TPTP Problem File: SLH0006^1.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_01153_044024__16301248_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1493 ( 542 unt; 215 typ; 0 def)
% Number of atoms : 3646 ( 945 equ; 0 cnn)
% Maximal formula atoms : 26 ( 2 avg)
% Number of connectives : 11488 ( 220 ~; 25 |; 293 &;9215 @)
% ( 0 <=>;1735 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 1052 (1052 >; 0 *; 0 +; 0 <<)
% Number of symbols : 205 ( 202 usr; 28 con; 0-5 aty)
% Number of variables : 3408 ( 209 ^;3067 !; 132 ?;3408 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:49:55.599
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_Mt__Nat__Onat_J,type,
produc4045820344675478307at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
set_se7970953024979822686et_nat: $tType ).
thf(ty_n_t__Set__Oset_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $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 ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (202)
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions,type,
assump5453534214990993103ptions: nat > nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_OL,type,
assump1710595444109740301irst_L: nat > nat > nat ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_Om,type,
assump1710595444109740334irst_m: nat > nat ).
thf(sy_c_Assumptions__and__Approximations_Osecond__assumptions,type,
assump2881078719466019805ptions: nat > nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Osecond__assumptions__axioms,type,
assump8934899134041091456axioms: nat > nat > $o ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_OGraphs,type,
clique5786534781347292306Graphs: set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
clique134924887794942129at_nat: set_nat_nat > set_nat_nat > set_set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Nat__Onat,type,
clique6722202388162463298od_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Nat__Onat_J,type,
clique8906516429304539640et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
clique1181040904276305582et_nat: set_set_set_nat > set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Obinprod_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
clique6888640300188662884et_nat: set_set_set_set_nat > set_set_set_set_nat > set_se7970953024979822686et_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,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodotl,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oplucking__step,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov,type,
clique5033774636164728513irst_v: set_set_nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov__gs,type,
clique8462013130872731469t_v_gs: set_set_set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Onumbers,type,
clique3652268606331196573umbers: nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU,type,
clique2699557479641037314nd_PLU: nat > nat > nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main__graph,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_OPLU__main__rel,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cap,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__neg__cup,type,
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thf(sy_c_Clique__Large__Monotone__Circuits_Osecond__assumptions_Odeviate__pos__cap,type,
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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_OFpow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite_Fpow_nat_nat: set_nat_nat > set_set_nat_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Nat__Onat_J,type,
finite_Fpow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite7717622420921165910et_nat: set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
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thf(sy_c_Finite__Set_Ocard_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
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__Int__Oint,type,
plus_plus_int: int > int > int ).
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__Int__Oint,type,
times_times_int: int > int > int ).
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__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Num_Onum_OBit0,type,
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thf(sy_c_Num_Onum_OBit1,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7376149671870096959at_nat: set_set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
bot_bo193956671110832956et_nat: set_set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
bot_bo7308840002416255730et_nat: set_se7970953024979822686et_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_less_set_nat_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le52856854838348540et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le1881650831954960232at_nat: ( $o > nat > nat ) > ( $o > nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Int__Oint_J,type,
ord_less_eq_o_int: ( $o > int ) > ( $o > int ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le5298321079317455902at_nat: ( $o > set_nat_nat ) > ( $o > set_nat_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le6539261115178940645et_nat: ( $o > set_set_nat ) > ( $o > set_set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le8326115459943588763et_nat: ( $o > set_set_set_nat ) > ( $o > set_set_set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
ord_le4828580971730773841et_nat: ( $o > set_set_set_set_nat ) > ( $o > set_set_set_set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J_J,type,
ord_le3972702709577541822et_nat: set_se7970953024979822686et_nat > set_se7970953024979822686et_nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
order_4901310360435744934at_nat: ( ( nat > nat ) > $o ) > nat > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
order_8228081171942161500at_nat: ( set_nat_nat > $o ) > set_nat_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
order_1279421399067128355et_nat: ( set_set_nat > $o ) > set_set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
order_3016652264488350681et_nat: ( set_set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
order_1601730647829663375et_nat: ( set_set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Nat__Onat,type,
produc2803780273060847707at_nat: set_set_set_nat > nat > produc4045820344675478307at_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Nat__Onat,type,
produc6523417423482510407at_nat: produc4045820344675478307at_nat > set_set_set_nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
collect_nat_nat: ( ( nat > nat ) > $o ) > set_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
collec7201453139178570183et_nat: ( set_set_set_nat > $o ) > set_set_set_set_nat ).
thf(sy_c_Set_OPow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
pow_nat_nat: set_nat_nat > set_set_nat_nat ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Nat__Onat_J,type,
pow_set_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
pow_set_set_nat: set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
pow_set_set_set_nat: set_set_set_set_nat > set_se7970953024979822686et_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_nat_nat_nat: ( ( nat > nat ) > nat ) > set_nat_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7432509271690132940et_nat: ( ( nat > nat ) > set_nat ) > set_nat_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_9186907679027735170et_nat: ( ( nat > nat ) > set_set_nat ) > set_nat_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_nat_nat_nat2: ( nat > nat > nat ) > set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2194112158459175443et_nat: ( nat > set_set_nat ) > set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_5738044413236618185et_nat: ( nat > set_set_set_nat ) > set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_1175979950292266990et_nat: ( set_nat_nat > set_set_set_nat ) > set_set_nat_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_8569768528772619084at_nat: ( set_nat > nat > nat ) > set_set_nat > set_nat_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
image_set_nat_nat: ( set_nat > nat ) > set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_6725021117256019401et_nat: ( set_nat > set_set_nat ) > set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Nat__Onat,type,
image_1454916318497077779at_nat: ( set_set_nat > nat ) > set_set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5842784325960735177et_nat: ( set_set_nat > set_nat ) > set_set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Nat__Onat,type,
image_7011612946075707337at_nat: ( set_set_set_nat > nat ) > set_set_set_set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_2225960715480453173et_nat: ( set_set_set_nat > set_set_nat ) > set_set_set_set_nat > set_set_set_nat ).
thf(sy_c_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__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member_set_nat_nat: set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
member2946998982187404937et_nat: set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
member3774042032884853055et_nat: set_set_set_set_nat > set_se7970953024979822686et_nat > $o ).
thf(sy_v_GS____,type,
gs: set_set_set_nat ).
thf(sy_v_G____,type,
g: set_set_nat ).
thf(sy_v_H____,type,
h: set_set_nat ).
thf(sy_v_K____,type,
k: set_nat > set_set_nat ).
thf(sy_v_X,type,
x: set_set_set_nat ).
thf(sy_v_Y,type,
y: set_set_set_nat ).
thf(sy_v_Z____,type,
z: set_set_set_nat ).
thf(sy_v_k,type,
k2: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_merge____,type,
merge: set_nat > set_nat > set_set_nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p,type,
p: nat ).
% Relevant facts (1274)
thf(fact_0_PLU__main__graph_Ocong,axiom,
clique711371890332037011_graph = clique711371890332037011_graph ).
% PLU_main_graph.cong
thf(fact_1__092_060open_062v_AH_A_092_060subseteq_062_Av_AG_092_060close_062,axiom,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ h ) @ ( clique5033774636164728513irst_v @ g ) ).
% \<open>v H \<subseteq> v G\<close>
thf(fact_2_contra,axiom,
~ ( member_set_set_nat @ g @ gs ) ).
% contra
thf(fact_3_HG,axiom,
ord_le6893508408891458716et_nat @ h @ g ).
% HG
thf(fact_4_PLU__main__rel_Ocong,axiom,
clique8954521387433384062in_rel = clique8954521387433384062in_rel ).
% PLU_main_rel.cong
thf(fact_5_vGk_I2_J,axiom,
( ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ g ) @ ( clique5033774636164728513irst_v @ g ) )
= g ) ).
% vGk(2)
thf(fact_6_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_7_subsetI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( member_set_set_nat @ X @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% subsetI
thf(fact_8_subsetI,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
=> ( member2946998982187404937et_nat @ X @ B ) )
=> ( ord_le572741076514265352et_nat @ A @ B ) ) ).
