TPTP Problem File: SLH0070^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Clique_and_Monotone_Circuits/0005_Clique_Large_Monotone_Circuits/prob_01100_041245__16293240_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1477 ( 696 unt; 206 typ; 0 def)
% Number of atoms : 2848 ( 984 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 9629 ( 197 ~; 25 |; 117 &;8278 @)
% ( 0 <=>;1012 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 661 ( 661 >; 0 *; 0 +; 0 <<)
% Number of symbols : 175 ( 174 usr; 27 con; 0-5 aty)
% Number of variables : 3224 ( 209 ^;2983 !; 32 ?;3224 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:49:42.144
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
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,
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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,
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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,
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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,
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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,
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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,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
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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__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (174)
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_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__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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__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_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_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__Real__Oreal,type,
inf_inf_real: real > real > real ).
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__Real__Oreal_J,type,
inf_inf_set_real: set_real > set_real > set_real ).
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 ).
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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__Real__Oreal,type,
member_real: real > set_real > $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 (1270)
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_second__assumptions__axioms,axiom,
assump2881078719466019805ptions @ l @ p @ k ).
% second_assumptions_axioms
thf(fact_4_sqcap__def,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118206219037_sqcap @ l @ p @ k @ X @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ X @ Y ) ) ) ).
% sqcap_def
thf(fact_5_second__assumptions_OPLU_Ocong,axiom,
clique2699557479641037314nd_PLU = clique2699557479641037314nd_PLU ).
% second_assumptions.PLU.cong
thf(fact_6_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_7_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_8_sqcup,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X @ Y ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% sqcup
thf(fact_9_PLU,axiom,
( ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) )
= z ) ).
% PLU
thf(fact_10_PLU__main__rel_Ocong,axiom,
clique8954521387433384062in_rel = clique8954521387433384062in_rel ).
% PLU_main_rel.cong
thf(fact_11_PLU__union,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ) ) ) ).
% PLU_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__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Z_An_O_APLU__main_A_IX_A_092_060odot_062l_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 @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) )
!= ( produc2803780273060847707at_nat @ Z @ N ) ) ).
% \<open>\<And>thesis. (\<And>Z n. PLU_main (X \<odot>l Y) = (Z, n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_17_deviate__pos__cup,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( clique3314026705536850673os_cup @ l @ p @ k @ X @ Y )
= bot_bo7198184520161983622et_nat ) ) ) ).
% deviate_pos_cup
thf(fact_18_sub,axiom,
ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% sub
thf(fact_19_sqcup__def,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118207531017_sqcup @ l @ p @ k @ X @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ) ).
% sqcup_def
thf(fact_20_XY_I1_J,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% XY(1)
thf(fact_21_XY_I2_J,axiom,
ord_le9131159989063066194et_nat @ y @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% XY(2)
thf(fact_22_res,axiom,
( ( clique429652266423863867U_main @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) )
= ( produc2803780273060847707at_nat @ z @ n ) ) ).
% res
thf(fact_23_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_24_second__assumptions_OPLU__main_Ocong,axiom,
clique429652266423863867U_main = clique429652266423863867U_main ).
% second_assumptions.PLU_main.cong
thf(fact_25_second__assumptions_Osqcup_Ocong,axiom,
clique2586627118207531017_sqcup = clique2586627118207531017_sqcup ).
% second_assumptions.sqcup.cong
thf(fact_26_second__assumptions_Odeviate__pos__cup,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( clique3314026705536850673os_cup @ L @ P @ K @ X @ Y )
= bot_bo7198184520161983622et_nat ) ) ) ) ).
% second_assumptions.deviate_pos_cup
thf(fact_27_second__assumptions_Osqcup__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique2586627118207531017_sqcup @ L @ P @ K @ X @ Y )
= ( clique2699557479641037314nd_PLU @ L @ P @ K @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ) ) ).
% second_assumptions.sqcup_def
thf(fact_28_second__assumptions_Osqcap_Ocong,axiom,
clique2586627118206219037_sqcap = clique2586627118206219037_sqcap ).
% second_assumptions.sqcap.cong
thf(fact_29_second__assumptions_Odeviate__pos__cup_Ocong,axiom,
clique3314026705536850673os_cup = clique3314026705536850673os_cup ).
% second_assumptions.deviate_pos_cup.cong
thf(fact_30_second__assumptions_Osqcup,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X @ Y ) @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) ) ) ) ) ).
% second_assumptions.sqcup
thf(fact_31_second__assumptions_OPLU__union,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P @ K @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) ) ) ) ) ).
% second_assumptions.PLU_union
thf(fact_32_second__assumptions_Osqcap__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique2586627118206219037_sqcap @ L @ P @ K @ X @ Y )
= ( clique2699557479641037314nd_PLU @ L @ P @ K @ ( clique7966186356931407165_odotl @ L @ K @ X @ Y ) ) ) ) ).
% second_assumptions.sqcap_def
thf(fact_33_joinl__join,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X @ Y ) @ ( clique5469973757772500719t_odot @ X @ Y ) ) ).
% joinl_join
thf(fact_34_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_35_Un__empty,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
= ( ( A = bot_bo7198184520161983622et_nat )
& ( B = bot_bo7198184520161983622et_nat ) ) ) ).
% Un_empty
thf(fact_36_Un__empty,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_37_Un__empty,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ( A = bot_bot_set_nat_nat )
& ( B = bot_bot_set_nat_nat ) ) ) ).
% Un_empty
thf(fact_38_Un__subset__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C )
= ( ( ord_le9131159989063066194et_nat @ A @ C )
& ( ord_le9131159989063066194et_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_39_Un__subset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C )
= ( ( ord_le6893508408891458716et_nat @ A @ C )
& ( ord_le6893508408891458716et_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_40_Un__subset__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C )
= ( ( ord_le9059583361652607317at_nat @ A @ C )
& ( ord_le9059583361652607317at_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_41_sup__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: nat,C2: set_set_set_nat,D: nat] :
( ( sup_su2809688525313048311at_nat @ ( produc2803780273060847707at_nat @ A2 @ B2 ) @ ( produc2803780273060847707at_nat @ C2 @ D ) )
= ( produc2803780273060847707at_nat @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) @ ( sup_sup_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_42_sup__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat,D: set_set_set_nat] :
( ( sup_su7058913318992137241et_nat @ ( produc8443863378681539197et_nat @ A2 @ B2 ) @ ( produc8443863378681539197et_nat @ C2 @ D ) )
= ( produc8443863378681539197et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) @ ( sup_su4213647025997063966et_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_43_sup__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_nat_nat,C2: set_set_set_nat,D: set_nat_nat] :
( ( sup_su3250215936735192988at_nat @ ( produc507788731782991104at_nat @ A2 @ B2 ) @ ( produc507788731782991104at_nat @ C2 @ D ) )
= ( produc507788731782991104at_nat @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) @ ( sup_sup_set_nat_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_44_sup__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,C2: set_set_set_nat,D: set_set_nat] :
( ( sup_su7542950870425426275et_nat @ ( produc1498124630991567047et_nat @ A2 @ B2 ) @ ( produc1498124630991567047et_nat @ C2 @ D ) )
= ( produc1498124630991567047et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_45_sup__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_set_set_nat,C2: set_nat_nat,D: set_set_set_nat] :
( ( sup_su1375613225412723612et_nat @ ( produc2168915450337631616et_nat @ A2 @ B2 ) @ ( produc2168915450337631616et_nat @ C2 @ D ) )
= ( produc2168915450337631616et_nat @ ( sup_sup_set_nat_nat @ A2 @ C2 ) @ ( sup_su4213647025997063966et_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_46_sup__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat,D: set_nat_nat] :
( ( sup_su1050037771541313183at_nat @ ( produc8650507651752646531at_nat @ A2 @ B2 ) @ ( produc8650507651752646531at_nat @ C2 @ D ) )
= ( produc8650507651752646531at_nat @ ( sup_sup_set_nat_nat @ A2 @ C2 ) @ ( sup_sup_set_nat_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_47_sup__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_set_nat,C2: set_nat_nat,D: set_set_nat] :
( ( sup_su5047128473845832934et_nat @ ( produc6308126451205306314et_nat @ A2 @ B2 ) @ ( produc6308126451205306314et_nat @ C2 @ D ) )
= ( produc6308126451205306314et_nat @ ( sup_sup_set_nat_nat @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_48_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_set_set_nat,C2: set_set_nat,D: set_set_set_nat] :
( ( sup_su8797241888131514979et_nat @ ( produc7315026656311086279et_nat @ A2 @ B2 ) @ ( produc7315026656311086279et_nat @ C2 @ D ) )
= ( produc7315026656311086279et_nat @ ( sup_sup_set_set_nat @ A2 @ C2 ) @ ( sup_su4213647025997063966et_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_49_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_nat_nat,C2: set_set_nat,D: set_nat_nat] :
( ( sup_su6693998056113403622at_nat @ ( produc8981877611209197642at_nat @ A2 @ B2 ) @ ( produc8981877611209197642at_nat @ C2 @ D ) )
= ( produc8981877611209197642at_nat @ ( sup_sup_set_set_nat @ A2 @ C2 ) @ ( sup_sup_set_nat_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_50_sup__Pair__Pair,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat,D: set_set_nat] :
( ( sup_su8202740707494180781et_nat @ ( produc9057842353944101649et_nat @ A2 @ B2 ) @ ( produc9057842353944101649et_nat @ C2 @ D ) )
= ( produc9057842353944101649et_nat @ ( sup_sup_set_set_nat @ A2 @ C2 ) @ ( sup_sup_set_set_nat @ B2 @ D ) ) ) ).
% sup_Pair_Pair
thf(fact_51_sup__bot__left,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_52_sup__bot__left,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_53_sup__bot__left,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_54_sup__bot__right,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= X2 ) ).
% sup_bot_right
thf(fact_55_sup__bot__right,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ bot_bot_set_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_56_sup__bot__right,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= X2 ) ).
% sup_bot_right
thf(fact_57_bot__eq__sup__iff,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_58_bot__eq__sup__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_59_bot__eq__sup__iff,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ X2 @ Y2 ) )
= ( ( X2 = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_60_sup__eq__bot__iff,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ X2 @ Y2 )
= bot_bo7198184520161983622et_nat )
= ( ( X2 = bot_bo7198184520161983622et_nat )
& ( Y2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_61_sup__eq__bot__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= bot_bot_set_set_nat )
= ( ( X2 = bot_bot_set_set_nat )
& ( Y2 = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_62_sup__eq__bot__iff,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ X2 @ Y2 )
= bot_bot_set_nat_nat )
= ( ( X2 = bot_bot_set_nat_nat )
& ( Y2 = bot_bot_set_nat_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_63_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A2 @ B2 )
= bot_bo7198184520161983622et_nat )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_64_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B2 )
= bot_bot_set_set_nat )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_65_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A2 @ B2 )
= bot_bot_set_nat_nat )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B2 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_66_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_67_sup__bot_Oleft__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_68_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_69_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_70_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_71_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_72_mem__Collect__eq,axiom,
! [A2: real,P2: real > $o] :
( ( member_real @ A2 @ ( collect_real @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_73_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_74_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_75_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_76_Collect__mem__eq,axiom,
! [A: set_real] :
( ( collect_real
@ ^ [X3: real] : ( member_real @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_77_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_78_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_79_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_80_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_81_subsetI,axiom,
! [A: set_real,B: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A )
=> ( member_real @ X4 @ B ) )
=> ( ord_less_eq_set_real @ A @ B ) ) ).
% subsetI
thf(fact_82_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_83_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_84_empty__Collect__eq,axiom,
! [P2: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P2 ) )
= ( ! [X3: set_set_nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_85_empty__Collect__eq,axiom,
! [P2: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P2 ) )
= ( ! [X3: set_nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_86_empty__Collect__eq,axiom,
! [P2: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P2 ) )
= ( ! [X3: nat > nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_87_Collect__empty__eq,axiom,
! [P2: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P2 )
= bot_bo7198184520161983622et_nat )
= ( ! [X3: set_set_nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_88_Collect__empty__eq,axiom,
! [P2: set_nat > $o] :
( ( ( collect_set_nat @ P2 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_89_Collect__empty__eq,axiom,
! [P2: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P2 )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_90_all__not__in__conv,axiom,
! [A: set_set_set_set_nat] :
( ( ! [X3: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ X3 @ A ) )
= ( A = bot_bo193956671110832956et_nat ) ) ).
% all_not_in_conv
thf(fact_91_all__not__in__conv,axiom,
! [A: set_real] :
( ( ! [X3: real] :
~ ( member_real @ X3 @ A ) )
= ( A = bot_bot_set_real ) ) ).
% all_not_in_conv
thf(fact_92_all__not__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ! [X3: set_set_nat] :
~ ( member_set_set_nat @ X3 @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_93_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_94_all__not__in__conv,axiom,
! [A: set_nat_nat] :
( ( ! [X3: nat > nat] :
~ ( member_nat_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_95_empty__iff,axiom,
! [C2: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ C2 @ bot_bo193956671110832956et_nat ) ).
% empty_iff
thf(fact_96_empty__iff,axiom,
! [C2: real] :
~ ( member_real @ C2 @ bot_bot_set_real ) ).
