
! [N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
= ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_1226_choose__mult,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ M @ N2 )
=> ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
= ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_1227_binomial__symmetric,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_1228__092_060P_062L_092_060G_062l__def,axiom,
( ( clique2294137941332549862_L_G_l @ l @ p @ k )
= ( collec7201453139178570183et_nat
@ ^ [X6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X6 @ ( clique7840962075309931874st_G_l @ l @ k ) )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X6 ) ) @ ( assump1710595444109740301irst_L @ l @ p ) ) ) ) ) ).
% \<P>L\<G>l_def
thf(fact_1229_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_1230_ACC__cf__def,axiom,
! [X: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ X )
= ( collect_nat_nat
@ ^ [F3: nat > nat] :
( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ k ) )
& ( clique3686358387679108662ccepts @ X @ ( clique5033774636164728462irst_C @ k @ F3 ) ) ) ) ) ).
% ACC_cf_def
thf(fact_1231_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1232_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1233_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1234_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P2 @ K2 )
& ( ord_less_nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1235_first__assumptions_OACC__cf__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ X )
= ( collect_nat_nat
@ ^ [F3: nat > nat] :
( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ K ) )
& ( clique3686358387679108662ccepts @ X @ ( clique5033774636164728462irst_C @ K @ F3 ) ) ) ) ) ) ).
% first_assumptions.ACC_cf_def
thf(fact_1236_first__assumptions_O_092_060P_062L_092_060G_062l__def,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique2294137941332549862_L_G_l @ L @ P @ K )
= ( collec7201453139178570183et_nat
@ ^ [X6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X6 @ ( clique7840962075309931874st_G_l @ L @ K ) )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X6 ) ) @ ( assump1710595444109740301irst_L @ L @ P ) ) ) ) ) ) ).
% first_assumptions.\<P>L\<G>l_def
thf(fact_1237_odot__def,axiom,
( clique5469973757772500719t_odot
= ( ^ [X6: set_set_set_nat,Y8: set_set_set_nat] :
( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D3: set_set_nat,E: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D3 @ E ) )
& ( member_set_set_nat @ D3 @ X6 )
& ( member_set_set_nat @ E @ Y8 ) ) ) ) ) ).
% odot_def
thf(fact_1238_first__assumptions_Oodot__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique5469973757772500719t_odot @ X @ Y2 )
= ( collect_set_set_nat
@ ^ [Uu: set_set_nat] :
? [D3: set_set_nat,E: set_set_nat] :
( ( Uu
= ( sup_sup_set_set_nat @ D3 @ E ) )
& ( member_set_set_nat @ D3 @ X )
& ( member_set_set_nat @ E @ Y2 ) ) ) ) ) ).
% first_assumptions.odot_def
thf(fact_1239_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ( member_nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1240_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( member_nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1241_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K @ M4 )
=> ? [N6: nat] :
( ( ord_less_nat @ M4 @ N6 )
& ( member_nat @ N6 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_1242_PLU__main,axiom,
! [X: set_set_set_nat,Z5: set_set_set_nat,N2: nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ( clique429652266423863867U_main @ l @ p @ k @ X )
= ( produc2803780273060847707at_nat @ Z5 @ N2 ) )
=> ( ( member2946998982187404937et_nat @ Z5 @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
& ( ( Z5 = bot_bo7198184520161983622et_nat )
= ( X = bot_bo7198184520161983622et_nat ) )
& ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ X ) ) @ ( clique3210737319928189260st_ACC @ k @ Z5 ) )
& ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ Z5 ) @ ( clique951075384711337423ACC_cf @ k @ X ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ) @ N2 ) ) ) ) ) ).
% PLU_main
thf(fact_1243_semiring__norm_I71_J,axiom,
! [M: num,N2: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_eq_num @ M @ N2 ) ) ).
% semiring_norm(71)
thf(fact_1244_semiring__norm_I78_J,axiom,
! [M: num,N2: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
= ( ord_less_num @ M @ N2 ) ) ).
% semiring_norm(78)
thf(fact_1245_semiring__norm_I68_J,axiom,
! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).
% semiring_norm(68)
thf(fact_1246_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1247_k2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ k ).
% k2
thf(fact_1248_l8,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ l ).
% l8
thf(fact_1249_l2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ l ).
% l2
thf(fact_1250_p,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ).
% p
thf(fact_1251_m2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( assump1710595444109740334irst_m @ k ) ).