% subsetI
thf(fact_9_subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_set_nat @ X @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% subsetI
thf(fact_10_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_11_subsetI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A )
=> ( member_nat_nat @ X @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% subsetI
thf(fact_12_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_13_subset__antisym,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( ord_le572741076514265352et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_14_subset__antisym,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_15_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_16_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_17_order__refl,axiom,
! [X2: set_set_nat] : ( ord_le6893508408891458716et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_18_order__refl,axiom,
! [X2: set_set_set_set_nat] : ( ord_le572741076514265352et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_19_order__refl,axiom,
! [X2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_20_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_21_order__refl,axiom,
! [X2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_22_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_23_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_24_order__refl,axiom,
! [X2: nat > nat] : ( ord_less_eq_nat_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_25_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_26_dual__order_Orefl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_27_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_28_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_29_dual__order_Orefl,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_30_dual__order_Orefl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_31_dual__order_Orefl,axiom,
! [A2: set_set_set_set_nat] : ( ord_le572741076514265352et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_32_dual__order_Orefl,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_33_v__mono,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_mono
thf(fact_34_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_35_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_36_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_37_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_38_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_39_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_40_subsetD,axiom,
! [A: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( member_nat_nat @ C @ A )
=> ( member_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_41_subsetD,axiom,
! [A: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_42_subsetD,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,C: set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( member2946998982187404937et_nat @ C @ A )
=> ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_43_subsetD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( member_set_set_nat @ C @ A )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_44_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_45_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_46_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_47_equalityE,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le572741076514265352et_nat @ A @ B )
=> ~ ( ord_le572741076514265352et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_48_equalityE,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ~ ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ~ ( ord_le9131159989063066194et_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_49_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_50_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ( member_nat_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_51_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( member_set_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_52_subset__eq,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A3: set_set_set_set_nat,B2: set_set_set_set_nat] :
! [X3: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X3 @ A3 )
=> ( member2946998982187404937et_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_53_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A3 )
=> ( member_set_set_nat @ X3 @ B2 ) ) ) ) ).
% subset_eq
thf(fact_54_sameprod__mono,axiom,
! [X4: set_nat_nat,Y: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X4 @ Y )
=> ( ord_le4954213926817602059at_nat @ ( clique134924887794942129at_nat @ X4 @ X4 ) @ ( clique134924887794942129at_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_55_sameprod__mono,axiom,
! [X4: set_set_nat,Y: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X4 @ Y )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X4 @ X4 ) @ ( clique8906516429304539640et_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_56_sameprod__mono,axiom,
! [X4: set_set_set_set_nat,Y: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ X4 @ Y )
=> ( ord_le3972702709577541822et_nat @ ( clique6888640300188662884et_nat @ X4 @ X4 ) @ ( clique6888640300188662884et_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_57_sameprod__mono,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y )
=> ( ord_le572741076514265352et_nat @ ( clique1181040904276305582et_nat @ X4 @ X4 ) @ ( clique1181040904276305582et_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_58_sameprod__mono,axiom,
! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) @ ( clique6722202388162463298od_nat @ Y @ Y ) ) ) ).
% sameprod_mono
thf(fact_59_order__antisym__conv,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_60_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_61_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_62_order__antisym__conv,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_63_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_64_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_65_order__antisym__conv,axiom,
! [Y2: set_set_set_set_nat,X2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ Y2 @ X2 )
=> ( ( ord_le572741076514265352et_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_66_order__antisym__conv,axiom,
! [Y2: set_set_set_nat,X2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ Y2 @ X2 )
=> ( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_67_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_68_linorder__le__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_69_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_70_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_71_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_72_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_74_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_75_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_76_ord__le__eq__subst,axiom,
! [A2: int,B3: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_77_ord__le__eq__subst,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_78_ord__le__eq__subst,axiom,
! [A2: nat,B3: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_79_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_80_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_81_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_82_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_83_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_84_ord__eq__le__subst,axiom,
! [A2: int,F: set_nat > int,B3: set_nat,C: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A2: set_nat,F: int > set_nat,B3: int,C: int] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_88_ord__eq__le__subst,axiom,
! [A2: nat > nat,F: nat > nat > nat,B3: nat,C: nat] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_89_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_90_linorder__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_91_order__eq__refl,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_92_order__eq__refl,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( X2 = Y2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_93_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_94_order__eq__refl,axiom,
! [X2: int,Y2: int] :
( ( X2 = Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_95_order__eq__refl,axiom,
! [X2: nat > nat,Y2: nat > nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_96_order__eq__refl,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( X2 = Y2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_97_order__eq__refl,axiom,
! [X2: set_set_set_set_nat,Y2: set_set_set_set_nat] :
( ( X2 = Y2 )
=> ( ord_le572741076514265352et_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_98_order__eq__refl,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( X2 = Y2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_99_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_100_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_101_order__subst2,axiom,
! [A2: int,B3: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_102_order__subst2,axiom,
! [A2: int,B3: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_103_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_104_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_106_order__subst2,axiom,
! [A2: int,B3: int,F: int > set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_107_order__subst2,axiom,
! [A2: set_nat,B3: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ ( F @ B3 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_108_order__subst2,axiom,
! [A2: nat,B3: nat,F: nat > nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat_nat @ ( F @ B3 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_109_order__subst1,axiom,
! [A2: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A2: nat,F: int > nat,B3: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A2: int,F: nat > int,B3: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A2: int,F: int > int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_113_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B3: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_114_order__subst1,axiom,
! [A2: set_nat,F: int > set_nat,B3: int,C: int] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_115_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_116_order__subst1,axiom,
! [A2: int,F: set_nat > int,B3: set_nat,C: set_nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_117_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_118_order__subst1,axiom,
! [A2: nat,F: ( nat > nat ) > nat,B3: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat_nat @ B3 @ C )
=> ( ! [X: nat > nat,Y3: nat > nat] :
( ( ord_less_eq_nat_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_119_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_120_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A4 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_121_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_122_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_123_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > nat,Z: nat > nat] : ( Y4 = Z ) )
= ( ^ [A4: nat > nat,B4: nat > nat] :
( ( ord_less_eq_nat_nat @ A4 @ B4 )
& ( ord_less_eq_nat_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_124_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_125_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_set_set_nat,Z: set_set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_set_set_nat,B4: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A4 @ B4 )
& ( ord_le572741076514265352et_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_126_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_set_nat,Z: set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A4 @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_127_le__fun__def,axiom,
( ord_less_eq_nat_nat
= ( ^ [F2: nat > nat,G2: nat > nat] :
! [X3: nat] : ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_128_le__funI,axiom,
! [F: nat > nat,G3: nat > nat] :
( ! [X: nat] : ( ord_less_eq_nat @ ( F @ X ) @ ( G3 @ X ) )
=> ( ord_less_eq_nat_nat @ F @ G3 ) ) ).
% le_funI
thf(fact_129_le__funE,axiom,
! [F: nat > nat,G3: nat > nat,X2: nat] :
( ( ord_less_eq_nat_nat @ F @ G3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G3 @ X2 ) ) ) ).
% le_funE
thf(fact_130_le__funD,axiom,
! [F: nat > nat,G3: nat > nat,X2: nat] :
( ( ord_less_eq_nat_nat @ F @ G3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G3 @ X2 ) ) ) ).