% empty_iff
thf(fact_97_empty__iff,axiom,
! [C2: set_set_nat] :
~ ( member_set_set_nat @ C2 @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_98_empty__iff,axiom,
! [C2: set_nat] :
~ ( member_set_nat @ C2 @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_99_empty__iff,axiom,
! [C2: nat > nat] :
~ ( member_nat_nat @ C2 @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_100_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_101_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_102_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_103_sup__left__idem,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_104_sup__left__idem,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_105_sup__left__idem,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% sup_left_idem
thf(fact_106_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_107_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_108_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_109_sup__idem,axiom,
! [X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_110_sup__idem,axiom,
! [X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_111_sup__idem,axiom,
! [X2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_112_sup_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_113_sup_Oidem,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_114_sup_Oidem,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_115_Un__iff,axiom,
! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) )
= ( ( member2946998982187404937et_nat @ C2 @ A )
| ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_116_Un__iff,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( sup_sup_set_real @ A @ B ) )
= ( ( member_real @ C2 @ A )
| ( member_real @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_117_Un__iff,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C2 @ A )
| ( member_set_set_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_118_Un__iff,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C2 @ A )
| ( member_nat_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_119_Un__iff,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C2 @ A )
| ( member_set_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_120_UnCI,axiom,
! [C2: set_set_set_nat,B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( ~ ( member2946998982187404937et_nat @ C2 @ B )
=> ( member2946998982187404937et_nat @ C2 @ A ) )
=> ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_121_UnCI,axiom,
! [C2: real,B: set_real,A: set_real] :
( ( ~ ( member_real @ C2 @ B )
=> ( member_real @ C2 @ A ) )
=> ( member_real @ C2 @ ( sup_sup_set_real @ A @ B ) ) ) ).
% UnCI
thf(fact_122_UnCI,axiom,
! [C2: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( ~ ( member_set_set_nat @ C2 @ B )
=> ( member_set_set_nat @ C2 @ A ) )
=> ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_123_UnCI,axiom,
! [C2: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( ~ ( member_nat_nat @ C2 @ B )
=> ( member_nat_nat @ C2 @ A ) )
=> ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_124_UnCI,axiom,
! [C2: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C2 @ B )
=> ( member_set_nat @ C2 @ A ) )
=> ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_125_Pair__le,axiom,
! [A2: num,B2: num,C2: num,D: num] :
( ( ord_le7298718801444597813um_num @ ( product_Pair_num_num @ A2 @ B2 ) @ ( product_Pair_num_num @ C2 @ D ) )
= ( ( ord_less_eq_num @ A2 @ C2 )
& ( ord_less_eq_num @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_126_Pair__le,axiom,
! [A2: num,B2: nat,C2: num,D: nat] :
( ( ord_le6590474864495760107um_nat @ ( product_Pair_num_nat @ A2 @ B2 ) @ ( product_Pair_num_nat @ C2 @ D ) )
= ( ( ord_less_eq_num @ A2 @ C2 )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_127_Pair__le,axiom,
! [A2: nat,B2: num,C2: nat,D: num] :
( ( ord_le9168388398137128427at_num @ ( product_Pair_nat_num @ A2 @ B2 ) @ ( product_Pair_nat_num @ C2 @ D ) )
= ( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_num @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_128_Pair__le,axiom,
! [A2: nat,B2: nat,C2: nat,D: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ C2 @ D ) )
= ( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_129_Pair__le,axiom,
! [A2: set_set_nat,B2: num,C2: set_set_nat,D: num] :
( ( ord_le4945134163061272151at_num @ ( produc3851147774108065007at_num @ A2 @ B2 ) @ ( produc3851147774108065007at_num @ C2 @ D ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
& ( ord_less_eq_num @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_130_Pair__le,axiom,
! [A2: set_set_nat,B2: nat,C2: set_set_nat,D: nat] :
( ( ord_le4236890226112434445at_nat @ ( produc7293815987952286373at_nat @ A2 @ B2 ) @ ( produc7293815987952286373at_nat @ C2 @ D ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_131_Pair__le,axiom,
! [A2: num,B2: set_set_nat,C2: num,D: set_set_nat] :
( ( ord_le8212681105015131479et_nat @ ( produc5263849632999935215et_nat @ A2 @ B2 ) @ ( produc5263849632999935215et_nat @ C2 @ D ) )
= ( ( ord_less_eq_num @ A2 @ C2 )
& ( ord_le6893508408891458716et_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_132_Pair__le,axiom,
! [A2: num,B2: nat > nat,C2: num,D: nat > nat] :
( ( ord_le1215176170919080154at_nat @ ( produc5125486429189257586at_nat @ A2 @ B2 ) @ ( produc5125486429189257586at_nat @ C2 @ D ) )
= ( ( ord_less_eq_num @ A2 @ C2 )
& ( ord_less_eq_nat_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_133_Pair__le,axiom,
! [A2: nat,B2: set_set_nat,C2: nat,D: set_set_nat] :
( ( ord_le1758471263074804493et_nat @ ( produc8033011827914384037et_nat @ A2 @ B2 ) @ ( produc8033011827914384037et_nat @ C2 @ D ) )
= ( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_le6893508408891458716et_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_134_Pair__le,axiom,
! [A2: nat,B2: nat > nat,C2: nat,D: nat > nat] :
( ( ord_le3929206603849117072at_nat @ ( produc7839516862119294504at_nat @ A2 @ B2 ) @ ( produc7839516862119294504at_nat @ C2 @ D ) )
= ( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat_nat @ B2 @ D ) ) ) ).
% Pair_le
thf(fact_135_empty__subsetI,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% empty_subsetI
thf(fact_136_empty__subsetI,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% empty_subsetI
thf(fact_137_empty__subsetI,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% empty_subsetI
thf(fact_138_subset__empty,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_139_subset__empty,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_140_subset__empty,axiom,
! [A: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_141_sup_Obounded__iff,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
& ( ord_le9131159989063066194et_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_142_sup_Obounded__iff,axiom,
! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
& ( ord_le6893508408891458716et_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_143_sup_Obounded__iff,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
& ( ord_le9059583361652607317at_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_144_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_145_sup_Obounded__iff,axiom,
! [B2: nat > nat,C2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat_nat @ B2 @ A2 )
& ( ord_less_eq_nat_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_146_le__sup__iff,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ Z2 )
= ( ( ord_le9131159989063066194et_nat @ X2 @ Z2 )
& ( ord_le9131159989063066194et_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_147_le__sup__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ Z2 )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Z2 )
& ( ord_le6893508408891458716et_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_148_le__sup__iff,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ Z2 )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Z2 )
& ( ord_le9059583361652607317at_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_149_le__sup__iff,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_150_le__sup__iff,axiom,
! [X2: nat > nat,Y2: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X2 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat_nat @ X2 @ Z2 )
& ( ord_less_eq_nat_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_151_sup__bot_Oright__neutral,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_152_sup__bot_Oright__neutral,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_153_sup__bot_Oright__neutral,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ bot_bot_set_nat_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_154_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( sup_su4213647025997063966et_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B2 = bot_bo7198184520161983622et_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_155_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_set_nat )
& ( B2 = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_156_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( sup_sup_set_nat_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B2 = bot_bot_set_nat_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_157_fst__bot,axiom,
( ( produc6523417423482510407at_nat @ bot_bo3510548743712556943at_nat )
= bot_bo7198184520161983622et_nat ) ).
% fst_bot
thf(fact_158_fst__sup,axiom,
! [X2: produc4045820344675478307at_nat,Y2: produc4045820344675478307at_nat] :
( ( produc6523417423482510407at_nat @ ( sup_su2809688525313048311at_nat @ X2 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ ( produc6523417423482510407at_nat @ X2 ) @ ( produc6523417423482510407at_nat @ Y2 ) ) ) ).
% fst_sup
thf(fact_159_bot__set__def,axiom,
( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ bot_bo6227097192321305471_nat_o ) ) ).
% bot_set_def
thf(fact_160_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_161_bot__set__def,axiom,
( bot_bot_set_nat_nat
= ( collect_nat_nat @ bot_bot_nat_nat_o ) ) ).
% bot_set_def
thf(fact_162_fst__mono,axiom,
! [X2: produc4045820344675478307at_nat,Y2: produc4045820344675478307at_nat] :
( ( ord_le2960050975727596227at_nat @ X2 @ Y2 )
=> ( ord_le9131159989063066194et_nat @ ( produc6523417423482510407at_nat @ X2 ) @ ( produc6523417423482510407at_nat @ Y2 ) ) ) ).
% fst_mono
thf(fact_163_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_164_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_165_set__eq__subset,axiom,
( ( ^ [Y3: set_set_nat,Z3: set_set_nat] : ( Y3 = Z3 ) )
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_166_set__eq__subset,axiom,
( ( ^ [Y3: set_nat_nat,Z3: set_nat_nat] : ( Y3 = Z3 ) )
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_167_subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_168_subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_169_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_170_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_171_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_172_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_173_subset__iff,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A3: set_set_set_set_nat,B3: set_set_set_set_nat] :
! [T: set_set_set_nat] :
( ( member2946998982187404937et_nat @ T @ A3 )
=> ( member2946998982187404937et_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_174_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [T: set_set_nat] :
( ( member_set_set_nat @ T @ A3 )
=> ( member_set_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_175_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [T: real] :
( ( member_real @ T @ A3 )
=> ( member_real @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_176_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_177_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [T: nat > nat] :
( ( member_nat_nat @ T @ A3 )
=> ( member_nat_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_178_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_179_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_180_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_181_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_182_subset__eq,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A3: set_set_set_set_nat,B3: set_set_set_set_nat] :
! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A3 )
=> ( member2946998982187404937et_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_183_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A3 )
=> ( member_set_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_184_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A3: set_real,B3: set_real] :
! [X3: real] :
( ( member_real @ X3 @ A3 )
=> ( member_real @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_185_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( member_set_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_186_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ( member_nat_nat @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_187_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_188_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_189_subsetD,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( member2946998982187404937et_nat @ C2 @ A )
=> ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_190_subsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ C2 @ A )
=> ( member_set_set_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_191_subsetD,axiom,
! [A: set_real,B: set_real,C2: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B ) ) ) ).
% subsetD
thf(fact_192_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C2 @ A )
=> ( member_set_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_193_subsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ C2 @ A )
=> ( member_nat_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_194_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_195_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_196_in__mono,axiom,
! [A: set_real,B: set_real,X2: real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( member_real @ X2 @ A )
=> ( member_real @ X2 @ B ) ) ) ).
% in_mono
thf(fact_197_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_198_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_199_ex__in__conv,axiom,
! [A: set_set_set_set_nat] :
( ( ? [X3: set_set_set_nat] : ( member2946998982187404937et_nat @ X3 @ A ) )
= ( A != bot_bo193956671110832956et_nat ) ) ).
% ex_in_conv
thf(fact_200_ex__in__conv,axiom,
! [A: set_real] :
( ( ? [X3: real] : ( member_real @ X3 @ A ) )
= ( A != bot_bot_set_real ) ) ).
% ex_in_conv
thf(fact_201_ex__in__conv,axiom,
! [A: set_set_set_nat] :
( ( ? [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A ) )
= ( A != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_202_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_203_ex__in__conv,axiom,
! [A: set_nat_nat] :
( ( ? [X3: nat > nat] : ( member_nat_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_204_equals0I,axiom,
! [A: set_set_set_set_nat] :
( ! [Y4: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ Y4 @ A )
=> ( A = bot_bo193956671110832956et_nat ) ) ).
% equals0I
thf(fact_205_equals0I,axiom,
! [A: set_real] :
( ! [Y4: real] :
~ ( member_real @ Y4 @ A )
=> ( A = bot_bot_set_real ) ) ).
% equals0I
thf(fact_206_equals0I,axiom,
! [A: set_set_set_nat] :
( ! [Y4: set_set_nat] :
~ ( member_set_set_nat @ Y4 @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_207_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y4: set_nat] :
~ ( member_set_nat @ Y4 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_208_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y4: nat > nat] :
~ ( member_nat_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_209_equals0D,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( A = bot_bo193956671110832956et_nat )
=> ~ ( member2946998982187404937et_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_210_equals0D,axiom,
! [A: set_real,A2: real] :
( ( A = bot_bot_set_real )
=> ~ ( member_real @ A2 @ A ) ) ).
% equals0D
thf(fact_211_equals0D,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( A = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_212_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_213_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_214_emptyE,axiom,
! [A2: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ A2 @ bot_bo193956671110832956et_nat ) ).
% emptyE
thf(fact_215_emptyE,axiom,
! [A2: real] :
~ ( member_real @ A2 @ bot_bot_set_real ) ).
% emptyE
thf(fact_216_emptyE,axiom,
! [A2: set_set_nat] :
~ ( member_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_217_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_218_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_219_sup__left__commute,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_220_sup__left__commute,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_221_sup__left__commute,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_222_sup_Oleft__commute,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ B2 @ ( sup_su4213647025997063966et_nat @ A2 @ C2 ) )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_223_sup_Oleft__commute,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ B2 @ ( sup_sup_set_nat_nat @ A2 @ C2 ) )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_224_sup_Oleft__commute,axiom,
! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ B2 @ ( sup_sup_set_set_nat @ A2 @ C2 ) )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_225_sup__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X3: set_set_set_nat,Y5: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y5 @ X3 ) ) ) ).