% m2
thf(fact_1252_m__def,axiom,
( ( assump1710595444109740334irst_m @ k )
= ( power_power_nat @ k @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% m_def
thf(fact_1253_kl2,axiom,
( k
= ( power_power_nat @ l @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% kl2
thf(fact_1254_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1255_semiring__norm_I76_J,axiom,
! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% semiring_norm(76)
thf(fact_1256_plucking__step_I4_J,axiom,
! [X: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ p @ X ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ Y2 ) @ ( clique951075384711337423ACC_cf @ k @ X ) ) ) ) @ ( power_power_nat @ ( minus_minus_nat @ k @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% plucking_step(4)
thf(fact_1257_second__assumptions__axioms_Ointro,axiom,
! [K: nat,L: nat] :
( ( K
= ( power_power_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L )
=> ( assump8934899134041091456axioms @ L @ K ) ) ) ).
% second_assumptions_axioms.intro
thf(fact_1258_second__assumptions__axioms__def,axiom,
( assump8934899134041091456axioms
= ( ^ [L2: nat,K2: nat] :
( ( K2
= ( power_power_nat @ L2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
& ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L2 ) ) ) ) ).
% second_assumptions_axioms_def
thf(fact_1259_second__assumptions_Ol8,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L ) ) ).
% second_assumptions.l8
thf(fact_1260_binomial__le__pow,axiom,
! [R: nat,N2: nat] :
( ( ord_less_eq_nat @ R @ N2 )
=> ( ord_less_eq_nat @ ( binomial @ N2 @ R ) @ ( power_power_nat @ N2 @ R ) ) ) ).
% binomial_le_pow
thf(fact_1261_binomial__le__pow2,axiom,
! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% binomial_le_pow2
thf(fact_1262_first__assumptions_Om2,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.m2
thf(fact_1263_first__assumptions_Op,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P ) ) ).
% first_assumptions.p
thf(fact_1264_first__assumptions_Ok2,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) ) ).
% first_assumptions.k2
thf(fact_1265_first__assumptions_Ol2,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) ) ).
% first_assumptions.l2
thf(fact_1266_first__assumptions_Ointro,axiom,
! [L: nat,P: nat,K: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L )
=> ( ( ord_less_nat @ L @ P )
=> ( ( ord_less_nat @ P @ K )
=> ( assump5453534214990993103ptions @ L @ P @ K ) ) ) ) ).
% first_assumptions.intro
thf(fact_1267_first__assumptions__def,axiom,
( assump5453534214990993103ptions
= ( ^ [L2: nat,P3: nat,K2: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L2 )
& ( ord_less_nat @ L2 @ P3 )
& ( ord_less_nat @ P3 @ K2 ) ) ) ) ).
% first_assumptions_def
thf(fact_1268_first__assumptions_Om__def,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( assump1710595444109740334irst_m @ K )
= ( power_power_nat @ K @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% first_assumptions.m_def
thf(fact_1269_second__assumptions_Okl2,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( K
= ( power_power_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% second_assumptions.kl2
thf(fact_1270_second__assumptions_Oplucking__step_I4_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y2: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) )
=> ( ( Y2
= ( clique4095374090462327202g_step @ P @ X ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ Y2 ) @ ( clique951075384711337423ACC_cf @ K @ X ) ) ) ) @ ( power_power_nat @ ( minus_minus_nat @ K @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% second_assumptions.plucking_step(4)
thf(fact_1271_second__assumptions_OPLU__main,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Z5: set_set_set_nat,N2: nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ( clique429652266423863867U_main @ L @ P @ K @ X )
= ( produc2803780273060847707at_nat @ Z5 @ N2 ) )
=> ( ( member2946998982187404937et_nat @ Z5 @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
& ( ( Z5 = bot_bo7198184520161983622et_nat )
= ( X = bot_bo7198184520161983622et_nat ) )
& ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ X ) ) @ ( clique3210737319928189260st_ACC @ K @ Z5 ) )
& ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ Z5 ) @ ( clique951075384711337423ACC_cf @ K @ X ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ K @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) @ N2 ) ) ) ) ) ) ).
% second_assumptions.PLU_main
% Conjectures (1)
thf(conj_0,conjecture,
( ( produc6523417423482510407at_nat @ ( clique429652266423863867U_main @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ x @ y ) ) )
= z ) ).
%------------------------------------------------------------------------------