% le_funD
thf(fact_131_antisym,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_132_antisym,axiom,
! [A2: set_nat_nat,B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_133_antisym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_134_antisym,axiom,
! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_135_antisym,axiom,
! [A2: nat > nat,B3: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_136_antisym,axiom,
! [A2: set_set_nat,B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_137_antisym,axiom,
! [A2: set_set_set_set_nat,B3: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A2 @ B3 )
=> ( ( ord_le572741076514265352et_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_138_antisym,axiom,
! [A2: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B3 )
=> ( ( ord_le9131159989063066194et_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% antisym
thf(fact_139_dual__order_Otrans,axiom,
! [B3: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B3 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_140_dual__order_Otrans,axiom,
! [B3: set_nat_nat,A2: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ C @ B3 )
=> ( ord_le9059583361652607317at_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_141_dual__order_Otrans,axiom,
! [B3: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_142_dual__order_Otrans,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( ord_less_eq_int @ C @ B3 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_143_dual__order_Otrans,axiom,
! [B3: nat > nat,A2: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat_nat @ C @ B3 )
=> ( ord_less_eq_nat_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_144_dual__order_Otrans,axiom,
! [B3: set_set_nat,A2: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ C @ B3 )
=> ( ord_le6893508408891458716et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_145_dual__order_Otrans,axiom,
! [B3: set_set_set_set_nat,A2: set_set_set_set_nat,C: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ B3 @ A2 )
=> ( ( ord_le572741076514265352et_nat @ C @ B3 )
=> ( ord_le572741076514265352et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_146_dual__order_Otrans,axiom,
! [B3: set_set_set_nat,A2: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ C @ B3 )
=> ( ord_le9131159989063066194et_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_147_dual__order_Oantisym,axiom,
! [B3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Oantisym,axiom,
! [B3: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B3 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_149_dual__order_Oantisym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_150_dual__order_Oantisym,axiom,
! [B3: int,A2: int] :
( ( ord_less_eq_int @ B3 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_151_dual__order_Oantisym,axiom,
! [B3: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ B3 @ A2 )
=> ( ( ord_less_eq_nat_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_152_dual__order_Oantisym,axiom,
! [B3: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B3 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_153_dual__order_Oantisym,axiom,
! [B3: set_set_set_set_nat,A2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ B3 @ A2 )
=> ( ( ord_le572741076514265352et_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_154_dual__order_Oantisym,axiom,
! [B3: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B3 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_155_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_156_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A4 )
& ( ord_le9059583361652607317at_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_157_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_158_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_159_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat > nat,Z: nat > nat] : ( Y4 = Z ) )
= ( ^ [A4: nat > nat,B4: nat > nat] :
( ( ord_less_eq_nat_nat @ B4 @ A4 )
& ( ord_less_eq_nat_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_160_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A4 )
& ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_161_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_set_set_nat,Z: set_set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_set_set_nat,B4: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ B4 @ A4 )
& ( ord_le572741076514265352et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_162_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_set_nat,Z: set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [A4: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B4 @ A4 )
& ( ord_le9131159989063066194et_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_163_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_164_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B3: int] :
( ! [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_165_order__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_166_order__trans,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,Z2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ Z2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_167_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_168_order__trans,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_169_order__trans,axiom,
! [X2: nat > nat,Y2: nat > nat,Z2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_170_order__trans,axiom,
! [X2: set_set_nat,Y2: set_set_nat,Z2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ Z2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_171_order__trans,axiom,
! [X2: set_set_set_set_nat,Y2: set_set_set_set_nat,Z2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ X2 @ Y2 )
=> ( ( ord_le572741076514265352et_nat @ Y2 @ Z2 )
=> ( ord_le572741076514265352et_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_172_order__trans,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,Z2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ Z2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_173_order_Otrans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_174_order_Otrans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_175_order_Otrans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_176_order_Otrans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_177_order_Otrans,axiom,
! [A2: nat > nat,B3: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat_nat @ B3 @ C )
=> ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_178_order_Otrans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_179_order_Otrans,axiom,
! [A2: set_set_set_set_nat,B3: set_set_set_set_nat,C: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A2 @ B3 )
=> ( ( ord_le572741076514265352et_nat @ B3 @ C )
=> ( ord_le572741076514265352et_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_180_order_Otrans,axiom,
! [A2: set_set_set_nat,B3: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B3 )
=> ( ( ord_le9131159989063066194et_nat @ B3 @ C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_181_order__antisym,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_182_order__antisym,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_183_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_184_order__antisym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_185_order__antisym,axiom,
! [X2: nat > nat,Y2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_186_order__antisym,axiom,
! [X2: set_set_nat,Y2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_187_order__antisym,axiom,
! [X2: set_set_set_set_nat,Y2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ X2 @ Y2 )
=> ( ( ord_le572741076514265352et_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_188_order__antisym,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_189_ord__le__eq__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_190_ord__le__eq__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_191_ord__le__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_192_ord__le__eq__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_193_ord__le__eq__trans,axiom,
! [A2: nat > nat,B3: nat > nat,C: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_194_ord__le__eq__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_195_ord__le__eq__trans,axiom,
! [A2: set_set_set_set_nat,B3: set_set_set_set_nat,C: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le572741076514265352et_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_196_ord__le__eq__trans,axiom,
! [A2: set_set_set_nat,B3: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_197_ord__eq__le__trans,axiom,
! [A2: set_nat,B3: set_nat,C: set_nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_198_ord__eq__le__trans,axiom,
! [A2: set_nat_nat,B3: set_nat_nat,C: set_nat_nat] :
( ( A2 = B3 )
=> ( ( ord_le9059583361652607317at_nat @ B3 @ C )
=> ( ord_le9059583361652607317at_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_199_ord__eq__le__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_200_ord__eq__le__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( A2 = B3 )
=> ( ( ord_less_eq_int @ B3 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_201_ord__eq__le__trans,axiom,
! [A2: nat > nat,B3: nat > nat,C: nat > nat] :
( ( A2 = B3 )
=> ( ( ord_less_eq_nat_nat @ B3 @ C )
=> ( ord_less_eq_nat_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_202_ord__eq__le__trans,axiom,
! [A2: set_set_nat,B3: set_set_nat,C: set_set_nat] :
( ( A2 = B3 )
=> ( ( ord_le6893508408891458716et_nat @ B3 @ C )
=> ( ord_le6893508408891458716et_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_203_ord__eq__le__trans,axiom,
! [A2: set_set_set_set_nat,B3: set_set_set_set_nat,C: set_set_set_set_nat] :
( ( A2 = B3 )
=> ( ( ord_le572741076514265352et_nat @ B3 @ C )
=> ( ord_le572741076514265352et_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_204_ord__eq__le__trans,axiom,
! [A2: set_set_set_nat,B3: set_set_set_nat,C: set_set_set_nat] :
( ( A2 = B3 )
=> ( ( ord_le9131159989063066194et_nat @ B3 @ C )
=> ( ord_le9131159989063066194et_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_205_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_206_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
= ( ^ [X3: set_nat_nat,Y5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X3 @ Y5 )
& ( ord_le9059583361652607317at_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_207_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_208_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [X3: int,Y5: int] :
( ( ord_less_eq_int @ X3 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_209_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > nat,Z: nat > nat] : ( Y4 = Z ) )
= ( ^ [X3: nat > nat,Y5: nat > nat] :
( ( ord_less_eq_nat_nat @ X3 @ Y5 )
& ( ord_less_eq_nat_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_210_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [X3: set_set_nat,Y5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X3 @ Y5 )
& ( ord_le6893508408891458716et_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_211_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_set_set_nat,Z: set_set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [X3: set_set_set_set_nat,Y5: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ X3 @ Y5 )
& ( ord_le572741076514265352et_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_212_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_set_nat,Z: set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [X3: set_set_set_nat,Y5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X3 @ Y5 )
& ( ord_le9131159989063066194et_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_213_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_214_le__cases3,axiom,
! [X2: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_215_nle__le,axiom,
! [A2: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_216_nle__le,axiom,
! [A2: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A2 )
& ( B3 != A2 ) ) ) ).
% nle_le
thf(fact_217_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_218_Collect__mono__iff,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) )
= ( ! [X3: nat > nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_219_Collect__mono__iff,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
= ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_220_Collect__mono__iff,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P ) @ ( collec7201453139178570183et_nat @ Q ) )
= ( ! [X3: set_set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_221_Collect__mono__iff,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) )
= ( ! [X3: set_set_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_222_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_223_set__eq__subset,axiom,
( ( ^ [Y4: set_nat_nat,Z: set_nat_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_224_set__eq__subset,axiom,
( ( ^ [Y4: set_set_nat,Z: set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_225_set__eq__subset,axiom,
( ( ^ [Y4: set_set_set_set_nat,Z: set_set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_set_set_nat,B2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A3 @ B2 )
& ( ord_le572741076514265352et_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_226_set__eq__subset,axiom,
( ( ^ [Y4: set_set_set_nat,Z: set_set_set_nat] : ( Y4 = Z ) )
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B2 )
& ( ord_le9131159989063066194et_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_227_mem__Collect__eq,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A2 @ ( collect_set_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_228_mem__Collect__eq,axiom,
! [A2: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A2 @ ( collect_nat_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_229_mem__Collect__eq,axiom,
! [A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( member2946998982187404937et_nat @ A2 @ ( collec7201453139178570183et_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_230_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_231_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_232_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_233_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_234_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_235_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_236_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_237_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_238_subset__trans,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C2 )
=> ( ord_le9059583361652607317at_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_239_subset__trans,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C2 )
=> ( ord_le6893508408891458716et_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_240_subset__trans,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,C2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( ord_le572741076514265352et_nat @ B @ C2 )
=> ( ord_le572741076514265352et_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_241_subset__trans,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C2 )
=> ( ord_le9131159989063066194et_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_242_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_243_Collect__mono,axiom,
! [P: ( nat > nat ) > $o,Q: ( nat > nat ) > $o] :
( ! [X: nat > nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le9059583361652607317at_nat @ ( collect_nat_nat @ P ) @ ( collect_nat_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_244_Collect__mono,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X: set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_245_Collect__mono,axiom,
! [P: set_set_set_nat > $o,Q: set_set_set_nat > $o] :
( ! [X: set_set_set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le572741076514265352et_nat @ ( collec7201453139178570183et_nat @ P ) @ ( collec7201453139178570183et_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_246_Collect__mono,axiom,
! [P: set_set_nat > $o,Q: set_set_nat > $o] :
( ! [X: set_set_nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le9131159989063066194et_nat @ ( collect_set_set_nat @ P ) @ ( collect_set_set_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_247_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_248_subset__refl,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A @ A ) ).