% sup_commute
thf(fact_226_sup__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [X3: set_nat_nat,Y5: set_nat_nat] : ( sup_sup_set_nat_nat @ Y5 @ X3 ) ) ) ).
% sup_commute
thf(fact_227_sup__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y5: set_set_nat] : ( sup_sup_set_set_nat @ Y5 @ X3 ) ) ) ).
% sup_commute
thf(fact_228_sup_Ocommute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_229_sup_Ocommute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( sup_sup_set_nat_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_230_sup_Ocommute,axiom,
( sup_sup_set_set_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] : ( sup_sup_set_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_231_sup__assoc,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ Z2 )
= ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_232_sup__assoc,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ Z2 )
= ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_233_sup__assoc,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ Z2 )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_234_sup_Oassoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ C2 )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_235_sup_Oassoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_236_sup_Oassoc,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_237_inf__sup__aci_I5_J,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [X3: set_set_set_nat,Y5: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_238_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat_nat
= ( ^ [X3: set_nat_nat,Y5: set_nat_nat] : ( sup_sup_set_nat_nat @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_239_inf__sup__aci_I5_J,axiom,
( sup_sup_set_set_nat
= ( ^ [X3: set_set_nat,Y5: set_set_nat] : ( sup_sup_set_set_nat @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_240_inf__sup__aci_I6_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ Z2 )
= ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_241_inf__sup__aci_I6_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ Z2 )
= ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_242_inf__sup__aci_I6_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ Z2 )
= ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_243_inf__sup__aci_I7_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_244_inf__sup__aci_I7_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat_nat @ Y2 @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_245_inf__sup__aci_I7_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_set_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_246_inf__sup__aci_I8_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_247_inf__sup__aci_I8_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) )
= ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_248_inf__sup__aci_I8_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_249_Un__left__commute,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) )
= ( sup_su4213647025997063966et_nat @ B @ ( sup_su4213647025997063966et_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_250_Un__left__commute,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) )
= ( sup_sup_set_nat_nat @ B @ ( sup_sup_set_nat_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_251_Un__left__commute,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) )
= ( sup_sup_set_set_nat @ B @ ( sup_sup_set_set_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_252_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_253_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_254_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_255_Un__commute,axiom,
( sup_su4213647025997063966et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( sup_su4213647025997063966et_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_256_Un__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] : ( sup_sup_set_nat_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_257_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] : ( sup_sup_set_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_258_Un__absorb,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_259_Un__absorb,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_260_Un__absorb,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_261_Un__assoc,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C )
= ( sup_su4213647025997063966et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_262_Un__assoc,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C )
= ( sup_sup_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_263_Un__assoc,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C )
= ( sup_sup_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_264_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_265_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_266_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_267_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_268_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_269_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_270_UnI2,axiom,
! [C2: set_set_set_nat,B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ B )
=> ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_271_UnI2,axiom,
! [C2: real,B: set_real,A: set_real] :
( ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( sup_sup_set_real @ A @ B ) ) ) ).
% UnI2
thf(fact_272_UnI2,axiom,
! [C2: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ B )
=> ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_273_UnI2,axiom,
! [C2: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( member_nat_nat @ C2 @ B )
=> ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_274_UnI2,axiom,
! [C2: set_nat,B: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ C2 @ B )
=> ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_275_UnI1,axiom,
! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ A )
=> ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_276_UnI1,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ ( sup_sup_set_real @ A @ B ) ) ) ).
% UnI1
thf(fact_277_UnI1,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ A )
=> ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_278_UnI1,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ A )
=> ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_279_UnI1,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ A )
=> ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_280_UnE,axiom,
! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ ( sup_su3906748206781935060et_nat @ A @ B ) )
=> ( ~ ( member2946998982187404937et_nat @ C2 @ A )
=> ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_281_UnE,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( sup_sup_set_real @ A @ B ) )
=> ( ~ ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B ) ) ) ).
% UnE
thf(fact_282_UnE,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ( ~ ( member_set_set_nat @ C2 @ A )
=> ( member_set_set_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_283_UnE,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ( ~ ( member_nat_nat @ C2 @ A )
=> ( member_nat_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_284_UnE,axiom,
! [C2: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C2 @ ( sup_sup_set_set_nat @ A @ B ) )
=> ( ~ ( member_set_nat @ C2 @ A )
=> ( member_set_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_285_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_286_first__assumptions_Ojoinl__join,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K @ X @ Y ) @ ( clique5469973757772500719t_odot @ X @ Y ) ) ) ).
% first_assumptions.joinl_join
thf(fact_287_Pair__mono,axiom,
! [X2: num,X5: num,Y2: num,Y6: num] :
( ( ord_less_eq_num @ X2 @ X5 )
=> ( ( ord_less_eq_num @ Y2 @ Y6 )
=> ( ord_le7298718801444597813um_num @ ( product_Pair_num_num @ X2 @ Y2 ) @ ( product_Pair_num_num @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_288_Pair__mono,axiom,
! [X2: num,X5: num,Y2: nat,Y6: nat] :
( ( ord_less_eq_num @ X2 @ X5 )
=> ( ( ord_less_eq_nat @ Y2 @ Y6 )
=> ( ord_le6590474864495760107um_nat @ ( product_Pair_num_nat @ X2 @ Y2 ) @ ( product_Pair_num_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_289_Pair__mono,axiom,
! [X2: nat,X5: nat,Y2: num,Y6: num] :
( ( ord_less_eq_nat @ X2 @ X5 )
=> ( ( ord_less_eq_num @ Y2 @ Y6 )
=> ( ord_le9168388398137128427at_num @ ( product_Pair_nat_num @ X2 @ Y2 ) @ ( product_Pair_nat_num @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_290_Pair__mono,axiom,
! [X2: nat,X5: nat,Y2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ X5 )
=> ( ( ord_less_eq_nat @ Y2 @ Y6 )
=> ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( product_Pair_nat_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_291_Pair__mono,axiom,
! [X2: set_set_nat,X5: set_set_nat,Y2: num,Y6: num] :
( ( ord_le6893508408891458716et_nat @ X2 @ X5 )
=> ( ( ord_less_eq_num @ Y2 @ Y6 )
=> ( ord_le4945134163061272151at_num @ ( produc3851147774108065007at_num @ X2 @ Y2 ) @ ( produc3851147774108065007at_num @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_292_Pair__mono,axiom,
! [X2: set_set_nat,X5: set_set_nat,Y2: nat,Y6: nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ X5 )
=> ( ( ord_less_eq_nat @ Y2 @ Y6 )
=> ( ord_le4236890226112434445at_nat @ ( produc7293815987952286373at_nat @ X2 @ Y2 ) @ ( produc7293815987952286373at_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_293_Pair__mono,axiom,
! [X2: num,X5: num,Y2: set_set_nat,Y6: set_set_nat] :
( ( ord_less_eq_num @ X2 @ X5 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ Y6 )
=> ( ord_le8212681105015131479et_nat @ ( produc5263849632999935215et_nat @ X2 @ Y2 ) @ ( produc5263849632999935215et_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_294_Pair__mono,axiom,
! [X2: num,X5: num,Y2: nat > nat,Y6: nat > nat] :
( ( ord_less_eq_num @ X2 @ X5 )
=> ( ( ord_less_eq_nat_nat @ Y2 @ Y6 )
=> ( ord_le1215176170919080154at_nat @ ( produc5125486429189257586at_nat @ X2 @ Y2 ) @ ( produc5125486429189257586at_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_295_Pair__mono,axiom,
! [X2: nat,X5: nat,Y2: set_set_nat,Y6: set_set_nat] :
( ( ord_less_eq_nat @ X2 @ X5 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ Y6 )
=> ( ord_le1758471263074804493et_nat @ ( produc8033011827914384037et_nat @ X2 @ Y2 ) @ ( produc8033011827914384037et_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_296_Pair__mono,axiom,
! [X2: nat,X5: nat,Y2: nat > nat,Y6: nat > nat] :
( ( ord_less_eq_nat @ X2 @ X5 )
=> ( ( ord_less_eq_nat_nat @ Y2 @ Y6 )
=> ( ord_le3929206603849117072at_nat @ ( produc7839516862119294504at_nat @ X2 @ Y2 ) @ ( produc7839516862119294504at_nat @ X5 @ Y6 ) ) ) ) ).
% Pair_mono
thf(fact_297_bot__prod__def,axiom,
( bot_bo2769642828321324397at_nat
= ( product_Pair_nat_nat @ bot_bot_nat @ bot_bot_nat ) ) ).
% bot_prod_def
thf(fact_298_bot__prod__def,axiom,
( bot_bo5242310025275135449at_nat
= ( produc7293815987952286373at_nat @ bot_bot_set_set_nat @ bot_bot_nat ) ) ).
% bot_prod_def
thf(fact_299_bot__prod__def,axiom,
( bot_bo2763891062237505497et_nat
= ( produc8033011827914384037et_nat @ bot_bot_nat @ bot_bot_set_set_nat ) ) ).
% bot_prod_def
thf(fact_300_bot__prod__def,axiom,
( bot_bo3510548743712556943at_nat
= ( produc2803780273060847707at_nat @ bot_bo7198184520161983622et_nat @ bot_bot_nat ) ) ).
% bot_prod_def
thf(fact_301_bot__prod__def,axiom,
( bot_bo3319466194874277650at_nat
= ( produc4994002345568745310at_nat @ bot_bot_set_nat_nat @ bot_bot_nat ) ) ).
% bot_prod_def
thf(fact_302_bot__prod__def,axiom,
( bot_bo3680126437115085455et_nat
= ( produc1530211740221758555et_nat @ bot_bot_nat @ bot_bo7198184520161983622et_nat ) ) ).
% bot_prod_def
thf(fact_303_bot__prod__def,axiom,
( bot_bo98289398700565778at_nat
= ( produc6017321930131081694at_nat @ bot_bot_nat @ bot_bot_set_nat_nat ) ) ).
% bot_prod_def
thf(fact_304_bot__prod__def,axiom,
( bot_bo6021600403123937349et_nat
= ( produc9057842353944101649et_nat @ bot_bot_set_set_nat @ bot_bot_set_set_nat ) ) ).
% bot_prod_def
thf(fact_305_bot__prod__def,axiom,
( bot_bo6248027959722598907et_nat
= ( produc1498124630991567047et_nat @ bot_bo7198184520161983622et_nat @ bot_bot_set_set_nat ) ) ).
% bot_prod_def
thf(fact_306_bot__prod__def,axiom,
( bot_bo7502318977428687611et_nat
= ( produc7315026656311086279et_nat @ bot_bot_set_set_nat @ bot_bo7198184520161983622et_nat ) ) ).
% bot_prod_def
thf(fact_307_sup_OcoboundedI2,axiom,
! [C2: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ B2 )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_308_sup_OcoboundedI2,axiom,
! [C2: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_309_sup_OcoboundedI2,axiom,
! [C2: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ B2 )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_310_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_311_sup_OcoboundedI2,axiom,
! [C2: nat > nat,B2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ C2 @ B2 )
=> ( ord_less_eq_nat_nat @ C2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_312_sup_OcoboundedI1,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_313_sup_OcoboundedI1,axiom,
! [C2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_314_sup_OcoboundedI1,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_315_sup_OcoboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_316_sup_OcoboundedI1,axiom,
! [C2: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ C2 @ A2 )
=> ( ord_less_eq_nat_nat @ C2 @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_317_sup_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_318_sup_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_319_sup_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_320_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_321_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] :
( ( sup_sup_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_322_sup_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_323_sup_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_324_sup_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_325_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_326_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( ( sup_sup_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_327_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_328_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_329_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_330_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_331_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_332_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_333_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_334_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_335_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_336_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_337_sup_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( A4
= ( sup_su4213647025997063966et_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_338_sup_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( A4
= ( sup_sup_set_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_339_sup_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( A4
= ( sup_sup_set_nat_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_340_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_341_sup_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( A4
= ( sup_sup_nat_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_342_sup_OboundedI,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_343_sup_OboundedI,axiom,
! [B2: set_set_nat,A2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_344_sup_OboundedI,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_345_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_346_sup_OboundedI,axiom,
! [B2: nat > nat,A2: nat > nat,C2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ C2 @ A2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_347_sup_OboundedE,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le9131159989063066194et_nat @ B2 @ A2 )
=> ~ ( ord_le9131159989063066194et_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_348_sup_OboundedE,axiom,
! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ~ ( ord_le6893508408891458716et_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_349_sup_OboundedE,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le9059583361652607317at_nat @ B2 @ A2 )
=> ~ ( ord_le9059583361652607317at_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_350_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_351_sup_OboundedE,axiom,
! [B2: nat > nat,C2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_352_sup__absorb2,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_353_sup__absorb2,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_354_sup__absorb2,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_355_sup__absorb2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_356_sup__absorb2,axiom,
! [X2: nat > nat,Y2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( sup_sup_nat_nat @ X2 @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_357_sup__absorb1,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X2 )
=> ( ( sup_su4213647025997063966et_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_358_sup__absorb1,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( sup_sup_set_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_359_sup__absorb1,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( ( sup_sup_set_nat_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_360_sup__absorb1,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_361_sup__absorb1,axiom,
! [Y2: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y2 @ X2 )
=> ( ( sup_sup_nat_nat @ X2 @ Y2 )
= X2 ) ) ).