% subset_refl
thf(fact_249_subset__refl,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ A @ A ) ).
% subset_refl
thf(fact_250_subset__refl,axiom,
! [A: set_set_set_set_nat] : ( ord_le572741076514265352et_nat @ A @ A ) ).
% subset_refl
thf(fact_251_subset__refl,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A @ A ) ).
% subset_refl
thf(fact_252_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_253_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B2: set_nat_nat] :
! [T: nat > nat] :
( ( member_nat_nat @ T @ A3 )
=> ( member_nat_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_254_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B2: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_255_subset__iff,axiom,
( ord_le572741076514265352et_nat
= ( ^ [A3: set_set_set_set_nat,B2: set_set_set_set_nat] :
! [T: set_set_set_nat] :
( ( member2946998982187404937et_nat @ T @ A3 )
=> ( member2946998982187404937et_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_256_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B2: set_set_set_nat] :
! [T: set_set_nat] :
( ( member_set_set_nat @ T @ A3 )
=> ( member_set_set_nat @ T @ B2 ) ) ) ) ).
% subset_iff
thf(fact_257_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_258_equalityD2,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_259_equalityD2,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_260_equalityD2,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( A = B )
=> ( ord_le572741076514265352et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_261_equalityD2,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ( ord_le9131159989063066194et_nat @ B @ A ) ) ).
% equalityD2
thf(fact_262_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_263_equalityD1,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( A = B )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_264_equalityD1,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( A = B )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_265_equalityD1,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( A = B )
=> ( ord_le572741076514265352et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_266_equalityD1,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( A = B )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% equalityD1
thf(fact_267_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X5: set_set_set_nat,G4: set_set_nat] :
? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X5 )
& ( ord_le6893508408891458716et_nat @ X3 @ G4 ) ) ) ) ).
% accepts_def
thf(fact_268_second__assumptions_Ov__sameprod__subset,axiom,
! [L: nat,P2: nat,K: nat,Vs: set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ) ).
% second_assumptions.v_sameprod_subset
thf(fact_269_v__gs__mono,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_mono
thf(fact_270_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_271_Greatest__equality,axiom,
! [P: set_nat_nat > $o,X2: set_nat_nat] :
( ( P @ X2 )
=> ( ! [Y3: set_nat_nat] :
( ( P @ Y3 )
=> ( ord_le9059583361652607317at_nat @ Y3 @ X2 ) )
=> ( ( order_8228081171942161500at_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_272_Greatest__equality,axiom,
! [P: int > $o,X2: int] :
( ( P @ X2 )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X2 ) )
=> ( ( order_Greatest_int @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_273_Greatest__equality,axiom,
! [P: ( nat > nat ) > $o,X2: nat > nat] :
( ( P @ X2 )
=> ( ! [Y3: nat > nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat_nat @ Y3 @ X2 ) )
=> ( ( order_4901310360435744934at_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_274_Greatest__equality,axiom,
! [P: set_set_nat > $o,X2: set_set_nat] :
( ( P @ X2 )
=> ( ! [Y3: set_set_nat] :
( ( P @ Y3 )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X2 ) )
=> ( ( order_1279421399067128355et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_275_Greatest__equality,axiom,
! [P: set_set_set_set_nat > $o,X2: set_set_set_set_nat] :
( ( P @ X2 )
=> ( ! [Y3: set_set_set_set_nat] :
( ( P @ Y3 )
=> ( ord_le572741076514265352et_nat @ Y3 @ X2 ) )
=> ( ( order_1601730647829663375et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_276_Greatest__equality,axiom,
! [P: set_set_set_nat > $o,X2: set_set_set_nat] :
( ( P @ X2 )
=> ( ! [Y3: set_set_set_nat] :
( ( P @ Y3 )
=> ( ord_le9131159989063066194et_nat @ Y3 @ X2 ) )
=> ( ( order_3016652264488350681et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_277_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_278_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X2 ) )
=> ( ! [X: set_nat] :
( ( P @ X )
=> ( ! [Y6: set_nat] :
( ( P @ Y6 )
=> ( ord_less_eq_set_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_279_GreatestI2__order,axiom,
! [P: set_nat_nat > $o,X2: set_nat_nat,Q: set_nat_nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_nat_nat] :
( ( P @ Y3 )
=> ( ord_le9059583361652607317at_nat @ Y3 @ X2 ) )
=> ( ! [X: set_nat_nat] :
( ( P @ X )
=> ( ! [Y6: set_nat_nat] :
( ( P @ Y6 )
=> ( ord_le9059583361652607317at_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_8228081171942161500at_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_280_GreatestI2__order,axiom,
! [P: int > $o,X2: int,Q: int > $o] :
( ( P @ X2 )
=> ( ! [Y3: int] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ Y3 @ X2 ) )
=> ( ! [X: int] :
( ( P @ X )
=> ( ! [Y6: int] :
( ( P @ Y6 )
=> ( ord_less_eq_int @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_281_GreatestI2__order,axiom,
! [P: ( nat > nat ) > $o,X2: nat > nat,Q: ( nat > nat ) > $o] :
( ( P @ X2 )
=> ( ! [Y3: nat > nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat_nat @ Y3 @ X2 ) )
=> ( ! [X: nat > nat] :
( ( P @ X )
=> ( ! [Y6: nat > nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_4901310360435744934at_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_282_GreatestI2__order,axiom,
! [P: set_set_nat > $o,X2: set_set_nat,Q: set_set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_set_nat] :
( ( P @ Y3 )
=> ( ord_le6893508408891458716et_nat @ Y3 @ X2 ) )
=> ( ! [X: set_set_nat] :
( ( P @ X )
=> ( ! [Y6: set_set_nat] :
( ( P @ Y6 )
=> ( ord_le6893508408891458716et_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1279421399067128355et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_283_GreatestI2__order,axiom,
! [P: set_set_set_set_nat > $o,X2: set_set_set_set_nat,Q: set_set_set_set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_set_set_set_nat] :
( ( P @ Y3 )
=> ( ord_le572741076514265352et_nat @ Y3 @ X2 ) )
=> ( ! [X: set_set_set_set_nat] :
( ( P @ X )
=> ( ! [Y6: set_set_set_set_nat] :
( ( P @ Y6 )
=> ( ord_le572741076514265352et_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1601730647829663375et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_284_GreatestI2__order,axiom,
! [P: set_set_set_nat > $o,X2: set_set_set_nat,Q: set_set_set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_set_set_nat] :
( ( P @ Y3 )
=> ( ord_le9131159989063066194et_nat @ Y3 @ X2 ) )
=> ( ! [X: set_set_set_nat] :
( ( P @ X )
=> ( ! [Y6: set_set_set_nat] :
( ( P @ Y6 )
=> ( ord_le9131159989063066194et_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_3016652264488350681et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_285_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_286_first__assumptions_Ov__mono,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le6893508408891458716et_nat @ G @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_287_acceptsI,axiom,
! [D: set_set_nat,G: set_set_nat,X4: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D @ G )
=> ( ( member_set_set_nat @ D @ X4 )
=> ( clique3686358387679108662ccepts @ X4 @ G ) ) ) ).
% acceptsI
thf(fact_288_increasing__def,axiom,
( measur1302623347068717141at_nat
= ( ^ [M: set_set_nat,Mu: set_nat > nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ M )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M )
=> ( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_289_increasing__def,axiom,
( measur1300132876559666865at_int
= ( ^ [M: set_set_nat,Mu: set_nat > int] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ M )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M )
=> ( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_int @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_290_increasing__def,axiom,
( measur5248428813077667851et_nat
= ( ^ [M: set_set_nat,Mu: set_nat > set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ M )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M )
=> ( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_291_increasing__def,axiom,
( measur1682793908422298635at_nat
= ( ^ [M: set_set_set_nat,Mu: set_set_nat > nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M )
=> ! [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ M )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_292_increasing__def,axiom,
( measur1680303437913248359at_int
= ( ^ [M: set_set_set_nat,Mu: set_set_nat > int] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M )
=> ! [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ M )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Y5 )
=> ( ord_less_eq_int @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_293_increasing__def,axiom,
( measur6851017734145829444at_nat
= ( ^ [M: set_set_nat,Mu: set_nat > nat > nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ M )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M )
=> ( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat_nat @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_294_increasing__def,axiom,
( measur496615480034414785et_nat
= ( ^ [M: set_set_nat,Mu: set_nat > set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ M )
=> ! [Y5: set_nat] :
( ( member_set_nat @ Y5 @ M )
=> ( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_le6893508408891458716et_nat @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_295_increasing__def,axiom,
( measur8307093849423850436at_nat
= ( ^ [M: set_set_nat_nat,Mu: set_nat_nat > nat] :
! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ M )
=> ! [Y5: set_nat_nat] :
( ( member_set_nat_nat @ Y5 @ M )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_296_increasing__def,axiom,
( measur8304603378914800160at_int
= ( ^ [M: set_set_nat_nat,Mu: set_nat_nat > int] :
! [X3: set_nat_nat] :
( ( member_set_nat_nat @ X3 @ M )
=> ! [Y5: set_nat_nat] :
( ( member_set_nat_nat @ Y5 @ M )
=> ( ( ord_le9059583361652607317at_nat @ X3 @ Y5 )
=> ( ord_less_eq_int @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_297_increasing__def,axiom,
( measur6219391137901972417et_nat
= ( ^ [M: set_set_set_nat,Mu: set_set_nat > set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ M )
=> ! [Y5: set_set_nat] :
( ( member_set_set_nat @ Y5 @ M )
=> ( ( ord_le6893508408891458716et_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( Mu @ X3 ) @ ( Mu @ Y5 ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_298_increasingD,axiom,
! [M2: set_set_nat,F: set_nat > nat,X2: set_nat,Y2: set_nat] :
( ( measur1302623347068717141at_nat @ M2 @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M2 )
=> ( ( member_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_299_increasingD,axiom,
! [M2: set_set_nat,F: set_nat > int,X2: set_nat,Y2: set_nat] :
( ( measur1300132876559666865at_int @ M2 @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M2 )
=> ( ( member_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_300_increasingD,axiom,
! [M2: set_set_nat,F: set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ( measur5248428813077667851et_nat @ M2 @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M2 )
=> ( ( member_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_301_increasingD,axiom,
! [M2: set_set_set_nat,F: set_set_nat > nat,X2: set_set_nat,Y2: set_set_nat] :
( ( measur1682793908422298635at_nat @ M2 @ F )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( member_set_set_nat @ X2 @ M2 )
=> ( ( member_set_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_302_increasingD,axiom,
! [M2: set_set_set_nat,F: set_set_nat > int,X2: set_set_nat,Y2: set_set_nat] :
( ( measur1680303437913248359at_int @ M2 @ F )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( member_set_set_nat @ X2 @ M2 )
=> ( ( member_set_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_303_increasingD,axiom,
! [M2: set_set_nat,F: set_nat > nat > nat,X2: set_nat,Y2: set_nat] :
( ( measur6851017734145829444at_nat @ M2 @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M2 )
=> ( ( member_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_nat_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_304_increasingD,axiom,
! [M2: set_set_nat,F: set_nat > set_set_nat,X2: set_nat,Y2: set_nat] :
( ( measur496615480034414785et_nat @ M2 @ F )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( member_set_nat @ X2 @ M2 )
=> ( ( member_set_nat @ Y2 @ M2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_305_increasingD,axiom,
! [M2: set_set_nat_nat,F: set_nat_nat > nat,X2: set_nat_nat,Y2: set_nat_nat] :
( ( measur8307093849423850436at_nat @ M2 @ F )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( member_set_nat_nat @ X2 @ M2 )
=> ( ( member_set_nat_nat @ Y2 @ M2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_306_increasingD,axiom,
! [M2: set_set_nat_nat,F: set_nat_nat > int,X2: set_nat_nat,Y2: set_nat_nat] :
( ( measur8304603378914800160at_int @ M2 @ F )
=> ( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( member_set_nat_nat @ X2 @ M2 )
=> ( ( member_set_nat_nat @ Y2 @ M2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_307_increasingD,axiom,
! [M2: set_set_set_nat,F: set_set_nat > set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ( measur6219391137901972417et_nat @ M2 @ F )
=> ( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( member_set_set_nat @ X2 @ M2 )
=> ( ( member_set_set_nat @ Y2 @ M2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% increasingD
thf(fact_308_le__left__mono,axiom,
! [X2: set_nat,Y2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ A2 ) ) ) ).
% le_left_mono
thf(fact_309_le__left__mono,axiom,
! [X2: set_nat_nat,Y2: set_nat_nat,A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X2 @ Y2 )
=> ( ( ord_le9059583361652607317at_nat @ Y2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ X2 @ A2 ) ) ) ).
% le_left_mono
thf(fact_310_le__left__mono,axiom,
! [X2: nat,Y2: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ A2 ) ) ) ).
% le_left_mono
thf(fact_311_le__left__mono,axiom,
! [X2: int,Y2: int,A2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ A2 )
=> ( ord_less_eq_int @ X2 @ A2 ) ) ) ).
% le_left_mono
thf(fact_312_le__left__mono,axiom,
! [X2: nat > nat,Y2: nat > nat,A2: nat > nat] :
( ( ord_less_eq_nat_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat_nat @ Y2 @ A2 )
=> ( ord_less_eq_nat_nat @ X2 @ A2 ) ) ) ).
% le_left_mono
thf(fact_313_le__left__mono,axiom,
! [X2: set_set_nat,Y2: set_set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X2 @ Y2 )
=> ( ( ord_le6893508408891458716et_nat @ Y2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ X2 @ A2 ) ) ) ).
% le_left_mono
thf(fact_314_le__left__mono,axiom,
! [X2: set_set_set_set_nat,Y2: set_set_set_set_nat,A2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ X2 @ Y2 )
=> ( ( ord_le572741076514265352et_nat @ Y2 @ A2 )
=> ( ord_le572741076514265352et_nat @ X2 @ A2 ) ) ) ).
% le_left_mono
thf(fact_315_le__left__mono,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X2 @ Y2 )
=> ( ( ord_le9131159989063066194et_nat @ Y2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ X2 @ A2 ) ) ) ).
% le_left_mono
thf(fact_316_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X5: $o > set_nat,Y7: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_set_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_317_le__rel__bool__arg__iff,axiom,
( ord_le5298321079317455902at_nat
= ( ^ [X5: $o > set_nat_nat,Y7: $o > set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le9059583361652607317at_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_318_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X5: $o > nat,Y7: $o > nat] :
( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_319_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_int
= ( ^ [X5: $o > int,Y7: $o > int] :
( ( ord_less_eq_int @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_int @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_320_le__rel__bool__arg__iff,axiom,
( ord_le1881650831954960232at_nat
= ( ^ [X5: $o > nat > nat,Y7: $o > nat > nat] :
( ( ord_less_eq_nat_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_less_eq_nat_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_321_le__rel__bool__arg__iff,axiom,
( ord_le6539261115178940645et_nat
= ( ^ [X5: $o > set_set_nat,Y7: $o > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le6893508408891458716et_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_322_le__rel__bool__arg__iff,axiom,
( ord_le4828580971730773841et_nat
= ( ^ [X5: $o > set_set_set_set_nat,Y7: $o > set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le572741076514265352et_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_323_le__rel__bool__arg__iff,axiom,
( ord_le8326115459943588763et_nat
= ( ^ [X5: $o > set_set_set_nat,Y7: $o > set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( X5 @ $false ) @ ( Y7 @ $false ) )
& ( ord_le9131159989063066194et_nat @ ( X5 @ $true ) @ ( Y7 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_324_verit__la__disequality,axiom,
! [A2: nat,B3: nat] :
( ( A2 = B3 )
| ~ ( ord_less_eq_nat @ A2 @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_325_verit__la__disequality,axiom,
! [A2: int,B3: int] :
( ( A2 = B3 )
| ~ ( ord_less_eq_int @ A2 @ B3 )
| ~ ( ord_less_eq_int @ B3 @ A2 ) ) ).