% sup_absorb1
thf(fact_362_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_363_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_364_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_365_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_366_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_367_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_368_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_369_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_370_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_371_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_372_sup__unique,axiom,
! [F: set_set_set_nat > set_set_set_nat > set_set_set_nat,X2: set_set_set_nat,Y2: set_set_set_nat] :
( ! [X4: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_set_nat,Y4: set_set_set_nat,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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_373_sup__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ! [X4: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_set_nat,Y4: set_set_nat,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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_374_sup__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X2: set_nat_nat,Y2: set_nat_nat] :
( ! [X4: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: set_nat_nat,Y4: set_nat_nat,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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_375_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X4 @ Y4 ) )
=> ( ! [X4: nat,Y4: nat,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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_376_sup__unique,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,X2: nat > nat,Y2: 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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_377_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_378_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_379_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_380_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_381_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_382_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_383_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_384_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_385_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_386_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_387_le__iff__sup,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X3: set_set_set_nat,Y5: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_388_le__iff__sup,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y5: set_set_nat] :
( ( sup_sup_set_set_nat @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_389_le__iff__sup,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y5: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_390_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( sup_sup_nat @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_391_le__iff__sup,axiom,
( ord_less_eq_nat_nat
= ( ^ [X3: nat > nat,Y5: nat > nat] :
( ( sup_sup_nat_nat @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_392_sup__least,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X2 )
=> ( ( ord_le9131159989063066194et_nat @ Z2 @ X2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_393_sup__least,axiom,
! [Y2: set_set_nat,X2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ Z2 @ X2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_394_sup__least,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ Z2 @ X2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_395_sup__least,axiom,
! [Y2: nat,X2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_396_sup__least,axiom,
! [Y2: nat > nat,X2: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ Y2 @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_397_sup__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) @ ( sup_su4213647025997063966et_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_398_sup__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) @ ( sup_sup_set_set_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_399_sup__mono,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B2 ) @ ( sup_sup_set_nat_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_400_sup__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_401_sup__mono,axiom,
! [A2: nat > nat,C2: nat > nat,B2: nat > nat,D: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ D )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ A2 @ B2 ) @ ( sup_sup_nat_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_402_sup_Omono,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,D: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ D @ B2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ C2 @ D ) @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_403_sup_Omono,axiom,
! [C2: set_set_nat,A2: set_set_nat,D: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ D @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ C2 @ D ) @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_404_sup_Omono,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,D: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ D @ B2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ C2 @ D ) @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_405_sup_Omono,axiom,
! [C2: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_406_sup_Omono,axiom,
! [C2: nat > nat,A2: nat > nat,D: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat_nat @ D @ B2 )
=> ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ C2 @ D ) @ ( sup_sup_nat_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_407_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_408_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_409_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_410_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_411_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_412_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_413_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_414_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_415_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_416_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_417_sup__ge2,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_418_sup__ge2,axiom,
! [Y2: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_419_sup__ge2,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_420_sup__ge2,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_421_sup__ge2,axiom,
! [Y2: nat > nat,X2: nat > nat] : ( ord_less_eq_nat_nat @ Y2 @ ( sup_sup_nat_nat @ X2 @ Y2 ) ) ).
% sup_ge2
thf(fact_422_sup__ge1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_423_sup__ge1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_424_sup__ge1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_425_sup__ge1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_426_sup__ge1,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ X2 @ Y2 ) ) ).
% sup_ge1
thf(fact_427_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_428_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_429_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_430_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_431_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_432_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_433_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_434_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_435_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_436_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_437_inf__sup__ord_I3_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_438_inf__sup__ord_I3_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_439_inf__sup__ord_I3_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_440_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_441_inf__sup__ord_I3_J,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ ( sup_sup_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_442_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ Y2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_443_inf__sup__ord_I4_J,axiom,
! [Y2: set_set_nat,X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ Y2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_444_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ Y2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_445_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X2: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_446_inf__sup__ord_I4_J,axiom,
! [Y2: nat > nat,X2: nat > nat] : ( ord_less_eq_nat_nat @ Y2 @ ( sup_sup_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_447_sup_Ostrict__coboundedI2,axiom,
! [C2: set_nat_nat,B2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ C2 @ B2 )
=> ( ord_less_set_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_448_sup_Ostrict__coboundedI2,axiom,
! [C2: set_set_nat,B2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ C2 @ B2 )
=> ( ord_less_set_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_449_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_450_sup_Ostrict__coboundedI2,axiom,
! [C2: set_set_set_nat,B2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ C2 @ B2 )
=> ( ord_le152980574450754630et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_451_sup_Ostrict__coboundedI2,axiom,
! [C2: real,B2: real,A2: real] :
( ( ord_less_real @ C2 @ B2 )
=> ( ord_less_real @ C2 @ ( sup_sup_real @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_452_sup_Ostrict__coboundedI1,axiom,
! [C2: set_nat_nat,A2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ C2 @ A2 )
=> ( ord_less_set_nat_nat @ C2 @ ( sup_sup_set_nat_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_453_sup_Ostrict__coboundedI1,axiom,
! [C2: set_set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ C2 @ A2 )
=> ( ord_less_set_set_nat @ C2 @ ( sup_sup_set_set_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_454_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C2 @ A2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_455_sup_Ostrict__coboundedI1,axiom,
! [C2: set_set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ C2 @ A2 )
=> ( ord_le152980574450754630et_nat @ C2 @ ( sup_su4213647025997063966et_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_456_sup_Ostrict__coboundedI1,axiom,
! [C2: real,A2: real,B2: real] :
( ( ord_less_real @ C2 @ A2 )
=> ( ord_less_real @ C2 @ ( sup_sup_real @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_457_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( A4
= ( sup_sup_set_nat_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_458_sup_Ostrict__order__iff,axiom,
( ord_less_set_set_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( A4
= ( sup_sup_set_set_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_459_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_460_sup_Ostrict__order__iff,axiom,
( ord_le152980574450754630et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( ( A4
= ( sup_su4213647025997063966et_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_461_sup_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [B4: real,A4: real] :
( ( A4
= ( sup_sup_real @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_462_sup_Ostrict__boundedE,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ ( sup_sup_set_nat_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_set_nat_nat @ B2 @ A2 )
=> ~ ( ord_less_set_nat_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_463_sup_Ostrict__boundedE,axiom,
! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( ord_less_set_set_nat @ ( sup_sup_set_set_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_set_set_nat @ B2 @ A2 )
=> ~ ( ord_less_set_set_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_464_sup_Ostrict__boundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_465_sup_Ostrict__boundedE,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ ( sup_su4213647025997063966et_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le152980574450754630et_nat @ B2 @ A2 )
=> ~ ( ord_le152980574450754630et_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_466_sup_Ostrict__boundedE,axiom,
! [B2: real,C2: real,A2: real] :
( ( ord_less_real @ ( sup_sup_real @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_real @ B2 @ A2 )
=> ~ ( ord_less_real @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_467_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_468_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_469_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_470_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_471_sup_Oabsorb4,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( sup_sup_real @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_472_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_473_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_474_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_475_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_476_sup_Oabsorb3,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( sup_sup_real @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_477_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_478_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_479_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_480_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_481_less__supI2,axiom,
! [X2: real,B2: real,A2: real] :
( ( ord_less_real @ X2 @ B2 )
=> ( ord_less_real @ X2 @ ( sup_sup_real @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_482_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_483_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_484_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_485_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_486_less__supI1,axiom,
! [X2: real,A2: real,B2: real] :
( ( ord_less_real @ X2 @ A2 )
=> ( ord_less_real @ X2 @ ( sup_sup_real @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_487_subset__Un__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_488_subset__Un__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( sup_sup_set_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_489_subset__Un__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_490_subset__UnE,axiom,
! [C: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) )
=> ~ ! [A5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A5 @ A )
=> ! [B5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B5 @ B )
=> ( C
!= ( sup_su4213647025997063966et_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_491_subset__UnE,axiom,
! [C: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
=> ~ ! [A5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A5 @ A )
=> ! [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B5 @ B )
=> ( C
!= ( sup_sup_set_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_492_subset__UnE,axiom,
! [C: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
=> ~ ! [A5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A5 @ A )
=> ! [B5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B5 @ B )
=> ( C
!= ( sup_sup_set_nat_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_493_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_494_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_495_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_496_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_497_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_498_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_499_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_500_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_501_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_502_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_503_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_504_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_505_Un__least,axiom,
! [A: set_set_set_nat,C: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_506_Un__least,axiom,
! [A: set_set_nat,C: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_507_Un__least,axiom,
! [A: set_nat_nat,C: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_508_Un__mono,axiom,
! [A: set_set_set_nat,C: set_set_set_nat,B: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ( ord_le9131159989063066194et_nat @ B @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_509_Un__mono,axiom,
! [A: set_set_nat,C: set_set_nat,B: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( ord_le6893508408891458716et_nat @ B @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_510_Un__mono,axiom,
! [A: set_nat_nat,C: set_nat_nat,B: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ( ord_le9059583361652607317at_nat @ B @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_511_Un__empty__right,axiom,
! [A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ bot_bo7198184520161983622et_nat )
= A ) ).
% Un_empty_right
thf(fact_512_Un__empty__right,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% Un_empty_right
thf(fact_513_Un__empty__right,axiom,
! [A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ bot_bot_set_nat_nat )
= A ) ).
% Un_empty_right
thf(fact_514_Un__empty__left,axiom,
! [B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ bot_bo7198184520161983622et_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_515_Un__empty__left,axiom,
! [B: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_516_Un__empty__left,axiom,
! [B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ bot_bot_set_nat_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_517_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_518_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_519_odotl__def,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k @ X @ Y )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X @ Y ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).
% odotl_def
thf(fact_520_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_521_ACC__union,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ).
% ACC_union
thf(fact_522_ACC__cf__union,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_union
thf(fact_523_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_524_ACC__cf__mono,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_mono
thf(fact_525_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_526_old_Oprod_Oinject,axiom,
! [A2: set_set_set_nat,B2: nat,A6: set_set_set_nat,B6: nat] :
( ( ( produc2803780273060847707at_nat @ A2 @ B2 )
= ( produc2803780273060847707at_nat @ A6 @ B6 ) )
= ( ( A2 = A6 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_527_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_528_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_529_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_530_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_531_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_532_inf__right__idem,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Y2 )
= ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_533_inf__right__idem,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_nat_nat @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_534_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_535_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_536_inf__left__idem,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_537_inf__left__idem,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) )
= ( inf_inf_set_nat_nat @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_538_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_539_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_540_inf__idem,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_541_inf__idem,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_542_inf_Oidem,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_543_inf_Oidem,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_544_Int__iff,axiom,
! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ ( inf_in2396666505901392698et_nat @ A @ B ) )
= ( ( member2946998982187404937et_nat @ C2 @ A )
& ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_545_Int__iff,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) )
= ( ( member_real @ C2 @ A )
& ( member_real @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_546_Int__iff,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C2 @ A )
& ( member_set_set_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_547_Int__iff,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C2 @ A )
& ( member_nat_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_548_IntI,axiom,
! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ A )
=> ( ( member2946998982187404937et_nat @ C2 @ B )
=> ( member2946998982187404937et_nat @ C2 @ ( inf_in2396666505901392698et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_549_IntI,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ A )
=> ( ( member_real @ C2 @ B )
=> ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) ) ) ) ).
% IntI
thf(fact_550_IntI,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ A )
=> ( ( member_set_set_nat @ C2 @ B )
=> ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_551_IntI,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ A )
=> ( ( member_nat_nat @ C2 @ B )
=> ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_552_ACC__odot,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ).
% ACC_odot
thf(fact_553_Lm,axiom,
ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lm
thf(fact_554_le__inf__iff,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) )
= ( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
& ( ord_le9131159989063066194et_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_555_le__inf__iff,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z2 ) )
= ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
& ( ord_le6893508408891458716et_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_556_le__inf__iff,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) )
= ( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
& ( ord_le9059583361652607317at_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_557_le__inf__iff,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_558_le__inf__iff,axiom,
! [X2: nat > nat,Y2: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_nat_nat @ X2 @ Y2 )
& ( ord_less_eq_nat_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_559_inf_Obounded__iff,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) )
= ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_560_inf_Obounded__iff,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_561_inf_Obounded__iff,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
& ( ord_le9059583361652607317at_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_562_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_563_inf_Obounded__iff,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat_nat @ A2 @ B2 )
& ( ord_less_eq_nat_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_564_inf__bot__left,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_left
thf(fact_565_inf__bot__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_566_inf__bot__left,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= bot_bot_set_nat_nat ) ).