% verit_la_disequality
thf(fact_326_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Y )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_327_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3686358387679108662ccepts @ X4 @ G )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X4 )
& ( ord_le6893508408891458716et_nat @ X3 @ G ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_328_first__assumptions_OacceptsI,axiom,
! [L: nat,P2: nat,K: nat,D: set_set_nat,G: set_set_nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le6893508408891458716et_nat @ D @ G )
=> ( ( member_set_set_nat @ D @ X4 )
=> ( clique3686358387679108662ccepts @ X4 @ G ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_329_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_330_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_331_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_332_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_333_verit__comp__simplify1_I2_J,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_334_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_335_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_set_set_nat] : ( ord_le572741076514265352et_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_336_verit__comp__simplify1_I2_J,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_337_second__assumptions_Oaxioms_I1_J,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( assump5453534214990993103ptions @ L @ P2 @ K ) ) ).
% second_assumptions.axioms(1)
thf(fact_338_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_339_first__assumptions_Ov__gs__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ X4 )
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ X4 ) ) ) ).
% first_assumptions.v_gs_def
thf(fact_340_second__assumptions__def,axiom,
( assump2881078719466019805ptions
= ( ^ [L2: nat,P3: nat,K2: nat] :
( ( assump5453534214990993103ptions @ L2 @ P3 @ K2 )
& ( assump8934899134041091456axioms @ L2 @ K2 ) ) ) ) ).
% second_assumptions_def
thf(fact_341_second__assumptions_Ointro,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( assump8934899134041091456axioms @ L @ K )
=> ( assump2881078719466019805ptions @ L @ P2 @ K ) ) ) ).
% second_assumptions.intro
thf(fact_342_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique951075384711337423ACC_cf @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_343_numbers2__mono,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X2 ) @ ( clique3652268606331196573umbers @ X2 ) ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ Y2 ) @ ( clique3652268606331196573umbers @ Y2 ) ) ) ) ).
% numbers2_mono
thf(fact_344__092_060open_062card_A_Iv_AG_J_A_061_Ak_A_092_060and_062_Av_AG_094_092_060two_062_A_061_AG_092_060close_062,axiom,
( ( ( finite_card_nat @ ( clique5033774636164728513irst_v @ g ) )
= k2 )
& ( ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ g ) @ ( clique5033774636164728513irst_v @ g ) )
= g ) ) ).
% \<open>card (v G) = k \<and> v G^\<two> = G\<close>
thf(fact_345_Fpow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_346_Fpow__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_le4954213926817602059at_nat @ ( finite_Fpow_nat_nat @ A ) @ ( finite_Fpow_nat_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_347_Fpow__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ ( finite_Fpow_set_nat @ A ) @ ( finite_Fpow_set_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_348_Fpow__mono,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ord_le3972702709577541822et_nat @ ( finite271239046940139020et_nat @ A ) @ ( finite271239046940139020et_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_349_Fpow__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le572741076514265352et_nat @ ( finite7717622420921165910et_nat @ A ) @ ( finite7717622420921165910et_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_350_image__eqI,axiom,
! [B3: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_351_image__eqI,axiom,
! [B3: nat,F: set_nat > nat,X2: set_nat,A: set_set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_nat @ B3 @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_352_image__eqI,axiom,
! [B3: set_nat,F: nat > set_nat,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_set_nat @ B3 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_353_image__eqI,axiom,
! [B3: nat,F: set_set_nat > nat,X2: set_set_nat,A: set_set_set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_set_set_nat @ X2 @ A )
=> ( member_nat @ B3 @ ( image_1454916318497077779at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_354_image__eqI,axiom,
! [B3: nat,F: ( nat > nat ) > nat,X2: nat > nat,A: set_nat_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_nat @ B3 @ ( image_nat_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_355_image__eqI,axiom,
! [B3: set_nat,F: set_nat > set_nat,X2: set_nat,A: set_set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_356_image__eqI,axiom,
! [B3: set_set_nat,F: nat > set_set_nat,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_set_set_nat @ B3 @ ( image_2194112158459175443et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_357_image__eqI,axiom,
! [B3: nat > nat,F: nat > nat > nat,X2: nat,A: set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_358_image__eqI,axiom,
! [B3: set_nat,F: set_set_nat > set_nat,X2: set_set_nat,A: set_set_set_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_set_set_nat @ X2 @ A )
=> ( member_set_nat @ B3 @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_359_image__eqI,axiom,
! [B3: set_nat,F: ( nat > nat ) > set_nat,X2: nat > nat,A: set_nat_nat] :
( ( B3
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A )
=> ( member_set_nat @ B3 @ ( image_7432509271690132940et_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_360_ACC__cf__mono,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ Y )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k2 @ X4 ) @ ( clique951075384711337423ACC_cf @ k2 @ Y ) ) ) ).
% ACC_cf_mono
thf(fact_361_vGk_I1_J,axiom,
( ( finite_card_nat @ ( clique5033774636164728513irst_v @ g ) )
= k2 ) ).
% vGk(1)
thf(fact_362_card__numbers,axiom,
! [N: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N ) )
= N ) ).
% card_numbers
thf(fact_363_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_364_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_365_rev__image__eqI,axiom,
! [X2: set_nat,A: set_set_nat,B3: nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_set_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_366_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_set_nat @ B3 @ ( image_nat_set_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_367_rev__image__eqI,axiom,
! [X2: set_set_nat,A: set_set_set_nat,B3: nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_1454916318497077779at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_368_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B3: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat @ B3 @ ( image_nat_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_369_rev__image__eqI,axiom,
! [X2: set_nat,A: set_set_nat,B3: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_set_nat @ B3 @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_370_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: set_set_nat,F: nat > set_set_nat] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_set_set_nat @ B3 @ ( image_2194112158459175443et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_371_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B3: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_nat_nat @ B3 @ ( image_nat_nat_nat2 @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_372_rev__image__eqI,axiom,
! [X2: set_set_nat,A: set_set_set_nat,B3: set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_set_nat @ B3 @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_373_rev__image__eqI,axiom,
! [X2: nat > nat,A: set_nat_nat,B3: set_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ X2 @ A )
=> ( ( B3
= ( F @ X2 ) )
=> ( member_set_nat @ B3 @ ( image_7432509271690132940et_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_374_ball__imageD,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_5842784325960735177et_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_375_ball__imageD,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_set_nat > $o] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ ( image_9186907679027735170et_nat @ F @ A ) )
=> ( P @ X ) )
=> ! [X6: nat > nat] :
( ( member_nat_nat @ X6 @ A )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_376_image__cong,axiom,
! [M2: set_set_set_nat,N2: set_set_set_nat,F: set_set_nat > set_nat,G3: set_set_nat > set_nat] :
( ( M2 = N2 )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ N2 )
=> ( ( F @ X )
= ( G3 @ X ) ) )
=> ( ( image_5842784325960735177et_nat @ F @ M2 )
= ( image_5842784325960735177et_nat @ G3 @ N2 ) ) ) ) ).
% image_cong
thf(fact_377_image__cong,axiom,
! [M2: set_nat_nat,N2: set_nat_nat,F: ( nat > nat ) > set_set_nat,G3: ( nat > nat ) > set_set_nat] :
( ( M2 = N2 )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ N2 )
=> ( ( F @ X )
= ( G3 @ X ) ) )
=> ( ( image_9186907679027735170et_nat @ F @ M2 )
= ( image_9186907679027735170et_nat @ G3 @ N2 ) ) ) ) ).