% inf_bot_left
thf(fact_567_inf__bot__right,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% inf_bot_right
thf(fact_568_inf__bot__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_569_inf__bot__right,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% inf_bot_right
thf(fact_570_inf__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: nat,C2: set_set_set_nat,D: nat] :
( ( inf_in8110749398470801937at_nat @ ( produc2803780273060847707at_nat @ A2 @ B2 ) @ ( produc2803780273060847707at_nat @ C2 @ D ) )
= ( produc2803780273060847707at_nat @ ( inf_in5711780100303410308et_nat @ A2 @ C2 ) @ ( inf_inf_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_571_inf__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat,D: set_set_set_nat] :
( ( inf_in7092662171172785971et_nat @ ( produc8443863378681539197et_nat @ A2 @ B2 ) @ ( produc8443863378681539197et_nat @ C2 @ D ) )
= ( produc8443863378681539197et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ C2 ) @ ( inf_in5711780100303410308et_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_572_inf__Pair__Pair,axiom,
! [A2: set_set_set_nat,B2: set_nat_nat,C2: set_set_set_nat,D: set_nat_nat] :
( ( inf_in6765570788667771830at_nat @ ( produc507788731782991104at_nat @ A2 @ B2 ) @ ( produc507788731782991104at_nat @ C2 @ D ) )
= ( produc507788731782991104at_nat @ ( inf_in5711780100303410308et_nat @ A2 @ C2 ) @ ( inf_inf_set_nat_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_573_inf__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_set_set_nat,C2: set_nat_nat,D: set_set_set_nat] :
( ( inf_in4890968077345302454et_nat @ ( produc2168915450337631616et_nat @ A2 @ B2 ) @ ( produc2168915450337631616et_nat @ C2 @ D ) )
= ( produc2168915450337631616et_nat @ ( inf_inf_set_nat_nat @ A2 @ C2 ) @ ( inf_in5711780100303410308et_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_574_inf__Pair__Pair,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat,D: set_nat_nat] :
( ( inf_in8487852146424888249at_nat @ ( produc8650507651752646531at_nat @ A2 @ B2 ) @ ( produc8650507651752646531at_nat @ C2 @ D ) )
= ( produc8650507651752646531at_nat @ ( inf_inf_set_nat_nat @ A2 @ C2 ) @ ( inf_inf_set_nat_nat @ B2 @ D ) ) ) ).
% inf_Pair_Pair
thf(fact_575_Int__subset__iff,axiom,
! [C: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( ord_le9131159989063066194et_nat @ C @ A )
& ( ord_le9131159989063066194et_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_576_Int__subset__iff,axiom,
! [C: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( ord_le6893508408891458716et_nat @ C @ A )
& ( ord_le6893508408891458716et_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_577_Int__subset__iff,axiom,
! [C: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C @ A )
& ( ord_le9059583361652607317at_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_578_inf__sup__absorb,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_579_inf__sup__absorb,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_580_inf__sup__absorb,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% inf_sup_absorb
thf(fact_581_sup__inf__absorb,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_582_sup__inf__absorb,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_583_sup__inf__absorb,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ X2 @ Y2 ) )
= X2 ) ).
% sup_inf_absorb
thf(fact_584_Int__Un__eq_I4_J,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ T2 @ ( inf_in5711780100303410308et_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_585_Int__Un__eq_I4_J,axiom,
! [T2: set_nat_nat,S: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T2 @ ( inf_inf_set_nat_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_586_Int__Un__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_set_nat @ T2 @ ( inf_inf_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_587_Int__Un__eq_I3_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ S @ ( inf_in5711780100303410308et_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_588_Int__Un__eq_I3_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S @ ( inf_inf_set_nat_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_589_Int__Un__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ S @ ( inf_inf_set_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_590_Int__Un__eq_I2_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_591_Int__Un__eq_I2_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_592_Int__Un__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_593_Int__Un__eq_I1_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_594_Int__Un__eq_I1_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_595_Int__Un__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_596_Un__Int__eq_I4_J,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ T2 @ ( sup_su4213647025997063966et_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_597_Un__Int__eq_I4_J,axiom,
! [T2: set_nat_nat,S: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T2 @ ( sup_sup_set_nat_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_598_Un__Int__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_set_nat @ T2 @ ( sup_sup_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_599_Un__Int__eq_I3_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ S @ ( sup_su4213647025997063966et_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_600_Un__Int__eq_I3_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S @ ( sup_sup_set_nat_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_601_Un__Int__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ S @ ( sup_sup_set_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_602_Un__Int__eq_I2_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_603_Un__Int__eq_I2_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_604_Un__Int__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_605_Un__Int__eq_I1_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_606_Un__Int__eq_I1_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_607_Un__Int__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_608_fst__inf,axiom,
! [X2: produc4045820344675478307at_nat,Y2: produc4045820344675478307at_nat] :
( ( produc6523417423482510407at_nat @ ( inf_in8110749398470801937at_nat @ X2 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ ( produc6523417423482510407at_nat @ X2 ) @ ( produc6523417423482510407at_nat @ Y2 ) ) ) ).
% fst_inf
thf(fact_609_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_610_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_611_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_612_Int__left__commute,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C ) )
= ( inf_in5711780100303410308et_nat @ B @ ( inf_in5711780100303410308et_nat @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_613_Int__left__commute,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C ) )
= ( inf_inf_set_nat_nat @ B @ ( inf_inf_set_nat_nat @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_614_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_615_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_616_Int__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_617_Int__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] : ( inf_inf_set_nat_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_618_Int__absorb,axiom,
! [A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_619_Int__absorb,axiom,
! [A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_620_Int__assoc,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C )
= ( inf_in5711780100303410308et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C ) ) ) ).
% Int_assoc
thf(fact_621_Int__assoc,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C )
= ( inf_inf_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C ) ) ) ).
% Int_assoc
thf(fact_622_IntD2,axiom,
! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ ( inf_in2396666505901392698et_nat @ A @ B ) )
=> ( member2946998982187404937et_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_623_IntD2,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) )
=> ( member_real @ C2 @ B ) ) ).
% IntD2
thf(fact_624_IntD2,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ( member_set_set_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_625_IntD2,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_626_IntD1,axiom,
! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ ( inf_in2396666505901392698et_nat @ A @ B ) )
=> ( member2946998982187404937et_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_627_IntD1,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) )
=> ( member_real @ C2 @ A ) ) ).
% IntD1
thf(fact_628_IntD1,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ( member_set_set_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_629_IntD1,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ( member_nat_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_630_IntE,axiom,
! [C2: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C2 @ ( inf_in2396666505901392698et_nat @ A @ B ) )
=> ~ ( ( member2946998982187404937et_nat @ C2 @ A )
=> ~ ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_631_IntE,axiom,
! [C2: real,A: set_real,B: set_real] :
( ( member_real @ C2 @ ( inf_inf_set_real @ A @ B ) )
=> ~ ( ( member_real @ C2 @ A )
=> ~ ( member_real @ C2 @ B ) ) ) ).
% IntE
thf(fact_632_IntE,axiom,
! [C2: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
=> ~ ( ( member_set_set_nat @ C2 @ A )
=> ~ ( member_set_set_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_633_IntE,axiom,
! [C2: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
=> ~ ( ( member_nat_nat @ C2 @ A )
=> ~ ( member_nat_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_634_inf__left__commute,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ Y2 @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_635_inf__left__commute,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat_nat @ Y2 @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_636_inf_Oleft__commute,axiom,
! [B2: set_set_set_nat,A2: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ B2 @ ( inf_in5711780100303410308et_nat @ A2 @ C2 ) )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_637_inf_Oleft__commute,axiom,
! [B2: set_nat_nat,A2: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ B2 @ ( inf_inf_set_nat_nat @ A2 @ C2 ) )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_638_inf__commute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X3: set_set_set_nat,Y5: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y5 @ X3 ) ) ) ).
% inf_commute
thf(fact_639_inf__commute,axiom,
( inf_inf_set_nat_nat
= ( ^ [X3: set_nat_nat,Y5: set_nat_nat] : ( inf_inf_set_nat_nat @ Y5 @ X3 ) ) ) ).
% inf_commute
thf(fact_640_inf_Ocommute,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_641_inf_Ocommute,axiom,
( inf_inf_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] : ( inf_inf_set_nat_nat @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_642_inf__assoc,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Z2 )
= ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_643_inf__assoc,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ Z2 )
= ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_644_inf_Oassoc,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_645_inf_Oassoc,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_646_inf__sup__aci_I1_J,axiom,
( inf_in5711780100303410308et_nat
= ( ^ [X3: set_set_set_nat,Y5: set_set_set_nat] : ( inf_in5711780100303410308et_nat @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_647_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat_nat
= ( ^ [X3: set_nat_nat,Y5: set_nat_nat] : ( inf_inf_set_nat_nat @ Y5 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_648_inf__sup__aci_I2_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Z2 )
= ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_649_inf__sup__aci_I2_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ Z2 )
= ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_650_inf__sup__aci_I3_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ Y2 @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_651_inf__sup__aci_I3_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat_nat @ Y2 @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_652_inf__sup__aci_I4_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) )
= ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_653_inf__sup__aci_I4_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) )
= ( inf_inf_set_nat_nat @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_654_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_655_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_656_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_657_inf__sup__ord_I2_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_658_inf__sup__ord_I2_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_659_inf__sup__ord_I2_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_660_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_661_inf__sup__ord_I2_J,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_662_inf__sup__ord_I1_J,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_663_inf__sup__ord_I1_J,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_664_inf__sup__ord_I1_J,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_665_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_666_inf__sup__ord_I1_J,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_667_inf__le1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_668_inf__le1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_669_inf__le1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_670_inf__le1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_671_inf__le1,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_672_inf__le2,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_673_inf__le2,axiom,
! [X2: set_set_nat,Y2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_674_inf__le2,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_675_inf__le2,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_676_inf__le2,axiom,
! [X2: nat > nat,Y2: nat > nat] : ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_inf__mono,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat,D: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ( ord_le9131159989063066194et_nat @ B2 @ D )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ ( inf_in5711780100303410308et_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_688_inf__mono,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat,D: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ D )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ ( inf_inf_set_set_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_689_inf__mono,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat,D: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ( ord_le9059583361652607317at_nat @ B2 @ D )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ ( inf_inf_set_nat_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_690_inf__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_691_inf__mono,axiom,
! [A2: nat > nat,C2: nat > nat,B2: nat > nat,D: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat_nat @ B2 @ D )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ ( inf_inf_nat_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_692_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_693_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_694_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_695_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_696_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_697_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_698_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_699_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_700_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_701_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_702_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_703_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_704_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_705_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_706_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_707_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_708_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_709_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_710_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_711_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_712_inf__unique,axiom,
! [F: set_set_set_nat > set_set_set_nat > set_set_set_nat,X2: set_set_set_nat,Y2: 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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_713_inf__unique,axiom,
! [F: set_set_nat > set_set_nat > set_set_nat,X2: set_set_nat,Y2: 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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_714_inf__unique,axiom,
! [F: set_nat_nat > set_nat_nat > set_nat_nat,X2: set_nat_nat,Y2: 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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_715_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: 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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_716_inf__unique,axiom,
! [F: ( nat > nat ) > ( nat > nat ) > nat > nat,X2: nat > nat,Y2: 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 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_717_le__iff__inf,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [X3: set_set_set_nat,Y5: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_718_le__iff__inf,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [X3: set_set_nat,Y5: set_set_nat] :
( ( inf_inf_set_set_nat @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_719_le__iff__inf,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [X3: set_nat_nat,Y5: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_720_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( inf_inf_nat @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_721_le__iff__inf,axiom,
( ord_less_eq_nat_nat
= ( ^ [X3: nat > nat,Y5: nat > nat] :
( ( inf_inf_nat_nat @ X3 @ Y5 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_722_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_723_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_724_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_725_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_726_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_727_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_728_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_729_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_730_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_731_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_732_inf__absorb1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_733_inf__absorb1,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( inf_inf_set_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_734_inf__absorb1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_735_inf__absorb1,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_736_inf__absorb1,axiom,
! [X2: nat > nat,Y2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( inf_inf_nat_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_737_inf__absorb2,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X2 )
=> ( ( inf_in5711780100303410308et_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_738_inf__absorb2,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( inf_inf_set_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_739_inf__absorb2,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( ( inf_inf_set_nat_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_740_inf__absorb2,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_741_inf__absorb2,axiom,
! [Y2: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y2 @ X2 )
=> ( ( inf_inf_nat_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_742_inf_OboundedE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ~ ( ord_le9131159989063066194et_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_743_inf_OboundedE,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ~ ( ord_le6893508408891458716et_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_744_inf_OboundedE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ~ ( ord_le9059583361652607317at_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_745_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_746_inf_OboundedE,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_747_inf_OboundedI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_748_inf_OboundedI,axiom,
! [A2: set_set_nat,B2: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_749_inf_OboundedI,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_750_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_751_inf_OboundedI,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ord_less_eq_nat_nat @ A2 @ ( inf_inf_nat_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_752_inf__greatest,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Z2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_753_inf__greatest,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_754_inf__greatest,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_755_inf__greatest,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_756_inf__greatest,axiom,
! [X2: nat > nat,Y2: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_757_inf_Oorder__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( A4
= ( inf_in5711780100303410308et_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_758_inf_Oorder__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( A4
= ( inf_inf_set_set_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_759_inf_Oorder__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( A4
= ( inf_inf_set_nat_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_760_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_761_inf_Oorder__iff,axiom,
( ord_less_eq_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] :
( A4
= ( inf_inf_nat_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_762_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_763_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_764_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_765_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_766_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_767_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_768_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_769_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_770_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_771_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_772_inf_Oabsorb__iff1,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_773_inf_Oabsorb__iff1,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( inf_inf_set_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_774_inf_Oabsorb__iff1,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_775_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_776_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat_nat
= ( ^ [A4: nat > nat,B4: nat > nat] :
( ( inf_inf_nat_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_777_inf_Oabsorb__iff2,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [B4: set_set_set_nat,A4: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_778_inf_Oabsorb__iff2,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [B4: set_set_nat,A4: set_set_nat] :
( ( inf_inf_set_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_779_inf_Oabsorb__iff2,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [B4: set_nat_nat,A4: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_780_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_781_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat_nat
= ( ^ [B4: nat > nat,A4: nat > nat] :
( ( inf_inf_nat_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_782_inf_OcoboundedI1,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_783_inf_OcoboundedI1,axiom,
! [A2: set_set_nat,C2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_784_inf_OcoboundedI1,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_785_inf_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_786_inf_OcoboundedI1,axiom,
! [A2: nat > nat,C2: nat > nat,B2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ C2 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_787_inf_OcoboundedI2,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ C2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_788_inf_OcoboundedI2,axiom,
! [B2: set_set_nat,C2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_789_inf_OcoboundedI2,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ C2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_790_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_791_inf_OcoboundedI2,axiom,
! [B2: nat > nat,C2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B2 @ C2 )
=> ( ord_less_eq_nat_nat @ ( inf_inf_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_792_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_793_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_794_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_795_less__infI1,axiom,
! [A2: real,X2: real,B2: real] :
( ( ord_less_real @ A2 @ X2 )
=> ( ord_less_real @ ( inf_inf_real @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_796_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_797_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_798_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_799_less__infI2,axiom,
! [B2: real,X2: real,A2: real] :
( ( ord_less_real @ B2 @ X2 )
=> ( ord_less_real @ ( inf_inf_real @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_800_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_801_inf_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_802_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_803_inf_Oabsorb3,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( inf_inf_real @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_804_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_805_inf_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_806_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_807_inf_Oabsorb4,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( inf_inf_real @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_808_inf_Ostrict__boundedE,axiom,
! [A2: set_nat_nat,B2: set_nat_nat,C2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_set_nat_nat @ A2 @ B2 )
=> ~ ( ord_less_set_nat_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_809_inf_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_810_inf_Ostrict__boundedE,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B2 @ C2 ) )
=> ~ ( ( ord_le152980574450754630et_nat @ A2 @ B2 )
=> ~ ( ord_le152980574450754630et_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_811_inf_Ostrict__boundedE,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_real @ A2 @ ( inf_inf_real @ B2 @ C2 ) )
=> ~ ( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_812_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat_nat
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( A4
= ( inf_inf_set_nat_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_813_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_814_inf_Ostrict__order__iff,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( A4
= ( inf_in5711780100303410308et_nat @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_815_inf_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [A4: real,B4: real] :
( ( A4
= ( inf_inf_real @ A4 @ B4 ) )
& ( A4 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_816_inf_Ostrict__coboundedI1,axiom,
! [A2: set_nat_nat,C2: set_nat_nat,B2: set_nat_nat] :
( ( ord_less_set_nat_nat @ A2 @ C2 )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_817_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_818_inf_Ostrict__coboundedI1,axiom,
! [A2: set_set_set_nat,C2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ C2 )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_819_inf_Ostrict__coboundedI1,axiom,
! [A2: real,C2: real,B2: real] :
( ( ord_less_real @ A2 @ C2 )
=> ( ord_less_real @ ( inf_inf_real @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_820_inf_Ostrict__coboundedI2,axiom,
! [B2: set_nat_nat,C2: set_nat_nat,A2: set_nat_nat] :
( ( ord_less_set_nat_nat @ B2 @ C2 )
=> ( ord_less_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_821_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_822_inf_Ostrict__coboundedI2,axiom,
! [B2: set_set_set_nat,C2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B2 @ C2 )
=> ( ord_le152980574450754630et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_823_inf_Ostrict__coboundedI2,axiom,
! [B2: real,C2: real,A2: real] :
( ( ord_less_real @ B2 @ C2 )
=> ( ord_less_real @ ( inf_inf_real @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_824_distrib__imp1,axiom,
! [X2: set_set_set_nat,Y2: 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 @ Y2 @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_825_distrib__imp1,axiom,
! [X2: set_nat_nat,Y2: 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 @ Y2 @ Z2 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_826_distrib__imp1,axiom,
! [X2: set_set_nat,Y2: 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 @ Y2 @ Z2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_827_distrib__imp2,axiom,
! [X2: set_set_set_nat,Y2: 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 @ Y2 @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_828_distrib__imp2,axiom,
! [X2: set_nat_nat,Y2: 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 @ Y2 @ Z2 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_829_distrib__imp2,axiom,
! [X2: set_set_nat,Y2: 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 @ Y2 @ Z2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_830_inf__sup__distrib1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_831_inf__sup__distrib1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_832_inf__sup__distrib1,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_833_inf__sup__distrib2,axiom,
! [Y2: set_set_set_nat,Z2: set_set_set_nat,X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) @ X2 )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y2 @ X2 ) @ ( inf_in5711780100303410308et_nat @ Z2 @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_834_inf__sup__distrib2,axiom,
! [Y2: set_nat_nat,Z2: set_nat_nat,X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) @ X2 )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y2 @ X2 ) @ ( inf_inf_set_nat_nat @ Z2 @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_835_inf__sup__distrib2,axiom,
! [Y2: set_set_nat,Z2: set_set_nat,X2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) @ X2 )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y2 @ X2 ) @ ( inf_inf_set_set_nat @ Z2 @ X2 ) ) ) ).
% inf_sup_distrib2
thf(fact_836_sup__inf__distrib1,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_837_sup__inf__distrib1,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_838_sup__inf__distrib1,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_839_sup__inf__distrib2,axiom,
! [Y2: set_set_set_nat,Z2: set_set_set_nat,X2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) @ X2 )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ Y2 @ X2 ) @ ( sup_su4213647025997063966et_nat @ Z2 @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_840_sup__inf__distrib2,axiom,
! [Y2: set_nat_nat,Z2: set_nat_nat,X2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) @ X2 )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ Y2 @ X2 ) @ ( sup_sup_set_nat_nat @ Z2 @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_841_sup__inf__distrib2,axiom,
! [Y2: set_set_nat,Z2: set_set_nat,X2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ Y2 @ Z2 ) @ X2 )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ Y2 @ X2 ) @ ( sup_sup_set_set_nat @ Z2 @ X2 ) ) ) ).
% sup_inf_distrib2
thf(fact_842_Int__mono,axiom,
! [A: set_set_set_nat,C: set_set_set_nat,B: set_set_set_nat,D2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ C )
=> ( ( ord_le9131159989063066194et_nat @ B @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_843_Int__mono,axiom,
! [A: set_set_nat,C: set_set_nat,B: set_set_nat,D2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ C )
=> ( ( ord_le6893508408891458716et_nat @ B @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_844_Int__mono,axiom,
! [A: set_nat_nat,C: set_nat_nat,B: set_nat_nat,D2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ C )
=> ( ( ord_le9059583361652607317at_nat @ B @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_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_855_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_856_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_857_Int__greatest,axiom,
! [C: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C @ A )
=> ( ( ord_le9131159989063066194et_nat @ C @ B )
=> ( ord_le9131159989063066194et_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_858_Int__greatest,axiom,
! [C: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C @ A )
=> ( ( ord_le6893508408891458716et_nat @ C @ B )
=> ( ord_le6893508408891458716et_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_859_Int__greatest,axiom,
! [C: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C @ A )
=> ( ( ord_le9059583361652607317at_nat @ C @ B )
=> ( ord_le9059583361652607317at_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_860_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_861_Int__Collect__mono,axiom,
! [A: set_real,B: set_real,P2: real > $o,Q: real > $o] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A )
=> ( ( P2 @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_real @ ( inf_inf_set_real @ A @ ( collect_real @ P2 ) ) @ ( inf_inf_set_real @ B @ ( collect_real @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_862_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_863_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_864_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_865_Int__emptyI,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 ) )
=> ( ( inf_in2396666505901392698et_nat @ A @ B )
= bot_bo193956671110832956et_nat ) ) ).
% Int_emptyI
thf(fact_866_Int__emptyI,axiom,
! [A: set_real,B: set_real] :
( ! [X4: real] :
( ( member_real @ X4 @ A )
=> ~ ( member_real @ X4 @ B ) )
=> ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real ) ) ).
% Int_emptyI
thf(fact_867_Int__emptyI,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 ) )
=> ( ( inf_in5711780100303410308et_nat @ A @ B )
= bot_bo7198184520161983622et_nat ) ) ).
% Int_emptyI
thf(fact_868_Int__emptyI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ~ ( member_set_nat @ X4 @ B ) )
=> ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_869_Int__emptyI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A )
=> ~ ( member_nat_nat @ X4 @ B ) )
=> ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat ) ) ).
% Int_emptyI
thf(fact_870_disjoint__iff,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ( inf_in2396666505901392698et_nat @ A @ B )
= bot_bo193956671110832956et_nat )
= ( ! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A )
=> ~ ( member2946998982187404937et_nat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_871_disjoint__iff,axiom,
! [A: set_real,B: set_real] :
( ( ( inf_inf_set_real @ A @ B )
= bot_bot_set_real )
= ( ! [X3: real] :
( ( member_real @ X3 @ A )
=> ~ ( member_real @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_872_disjoint__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ~ ( member_set_set_nat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_873_disjoint__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ~ ( member_set_nat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_874_disjoint__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ~ ( member_nat_nat @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_875_Int__empty__left,axiom,
! [B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ B )
= bot_bo7198184520161983622et_nat ) ).
% Int_empty_left
thf(fact_876_Int__empty__left,axiom,
! [B: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ B )
= bot_bot_set_set_nat ) ).
% Int_empty_left
thf(fact_877_Int__empty__left,axiom,
! [B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ B )
= bot_bot_set_nat_nat ) ).
% Int_empty_left
thf(fact_878_Int__empty__right,axiom,
! [A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% Int_empty_right
thf(fact_879_Int__empty__right,axiom,
! [A: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% Int_empty_right
thf(fact_880_Int__empty__right,axiom,
! [A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% Int_empty_right
thf(fact_881_disjoint__iff__not__equal,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ A @ B )
= bot_bo7198184520161983622et_nat )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ! [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ B )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_882_disjoint__iff__not__equal,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ B )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_883_disjoint__iff__not__equal,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ A @ B )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ! [Y5: nat > nat] :
( ( member_nat_nat @ Y5 @ B )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_884_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_885_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_886_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_887_psubset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_888_psubset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_889_psubset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_890_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_891_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_892_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_893_psubset__subset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_894_psubset__subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_895_psubset__subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% psubset_subset_trans
thf(fact_896_subset__not__subset__eq,axiom,
( ord_le152980574450754630et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B3 )
& ~ ( ord_le9131159989063066194et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_897_subset__not__subset__eq,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
& ~ ( ord_le6893508408891458716et_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_898_subset__not__subset__eq,axiom,
( ord_less_set_nat_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B3 )
& ~ ( ord_le9059583361652607317at_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_899_subset__psubset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_900_subset__psubset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_set_set_nat @ B @ C )
=> ( ord_less_set_set_nat @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_901_subset__psubset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_set_nat_nat @ B @ C )
=> ( ord_less_set_nat_nat @ A @ C ) ) ) ).
% subset_psubset_trans
thf(fact_902_subset__iff__psubset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_903_subset__iff__psubset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B3: set_set_nat] :
( ( ord_less_set_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_904_subset__iff__psubset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B3: set_nat_nat] :
( ( ord_less_set_nat_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_905_not__psubset__empty,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ bot_bot_set_set_nat ) ).