% image_cong
thf(fact_378_bex__imageD,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_5842784325960735177et_nat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_379_bex__imageD,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,P: set_set_nat > $o] :
( ? [X6: set_set_nat] :
( ( member_set_set_nat @ X6 @ ( image_9186907679027735170et_nat @ F @ A ) )
& ( P @ X6 ) )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_380_image__iff,axiom,
! [Z2: set_set_nat,F: ( nat > nat ) > set_set_nat,A: set_nat_nat] :
( ( member_set_set_nat @ Z2 @ ( image_9186907679027735170et_nat @ F @ A ) )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_381_image__iff,axiom,
! [Z2: set_nat,F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( member_set_nat @ Z2 @ ( image_5842784325960735177et_nat @ F @ A ) )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
& ( Z2
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_382_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_383_imageI,axiom,
! [X2: set_nat,A: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_set_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_384_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A ) ) ) ).
% imageI
thf(fact_385_imageI,axiom,
! [X2: set_set_nat,A: set_set_set_nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_1454916318497077779at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_386_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_387_imageI,axiom,
! [X2: set_nat,A: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_388_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > set_set_nat] :
( ( member_nat @ X2 @ A )
=> ( member_set_set_nat @ ( F @ X2 ) @ ( image_2194112158459175443et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_389_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_nat_nat_nat2 @ F @ A ) ) ) ).
% imageI
thf(fact_390_imageI,axiom,
! [X2: set_set_nat,A: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_5842784325960735177et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_391_imageI,axiom,
! [X2: nat > nat,A: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ X2 @ A )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_7432509271690132940et_nat @ F @ A ) ) ) ).
% imageI
thf(fact_392_image__Fpow__mono,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ ( image_set_nat_nat @ F ) @ ( finite_Fpow_set_nat @ A ) ) @ ( finite_Fpow_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_393_image__Fpow__mono,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A ) @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_2225960715480453173et_nat @ ( image_5842784325960735177et_nat @ F ) @ ( finite7717622420921165910et_nat @ A ) ) @ ( finite_Fpow_set_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_394_image__Fpow__mono,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_9186907679027735170et_nat @ F @ A ) @ B )
=> ( ord_le572741076514265352et_nat @ ( image_1175979950292266990et_nat @ ( image_9186907679027735170et_nat @ F ) @ ( finite_Fpow_nat_nat @ A ) ) @ ( finite7717622420921165910et_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_395_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_396_all__subset__image,axiom,
! [F: set_nat > nat,A: set_set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( P @ ( image_set_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_397_all__subset__image,axiom,
! [F: nat > set_nat,A: set_nat,P: set_set_nat > $o] :
( ( ! [B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_set_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_398_all__subset__image,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ A )
=> ( P @ ( image_nat_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_399_all__subset__image,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ A )
=> ( P @ ( image_1454916318497077779at_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_400_all__subset__image,axiom,
! [F: nat > nat > nat,A: set_nat,P: set_nat_nat > $o] :
( ( ! [B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_401_all__subset__image,axiom,
! [F: set_nat > set_nat,A: set_set_nat,P: set_set_nat > $o] :
( ( ! [B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( P @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_402_all__subset__image,axiom,
! [F: nat > set_set_nat,A: set_nat,P: set_set_set_nat > $o] :
( ( ! [B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( P @ ( image_2194112158459175443et_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_403_all__subset__image,axiom,
! [F: set_set_set_nat > nat,A: set_set_set_set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_7011612946075707337at_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ B2 @ A )
=> ( P @ ( image_7011612946075707337at_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_404_all__subset__image,axiom,
! [F: set_nat > nat > nat,A: set_set_nat,P: set_nat_nat > $o] :
( ( ! [B2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B2 @ ( image_8569768528772619084at_nat @ F @ A ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( P @ ( image_8569768528772619084at_nat @ F @ B2 ) ) ) ) ) ).
% all_subset_image
thf(fact_405_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_406_subset__image__iff,axiom,
! [B: set_nat,F: set_nat > nat,A: set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A )
& ( B
= ( image_set_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_407_subset__image__iff,axiom,
! [B: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_408_subset__image__iff,axiom,
! [B: set_nat,F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A )
& ( B
= ( image_nat_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_409_subset__image__iff,axiom,
! [B: set_nat,F: set_set_nat > nat,A: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A )
& ( B
= ( image_1454916318497077779at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_410_subset__image__iff,axiom,
! [B: set_nat_nat,F: nat > nat > nat,A: set_nat] :
( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_411_subset__image__iff,axiom,
! [B: set_set_nat,F: set_nat > set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A )
& ( B
= ( image_7916887816326733075et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_412_subset__image__iff,axiom,
! [B: set_set_set_nat,F: nat > set_set_nat,A: set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_2194112158459175443et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_413_subset__image__iff,axiom,
! [B: set_nat,F: set_set_set_nat > nat,A: set_set_set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_7011612946075707337at_nat @ F @ A ) )
= ( ? [AA: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ AA @ A )
& ( B
= ( image_7011612946075707337at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_414_subset__image__iff,axiom,
! [B: set_nat_nat,F: set_nat > nat > nat,A: set_set_nat] :
( ( ord_le9059583361652607317at_nat @ B @ ( image_8569768528772619084at_nat @ F @ A ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A )
& ( B
= ( image_8569768528772619084at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_415_image__subset__iff,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A ) @ B )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A )
=> ( member_set_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_416_image__subset__iff,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_9186907679027735170et_nat @ F @ A ) @ B )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A )
=> ( member_set_set_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_417_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_418_subset__imageE,axiom,
! [B: set_nat,F: set_nat > nat,A: set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A )
=> ( B
!= ( image_set_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_419_subset__imageE,axiom,
! [B: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_set_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_420_subset__imageE,axiom,
! [B: set_nat,F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_421_subset__imageE,axiom,
! [B: set_nat,F: set_set_nat > nat,A: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A ) )
=> ~ ! [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A )
=> ( B
!= ( image_1454916318497077779at_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_422_subset__imageE,axiom,
! [B: set_nat_nat,F: nat > nat > nat,A: set_nat] :
( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat_nat2 @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_423_subset__imageE,axiom,
! [B: set_set_nat,F: set_nat > set_nat,A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A )
=> ( B
!= ( image_7916887816326733075et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_424_subset__imageE,axiom,
! [B: set_set_set_nat,F: nat > set_set_nat,A: set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_2194112158459175443et_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_425_subset__imageE,axiom,
! [B: set_nat,F: set_set_set_nat > nat,A: set_set_set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_7011612946075707337at_nat @ F @ A ) )
=> ~ ! [C3: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ C3 @ A )
=> ( B
!= ( image_7011612946075707337at_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_426_subset__imageE,axiom,
! [B: set_nat_nat,F: set_nat > nat > nat,A: set_set_nat] :
( ( ord_le9059583361652607317at_nat @ B @ ( image_8569768528772619084at_nat @ F @ A ) )
=> ~ ! [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A )
=> ( B
!= ( image_8569768528772619084at_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_427_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_428_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > nat,B: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_429_image__subsetI,axiom,
! [A: set_nat,F: nat > set_nat,B: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_430_image__subsetI,axiom,
! [A: set_set_set_nat,F: set_set_nat > nat,B: set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_431_image__subsetI,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat,B: set_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_432_image__subsetI,axiom,
! [A: set_nat,F: nat > nat > nat,B: set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat_nat @ ( F @ X ) @ B ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_433_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > set_nat,B: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_434_image__subsetI,axiom,
! [A: set_nat,F: nat > set_set_nat,B: set_set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_set_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_435_image__subsetI,axiom,
! [A: set_set_set_set_nat,F: set_set_set_nat > nat,B: set_nat] :
( ! [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_7011612946075707337at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_436_image__subsetI,axiom,
! [A: set_set_nat,F: set_nat > nat > nat,B: set_nat_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( member_nat_nat @ ( F @ X ) @ B ) )
=> ( ord_le9059583361652607317at_nat @ ( image_8569768528772619084at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_437_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_438_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_439_image__mono,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_nat > nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A ) @ ( image_set_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_440_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A ) @ ( image_nat_nat_nat2 @ F @ B ) ) ) ).