% not_psubset_empty
thf(fact_906_not__psubset__empty,axiom,
! [A: set_nat_nat] :
~ ( ord_less_set_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% not_psubset_empty
thf(fact_907_not__psubset__empty,axiom,
! [A: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% not_psubset_empty
thf(fact_908_Un__Int__crazy,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ B @ C ) ) @ ( inf_in5711780100303410308et_nat @ C @ A ) )
= ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ B @ C ) ) @ ( sup_su4213647025997063966et_nat @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_909_Un__Int__crazy,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: 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 @ C ) ) @ ( inf_inf_set_nat_nat @ C @ A ) )
= ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ B @ C ) ) @ ( sup_sup_set_nat_nat @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_910_Un__Int__crazy,axiom,
! [A: set_set_nat,B: set_set_nat,C: 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 @ C ) ) @ ( inf_inf_set_set_nat @ C @ A ) )
= ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ B @ C ) ) @ ( sup_sup_set_set_nat @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_911_Int__Un__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ ( inf_in5711780100303410308et_nat @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_912_Int__Un__distrib,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ ( inf_inf_set_nat_nat @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_913_Int__Un__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ ( inf_inf_set_set_nat @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_914_Un__Int__distrib,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C ) )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ ( sup_su4213647025997063966et_nat @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_915_Un__Int__distrib,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C ) )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ ( sup_sup_set_nat_nat @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_916_Un__Int__distrib,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C ) )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ A @ B ) @ ( sup_sup_set_set_nat @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_917_Int__Un__distrib2,axiom,
! [B: set_set_set_nat,C: set_set_set_nat,A: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ B @ C ) @ A )
= ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ B @ A ) @ ( inf_in5711780100303410308et_nat @ C @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_918_Int__Un__distrib2,axiom,
! [B: set_nat_nat,C: set_nat_nat,A: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ B @ C ) @ A )
= ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ B @ A ) @ ( inf_inf_set_nat_nat @ C @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_919_Int__Un__distrib2,axiom,
! [B: set_set_nat,C: set_set_nat,A: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ C ) @ A )
= ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ A ) @ ( inf_inf_set_set_nat @ C @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_920_Un__Int__distrib2,axiom,
! [B: set_set_set_nat,C: set_set_set_nat,A: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ B @ C ) @ A )
= ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ B @ A ) @ ( sup_su4213647025997063966et_nat @ C @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_921_Un__Int__distrib2,axiom,
! [B: set_nat_nat,C: set_nat_nat,A: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ B @ C ) @ A )
= ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ B @ A ) @ ( sup_sup_set_nat_nat @ C @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_922_Un__Int__distrib2,axiom,
! [B: set_set_nat,C: set_set_nat,A: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ B @ C ) @ A )
= ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ B @ A ) @ ( sup_sup_set_set_nat @ C @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_923_first__assumptions_OACC__odot,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K @ X ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_odot
thf(fact_924_distrib__inf__le,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ X2 @ Y2 ) @ ( inf_in5711780100303410308et_nat @ X2 @ Z2 ) ) @ ( inf_in5711780100303410308et_nat @ X2 @ ( sup_su4213647025997063966et_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_925_distrib__inf__le,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_set_nat @ X2 @ Z2 ) ) @ ( inf_inf_set_set_nat @ X2 @ ( sup_sup_set_set_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_926_distrib__inf__le,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat_nat @ X2 @ Z2 ) ) @ ( inf_inf_set_nat_nat @ X2 @ ( sup_sup_set_nat_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_927_distrib__inf__le,axiom,
! [X2: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z2 ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_928_distrib__inf__le,axiom,
! [X2: nat > nat,Y2: nat > nat,Z2: nat > nat] : ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ ( inf_inf_nat_nat @ X2 @ Y2 ) @ ( inf_inf_nat_nat @ X2 @ Z2 ) ) @ ( inf_inf_nat_nat @ X2 @ ( sup_sup_nat_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_929_distrib__sup__le,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ ( inf_in5711780100303410308et_nat @ Y2 @ Z2 ) ) @ ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ X2 @ Y2 ) @ ( sup_su4213647025997063966et_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_930_distrib__sup__le,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ X2 @ ( inf_inf_set_set_nat @ Y2 @ Z2 ) ) @ ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_set_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_931_distrib__sup__le,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ X2 @ ( inf_inf_set_nat_nat @ Y2 @ Z2 ) ) @ ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_932_distrib__sup__le,axiom,
! [X2: nat,Y2: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_933_distrib__sup__le,axiom,
! [X2: nat > nat,Y2: nat > nat,Z2: nat > nat] : ( ord_less_eq_nat_nat @ ( sup_sup_nat_nat @ X2 @ ( inf_inf_nat_nat @ Y2 @ Z2 ) ) @ ( inf_inf_nat_nat @ ( sup_sup_nat_nat @ X2 @ Y2 ) @ ( sup_sup_nat_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_934_Un__Int__assoc__eq,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C )
= ( inf_in5711780100303410308et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C ) ) )
= ( ord_le9131159989063066194et_nat @ C @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_935_Un__Int__assoc__eq,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C )
= ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C ) ) )
= ( ord_le6893508408891458716et_nat @ C @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_936_Un__Int__assoc__eq,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C )
= ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C ) ) )
= ( ord_le9059583361652607317at_nat @ C @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_937_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_938_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_939_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_940_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ Y )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X ) @ ( clique951075384711337423ACC_cf @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_941_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_942_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X ) @ ( clique951075384711337423ACC_cf @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_943_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_944_first__assumptions_OACC__union,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K @ X ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_union
thf(fact_945_old_Oprod_Oexhaust,axiom,
! [Y2: produc4045820344675478307at_nat] :
~ ! [A7: set_set_set_nat,B7: nat] :
( Y2
!= ( produc2803780273060847707at_nat @ A7 @ B7 ) ) ).
% old.prod.exhaust
thf(fact_946_surj__pair,axiom,
! [P: produc4045820344675478307at_nat] :
? [X4: set_set_set_nat,Y4: nat] :
( P
= ( produc2803780273060847707at_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_947_prod__cases,axiom,
! [P2: produc4045820344675478307at_nat > $o,P: produc4045820344675478307at_nat] :
( ! [A7: set_set_set_nat,B7: nat] : ( P2 @ ( produc2803780273060847707at_nat @ A7 @ B7 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_948_Pair__inject,axiom,
! [A2: set_set_set_nat,B2: nat,A6: set_set_set_nat,B6: nat] :
( ( ( produc2803780273060847707at_nat @ A2 @ B2 )
= ( produc2803780273060847707at_nat @ A6 @ B6 ) )
=> ~ ( ( A2 = A6 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_949_first__assumptions_Oodotl__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique7966186356931407165_odotl @ L @ K @ X @ Y )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X @ Y ) @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ).
% first_assumptions.odotl_def
thf(fact_950_fst__eqD,axiom,
! [X2: set_set_set_nat,Y2: nat,A2: set_set_set_nat] :
( ( ( produc6523417423482510407at_nat @ ( produc2803780273060847707at_nat @ X2 @ Y2 ) )
= A2 )
=> ( X2 = A2 ) ) ).
% fst_eqD
thf(fact_951_fst__conv,axiom,
! [X1: set_set_set_nat,X22: nat] :
( ( produc6523417423482510407at_nat @ ( produc2803780273060847707at_nat @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_952_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_953_sqcup__sub,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X @ Y ) ) ) ) ) ).
% sqcup_sub
thf(fact_954_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_955_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ bot_bo7198184520161983622et_nat @ X2 )
= bot_bo7198184520161983622et_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_956_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_957_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ bot_bot_set_nat_nat @ X2 )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_958_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ X2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_959_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_960_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ X2 @ bot_bot_set_nat_nat )
= bot_bot_set_nat_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_961_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_962_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_963_ACC__cf__odot,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X @ Y ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X ) @ ( clique951075384711337423ACC_cf @ k @ Y ) ) ) ).
% ACC_cf_odot
thf(fact_964_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_965_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_966_dual__order_Orefl,axiom,
! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_967_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_968_dual__order_Orefl,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_969_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_970_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_971_order__refl,axiom,
! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% order_refl
thf(fact_972_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_973_order__refl,axiom,
! [X2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_974_psubsetD,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le52856854838348540et_nat @ A @ B )
=> ( ( member2946998982187404937et_nat @ C2 @ A )
=> ( member2946998982187404937et_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_975_psubsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: nat > nat] :
( ( ord_less_set_nat_nat @ A @ B )
=> ( ( member_nat_nat @ C2 @ A )
=> ( member_nat_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_976_psubsetD,axiom,
! [A: set_real,B: set_real,C2: real] :
( ( ord_less_set_real @ A @ B )
=> ( ( member_real @ C2 @ A )
=> ( member_real @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_977_psubsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( member_set_set_nat @ C2 @ A )
=> ( member_set_set_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_978_psubset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A @ B )
=> ( ( ord_le152980574450754630et_nat @ B @ C )
=> ( ord_le152980574450754630et_nat @ A @ C ) ) ) ).
% psubset_trans
thf(fact_979_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_980_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_981_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_982_diff__shunt__var,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ( minus_2447799839930672331et_nat @ X2 @ Y2 )
= bot_bo7198184520161983622et_nat )
= ( ord_le9131159989063066194et_nat @ X2 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_983_diff__shunt__var,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ X2 @ Y2 )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ X2 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_984_diff__shunt__var,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ X2 @ Y2 )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ X2 @ Y2 ) ) ).
% diff_shunt_var
thf(fact_985_order__antisym__conv,axiom,
! [Y2: set_set_nat,X2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_986_order__antisym__conv,axiom,
! [Y2: set_nat_nat,X2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_987_order__antisym__conv,axiom,
! [Y2: num,X2: num] :
( ( ord_less_eq_num @ Y2 @ X2 )
=> ( ( ord_less_eq_num @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_988_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_989_order__antisym__conv,axiom,
! [Y2: nat > nat,X2: nat > nat] :
( ( ord_less_eq_nat_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_990_linorder__le__cases,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_991_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_992_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_993_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_994_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_995_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_996_ord__le__eq__subst,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > num,C2: num] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [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 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_997_ord__le__eq__subst,axiom,
! [A2: set_set_nat,B2: set_set_nat,F: set_set_nat > nat,C2: nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [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 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_998_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > set_set_nat,C2: set_set_nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_999_ord__le__eq__subst,axiom,
! [A2: num,B2: num,F: num > nat > nat,C2: nat > nat] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [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 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1000_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_set_nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1001_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat > nat,C2: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [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 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1002_ord__eq__le__subst,axiom,
! [A2: num,F: num > num,B2: num,C2: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1003_ord__eq__le__subst,axiom,
! [A2: nat,F: num > nat,B2: num,C2: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1004_ord__eq__le__subst,axiom,
! [A2: num,F: nat > num,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_num @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1005_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1006_ord__eq__le__subst,axiom,
! [A2: num,F: set_set_nat > num,B2: set_set_nat,C2: set_set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ! [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 @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1007_ord__eq__le__subst,axiom,
! [A2: nat,F: set_set_nat > nat,B2: set_set_nat,C2: set_set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ C2 )
=> ( ! [X4: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1008_ord__eq__le__subst,axiom,
! [A2: set_set_nat,F: num > set_set_nat,B2: num,C2: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1009_ord__eq__le__subst,axiom,
! [A2: nat > nat,F: num > nat > nat,B2: num,C2: num] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_num @ B2 @ C2 )
=> ( ! [X4: num,Y4: num] :
( ( ord_less_eq_num @ X4 @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1010_ord__eq__le__subst,axiom,
! [A2: set_set_nat,F: nat > set_set_nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1011_ord__eq__le__subst,axiom,
! [A2: nat > nat,F: nat > nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1012_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_1013_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_1014_first__assumptions_OACC__cf__odot,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X @ Y ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X ) @ ( clique951075384711337423ACC_cf @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_odot
thf(fact_1015_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_1016_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_1017_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_1018_second__assumptions_Osqcup__sub,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( member2946998982187404937et_nat @ X @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P @ K ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X @ Y ) ) ) ) ) ) ).
% second_assumptions.sqcup_sub
thf(fact_1019_deviate__pos__cup__def,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705536850673os_cup @ l @ p @ k @ X @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X @ Y ) ) ) ) ).
% deviate_pos_cup_def
thf(fact_1020_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_1021_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_1022_deviate__neg__cap__def,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987438064265eg_cap @ l @ p @ k @ X @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X @ Y ) ) @ ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X @ Y ) ) ) ) ).
% deviate_neg_cap_def
thf(fact_1023_deviate__pos__cap__def,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705535538693os_cap @ l @ p @ k @ X @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ ( clique2586627118206219037_sqcap @ l @ p @ k @ X @ Y ) ) ) ) ).
% deviate_pos_cap_def
thf(fact_1024_deviate__neg__cup__def,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987439376245eg_cup @ l @ p @ k @ X @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( clique2586627118207531017_sqcup @ l @ p @ k @ X @ Y ) ) @ ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ) ) ).
% deviate_neg_cup_def
thf(fact_1025_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_1026_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_1027_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_1028_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_1029_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_1030_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1031_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_1032_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1033_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_1034_less__not__refl3,axiom,
! [S2: nat,T3: nat] :
( ( ord_less_nat @ S2 @ T3 )
=> ( S2 != T3 ) ) ).
% less_not_refl3
thf(fact_1035_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1036_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_1037_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_1038_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1039_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_1040_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_1041_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1049_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_second__assumptions_Odeviate__pos__cup__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique3314026705536850673os_cup @ L @ P @ K @ X @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X @ Y ) ) ) ) ) ).
% second_assumptions.deviate_pos_cup_def
thf(fact_1057_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1058_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_1059_second__assumptions_Odeviate__pos__cap__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique3314026705535538693os_cap @ L @ P @ K @ X @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118206219037_sqcap @ L @ P @ K @ X @ Y ) ) ) ) ) ).
% second_assumptions.deviate_pos_cap_def
thf(fact_1060_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_1061_finite__ACC,axiom,
! [X: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X ) ) ).
% finite_ACC
thf(fact_1062_second__assumptions_Odeviate__neg__cup_Ocong,axiom,
clique1591571987439376245eg_cup = clique1591571987439376245eg_cup ).
% second_assumptions.deviate_neg_cup.cong
thf(fact_1063_second__assumptions_Odeviate__pos__cap_Ocong,axiom,
clique3314026705535538693os_cap = clique3314026705535538693os_cap ).
% second_assumptions.deviate_pos_cap.cong
thf(fact_1064_second__assumptions_Odeviate__neg__cap_Ocong,axiom,
clique1591571987438064265eg_cap = clique1591571987438064265eg_cap ).
% second_assumptions.deviate_neg_cap.cong
thf(fact_1065_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_1066_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_1067_second__assumptions_Odeviate__neg__cup__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique1591571987439376245eg_cup @ L @ P @ K @ X @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118207531017_sqcup @ L @ P @ K @ X @ Y ) ) @ ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X @ Y ) ) ) ) ) ).
% second_assumptions.deviate_neg_cup_def
thf(fact_1068_second__assumptions_Odeviate__neg__cap__def,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P @ K )
=> ( ( clique1591571987438064265eg_cap @ L @ P @ K @ X @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118206219037_sqcap @ L @ P @ K @ X @ Y ) ) @ ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X @ Y ) ) ) ) ) ).
% second_assumptions.deviate_neg_cap_def
thf(fact_1069_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_1070_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_1071_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_1072_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_1073_v__gs__union,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_union
thf(fact_1074_v__gs__mono,axiom,
! [X: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_mono
thf(fact_1075_finV_I1_J,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ x ) ).
% finV(1)
thf(fact_1076_finV_I2_J,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ y ) ).
% finV(2)
thf(fact_1077_finvXY,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) ) ).
% finvXY
thf(fact_1078_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_1079_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_1080_first__assumptions_Ov__gs__union,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X @ Y ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ).
% first_assumptions.v_gs_union
thf(fact_1081_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_1082_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ( ord_le9131159989063066194et_nat @ X @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_1083_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_1084_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_1085_first__assumptions_OL_Ocong,axiom,
assump1710595444109740301irst_L = assump1710595444109740301irst_L ).
% first_assumptions.L.cong
thf(fact_1086_first__assumptions_Ok,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_1087_first__assumptions_Okp,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ P @ K ) ) ).
% first_assumptions.kp
thf(fact_1088_first__assumptions_Opl,axiom,
! [L: nat,P: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P @ K )
=> ( ord_less_nat @ L @ P ) ) ).
% first_assumptions.pl
thf(fact_1089_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_1090_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_1091_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_1092_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_1093_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_1094__092_060open_062card_A_Iv__gs_A_IX_A_092_060odot_062l_AY_J_J_A_092_060le_062_Acard_A_Iv__gs_A_IX_A_092_060odot_062_AY_J_J_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) ) ) ).
% \<open>card (v_gs (X \<odot>l Y)) \<le> card (v_gs (X \<odot> Y))\<close>
thf(fact_1095_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_1096_plucking__step_I3_J,axiom,
! [X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ X ) ) @ ( clique3210737319928189260st_ACC @ k @ Y ) ) ) ) ) ).
% plucking_step(3)
thf(fact_1097_plucking__step_I2_J,axiom,
! [X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X ) )
=> ( ord_le9131159989063066194et_nat @ Y @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ) ) ).
% plucking_step(2)
thf(fact_1098_plucking__step_I5_J,axiom,
! [X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X ) )
=> ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ).
% plucking_step(5)
thf(fact_1099_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_1100_second__assumptions_Oplucking__step_I2_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P @ X ) )
=> ( ord_le9131159989063066194et_nat @ Y @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ) ) ).
% second_assumptions.plucking_step(2)
thf(fact_1101_second__assumptions_Oplucking__step_I5_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P @ X ) )
=> ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% second_assumptions.plucking_step(5)
thf(fact_1102_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_1103_second__assumptions_Oplucking__step_I3_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P @ X ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ X ) ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ) ) ).
% second_assumptions.plucking_step(3)
thf(fact_1104_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_1105__092_060open_062card_A_Iv__gs_AX_J_A_K_Acard_A_Iv__gs_AY_J_A_092_060le_062_AL_A_K_AL_092_060close_062,axiom,
ord_less_eq_nat @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) ) ) @ ( times_times_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( assump1710595444109740301irst_L @ l @ p ) ) ).
% \<open>card (v_gs X) * card (v_gs Y) \<le> L * L\<close>
thf(fact_1106__092_060open_062card_A_Iv__gs_A_IX_A_092_060odot_062_AY_J_J_A_092_060le_062_Acard_A_Iv__gs_AX_J_A_K_Acard_A_Iv__gs_AY_J_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) ) ) ).
% \<open>card (v_gs (X \<odot> Y)) \<le> card (v_gs X) * card (v_gs Y)\<close>
thf(fact_1107_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_1108_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_1109_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_1110_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_1111_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_1112_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1113_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1114_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_1115__092_060open_062n_A_K_A_Ip_A_N_A1_J_A_092_060le_062_Acard_A_Iv__gs_A_IX_A_092_060odot_062l_AY_J_J_092_060close_062,axiom,
ord_less_eq_nat @ ( times_times_nat @ n @ ( minus_minus_nat @ p @ one_one_nat ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) ) ) ).
% \<open>n * (p - 1) \<le> card (v_gs (X \<odot>l Y))\<close>
thf(fact_1116_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_1117_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_1118_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_1119_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_1120_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1121_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1122_plucking__step_I1_J,axiom,
! [X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) @ p ) @ one_one_nat ) ) ) ) ) ).
% plucking_step(1)
thf(fact_1123_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1124_second__assumptions_Oplucking__step_I1_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P @ X ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X ) ) @ P ) @ one_one_nat ) ) ) ) ) ) ).
% second_assumptions.plucking_step(1)
thf(fact_1125_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_1126_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_1127_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_1128_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_1129_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_1130_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_1131_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_1132_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_1133_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1134_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1135_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_1136_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_1137_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_1138_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_1139_diff__add__inverse2,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
= M ) ).
% diff_add_inverse2
thf(fact_1140_diff__add__inverse,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
= M ) ).
% diff_add_inverse
thf(fact_1141_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_1142_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_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_le__add2,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% le_add2
thf(fact_1152_le__add1,axiom,
! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% le_add1
thf(fact_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_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_1163_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_1164_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_1165_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_1166_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_1167_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_1168_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_1169_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_1170_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_1171_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_1172_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_1173_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_1174_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_1175_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_1176_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_1177_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_1178_pointwise__minimal__pointwise__maximal_I2_J,axiom,
! [S2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat_nat )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S2 )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S2 )
=> ( ( ord_less_eq_nat_nat @ X4 @ Xa )
| ( ord_less_eq_nat_nat @ Xa @ X4 ) ) ) )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S2 )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S2 )
=> ( ord_less_eq_nat_nat @ Xa2 @ X4 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(2)
thf(fact_1179_pointwise__minimal__pointwise__maximal_I1_J,axiom,
! [S2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat_nat )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S2 )
=> ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S2 )
=> ( ( ord_less_eq_nat_nat @ X4 @ Xa )
| ( ord_less_eq_nat_nat @ Xa @ X4 ) ) ) )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ S2 )
& ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S2 )
=> ( ord_less_eq_nat_nat @ X4 @ Xa2 ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(1)
thf(fact_1180_main,axiom,
( ( member2946998982187404937et_nat @ z @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) )
& ( ( z = bot_bo7198184520161983622et_nat )
= ( ( clique7966186356931407165_odotl @ l @ k @ x @ y )
= bot_bo7198184520161983622et_nat ) )
& ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) ) ) @ ( clique3210737319928189260st_ACC @ k @ z ) )
& ( 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 @ z ) @ ( clique951075384711337423ACC_cf @ k @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ) @ n ) ) ) ).
% main
thf(fact_1181_card__POS,axiom,
( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ k ) )
= ( binomial @ ( assump1710595444109740334irst_m @ k ) @ k ) ) ).
% card_POS
thf(fact_1182_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_1183_semiring__norm_I68_J,axiom,
! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).
% semiring_norm(68)
thf(fact_1184_k2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ k ).
% k2
thf(fact_1185_l8,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ l ).
% l8
thf(fact_1186_l2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ l ).
% l2
thf(fact_1187_p,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ).
% p
thf(fact_1188_m2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( assump1710595444109740334irst_m @ k ) ).
% m2
thf(fact_1189_m__def,axiom,
( ( assump1710595444109740334irst_m @ k )
= ( power_power_nat @ k @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% m_def
thf(fact_1190_kl2,axiom,
( k
= ( power_power_nat @ l @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% kl2
thf(fact_1191_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1192__092_060open_062L_A_K_AL_A_061_AL_092_060_094sup_0622_092_060close_062,axiom,
( ( times_times_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( assump1710595444109740301irst_L @ l @ p ) )
= ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% \<open>L * L = L\<^sup>2\<close>
thf(fact_1193_n,axiom,
ord_less_nat @ n @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% n
thf(fact_1194__092_060open_062n_A_K_A_Ip_A_N_A1_J_A_092_060le_062_AL_092_060_094sup_0622_092_060close_062,axiom,
ord_less_eq_nat @ ( times_times_nat @ n @ ( minus_minus_nat @ p @ one_one_nat ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% \<open>n * (p - 1) \<le> L\<^sup>2\<close>
thf(fact_1195_card__join,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% card_join
thf(fact_1196_card,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% card
thf(fact_1197_plucking__step_I4_J,axiom,
! [X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( 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 @ Y ) @ ( clique951075384711337423ACC_cf @ k @ X ) ) ) ) @ ( power_power_nat @ ( minus_minus_nat @ k @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% plucking_step(4)
thf(fact_1198_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_1199_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_1200_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_1201_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_1202_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_1203_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_1204_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_1205_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_1206_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_1207_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_1208_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_1209_second__assumptions_Oplucking__step_I4_J,axiom,
! [L: nat,P: nat,K: nat,X: set_set_set_nat,Y: 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 ) ) )
=> ( ( Y
= ( 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 @ Y ) @ ( clique951075384711337423ACC_cf @ K @ X ) ) ) ) @ ( power_power_nat @ ( minus_minus_nat @ K @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% second_assumptions.plucking_step(4)
thf(fact_1210_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
thf(fact_1211_power2__nat__le__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% power2_nat_le_imp_le
thf(fact_1212_power2__nat__le__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% power2_nat_le_eq_le
thf(fact_1213_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_1214_less__exp,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% less_exp
thf(fact_1215_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1216_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B2 @ N ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1217_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N: nat] :
( ( ord_less_nat @ ( power_power_nat @ B2 @ N ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1218_two__realpow__ge__one,axiom,
! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% two_realpow_ge_one
thf(fact_1219_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_1220_binomial__strict__antimono,axiom,
! [K: nat,K3: nat,N2: nat] :
( ( ord_less_nat @ K @ K3 )
=> ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
=> ( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ord_less_nat @ ( binomial @ N2 @ K3 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_1221_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_1222_complete__real,axiom,
! [S: set_real] :
( ? [X8: real] : ( member_real @ X8 @ S )
=> ( ? [Z6: real] :
! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z6 ) )
=> ? [Y4: real] :
( ! [X8: real] :
( ( member_real @ X8 @ S )
=> ( ord_less_eq_real @ X8 @ Y4 ) )
& ! [Z6: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ( ord_less_eq_real @ X4 @ Z6 ) )
=> ( ord_less_eq_real @ Y4 @ Z6 ) ) ) ) ) ).
% complete_real
thf(fact_1223_sum__le__prod1,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ one_one_real )
=> ( ( ord_less_eq_real @ B2 @ one_one_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ A2 @ B2 ) ) ) ) ) ).
% sum_le_prod1
thf(fact_1224_L2__set__mult__ineq__lemma,axiom,
! [A2: real,C2: real,B2: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A2 @ C2 ) ) @ ( times_times_real @ B2 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_1225_binomial__maximum_H,axiom,
! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% binomial_maximum'
thf(fact_1226_binomial__mono,axiom,
! [K: nat,K3: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) @ N2 )
=> ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K3 ) ) ) ) ).
% binomial_mono
thf(fact_1227_binomial__strict__mono,axiom,
! [K: nat,K3: nat,N2: nat] :
( ( ord_less_nat @ K @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) @ N2 )
=> ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K3 ) ) ) ) ).
% binomial_strict_mono
thf(fact_1228__C_K_C,axiom,
ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( minus_minus_nat @ k @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ) ).
% "*"
thf(fact_1229_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_1230_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1231_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1232_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1233_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1234_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1235_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1236_add__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1237_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1238_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1239_mult__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1240_mult__cancel1,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N2 ) )
= ( ( M = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1241_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1242_mult__is__0,axiom,
! [M: nat,N2: nat] :
( ( ( times_times_nat @ M @ N2 )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1243_add__gr__0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1244_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1245_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1246_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_1247_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1248_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1249_nat__0__less__mult__iff,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1250_mult__less__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1251_nat__zero__less__power__iff,axiom,
! [X2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X2 )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1252_binomial__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( binomial @ N2 @ K )
= zero_zero_nat )
= ( ord_less_nat @ N2 @ K ) ) ).
% binomial_eq_0_iff
thf(fact_1253_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1254_mult__le__cancel2,axiom,
! [M: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1255_zero__less__binomial__iff,axiom,
! [N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
= ( ord_less_eq_nat @ K @ N2 ) ) ).
% zero_less_binomial_iff
thf(fact_1256_zero__less__binomial,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).
% zero_less_binomial
thf(fact_1257_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1258_ex__least__nat__le,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ N2 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1259_mult__eq__self__implies__10,axiom,
! [M: nat,N2: nat] :
( ( M
= ( times_times_nat @ M @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1260_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1261_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1262_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1263_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1264_diff__add__0,axiom,
! [N2: nat,M: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1265_add__eq__self__zero,axiom,
! [M: nat,N2: nat] :
( ( ( plus_plus_nat @ M @ N2 )
= M )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1266_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1267_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1268_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1269_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
% Conjectures (1)
thf(conj_0,conjecture,
member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k @ ( clique7966186356931407165_odotl @ l @ k @ x @ y ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k ) ).
%------------------------------------------------------------------------------