% image_mono
thf(fact_441_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A ) @ ( image_2194112158459175443et_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_442_image__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: ( nat > nat ) > nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A ) @ ( image_nat_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_443_image__mono,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A ) @ ( image_7916887816326733075et_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_444_image__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,F: set_set_nat > nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A ) @ ( image_1454916318497077779at_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_445_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > set_set_set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le572741076514265352et_nat @ ( image_5738044413236618185et_nat @ F @ A ) @ ( image_5738044413236618185et_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_446_image__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7432509271690132940et_nat @ F @ A ) @ ( image_7432509271690132940et_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_447_second__assumptions_Oaxioms_I2_J,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( assump8934899134041091456axioms @ L @ K ) ) ).
% second_assumptions.axioms(2)
thf(fact_448_ACC__cf___092_060F_062,axiom,
! [X4: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k2 @ X4 ) @ ( clique2971579238625216137irst_F @ k2 ) ) ).
% ACC_cf_\<F>
thf(fact_449_G_I1_J,axiom,
member_set_set_nat @ g @ ( clique3326749438856946062irst_K @ k2 ) ).
% G(1)
thf(fact_450_vm,axiom,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ g ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ).
% vm
thf(fact_451_ex__card,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ A ) )
=> ? [S: set_nat] :
( ( ord_less_eq_set_nat @ S @ A )
& ( ( finite_card_nat @ S )
= N ) ) ) ).
% ex_card
thf(fact_452_ex__card,axiom,
! [N: nat,A: set_nat_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ A ) )
=> ? [S: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S @ A )
& ( ( finite_card_nat_nat @ S )
= N ) ) ) ).
% ex_card
thf(fact_453_ex__card,axiom,
! [N: nat,A: set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ A ) )
=> ? [S: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ A )
& ( ( finite_card_set_nat @ S )
= N ) ) ) ).
% ex_card
thf(fact_454_ex__card,axiom,
! [N: nat,A: set_set_set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite8805468973633305546et_nat @ A ) )
=> ? [S: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ S @ A )
& ( ( finite8805468973633305546et_nat @ S )
= N ) ) ) ).
% ex_card
thf(fact_455_ex__card,axiom,
! [N: nat,A: set_set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ A ) )
=> ? [S: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S @ A )
& ( ( finite1149291290879098388et_nat @ S )
= N ) ) ) ).
% ex_card
thf(fact_456_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_457_v___092_060G_062__2,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% v_\<G>_2
thf(fact_458_surj__card__le,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_459_surj__card__le,axiom,
! [A: set_set_nat,B: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_set_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_460_surj__card__le,axiom,
! [A: set_nat,B: set_set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_461_surj__card__le,axiom,
! [A: set_nat_nat,B: set_nat,F: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_462_surj__card__le,axiom,
! [A: set_set_set_nat,B: set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite1149291290879098388et_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_463_surj__card__le,axiom,
! [A: set_nat,B: set_nat_nat,F: nat > nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_464_surj__card__le,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_set_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_465_surj__card__le,axiom,
! [A: set_nat,B: set_set_set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_466_surj__card__le,axiom,
! [A: set_set_nat,B: set_nat_nat,F: set_nat > nat > nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ord_le9059583361652607317at_nat @ B @ ( image_8569768528772619084at_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ B ) @ ( finite_card_set_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_467_surj__card__le,axiom,
! [A: set_nat_nat,B: set_set_nat,F: ( nat > nat ) > set_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7432509271690132940et_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_nat_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_468_ACC__cf__I,axiom,
! [F3: nat > nat,X4: set_set_set_nat] :
( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ k2 ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ k2 @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ k2 @ X4 ) ) ) ) ).
% ACC_cf_I
thf(fact_469_image__Pow__mono,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A ) @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ ( image_set_nat_nat @ F ) @ ( pow_set_nat @ A ) ) @ ( pow_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_470_image__Pow__mono,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A ) @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_2225960715480453173et_nat @ ( image_5842784325960735177et_nat @ F ) @ ( pow_set_set_nat @ A ) ) @ ( pow_set_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_471_image__Pow__mono,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_9186907679027735170et_nat @ F @ A ) @ B )
=> ( ord_le572741076514265352et_nat @ ( image_1175979950292266990et_nat @ ( image_9186907679027735170et_nat @ F ) @ ( pow_nat_nat @ A ) ) @ ( pow_set_set_nat @ B ) ) ) ).
% image_Pow_mono
thf(fact_472_first__assumptions_Ofinite__numbers2,axiom,
! [L: nat,P2: nat,K: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ) ).
% first_assumptions.finite_numbers2
thf(fact_473_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k2 ) ).
% finite_\<F>
thf(fact_474_finite__numbers,axiom,
! [N: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ).
% finite_numbers
thf(fact_475_finite__vG,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ).
% finite_vG
thf(fact_476_finite__members___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ).
% finite_members_\<G>
thf(fact_477_G0,axiom,
member_set_set_nat @ g @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% G0
thf(fact_478__092_060K_062___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% \<K>_\<G>
thf(fact_479_v___092_060G_062,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ).
% v_\<G>
thf(fact_480_finite__imageI,axiom,
! [F3: set_nat,H2: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( finite_finite_nat @ ( image_nat_nat @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_481_finite__imageI,axiom,
! [F3: set_set_nat,H2: set_nat > nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( finite_finite_nat @ ( image_set_nat_nat @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_482_finite__imageI,axiom,
! [F3: set_nat,H2: nat > set_nat] :
( ( finite_finite_nat @ F3 )
=> ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_483_finite__imageI,axiom,
! [F3: set_set_nat,H2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( finite1152437895449049373et_nat @ ( image_7916887816326733075et_nat @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_484_finite__imageI,axiom,
! [F3: set_nat_nat,H2: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ F3 )
=> ( finite_finite_nat @ ( image_nat_nat_nat @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_485_finite__imageI,axiom,
! [F3: set_nat,H2: nat > nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( finite2115694454571419734at_nat @ ( image_nat_nat_nat2 @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_486_finite__imageI,axiom,
! [F3: set_nat,H2: nat > set_set_nat] :
( ( finite_finite_nat @ F3 )
=> ( finite6739761609112101331et_nat @ ( image_2194112158459175443et_nat @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_487_finite__imageI,axiom,
! [F3: set_set_set_nat,H2: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( finite_finite_nat @ ( image_1454916318497077779at_nat @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_488_finite__imageI,axiom,
! [F3: set_set_nat,H2: set_nat > nat > nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( finite2115694454571419734at_nat @ ( image_8569768528772619084at_nat @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_489_finite__imageI,axiom,
! [F3: set_set_nat,H2: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( finite6739761609112101331et_nat @ ( image_6725021117256019401et_nat @ H2 @ F3 ) ) ) ).
% finite_imageI
thf(fact_490_PowI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( member_set_nat @ A @ ( pow_nat @ B ) ) ) ).
% PowI
thf(fact_491_PowI,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( member_set_nat_nat @ A @ ( pow_nat_nat @ B ) ) ) ).
% PowI
thf(fact_492_PowI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( member_set_set_nat @ A @ ( pow_set_nat @ B ) ) ) ).
% PowI
thf(fact_493_PowI,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( member3774042032884853055et_nat @ A @ ( pow_set_set_set_nat @ B ) ) ) ).
% PowI
thf(fact_494_PowI,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( member2946998982187404937et_nat @ A @ ( pow_set_set_nat @ B ) ) ) ).
% PowI
thf(fact_495_Pow__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_496_Pow__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( member_set_nat_nat @ A @ ( pow_nat_nat @ B ) )
= ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_497_Pow__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( member_set_set_nat @ A @ ( pow_set_nat @ B ) )
= ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_498_Pow__iff,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ A @ ( pow_set_set_set_nat @ B ) )
= ( ord_le572741076514265352et_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_499_Pow__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A @ ( pow_set_set_nat @ B ) )
= ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% Pow_iff
thf(fact_500_finite__v__gs,axiom,
! [X4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ).
% finite_v_gs
thf(fact_501_finite__ACC,axiom,
! [X4: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k2 @ X4 ) ) ).
% finite_ACC
thf(fact_502_finite__Pow__iff,axiom,
! [A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ ( pow_nat_nat @ A ) )
= ( finite2115694454571419734at_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_503_finite__Pow__iff,axiom,
! [A: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ ( pow_set_set_nat @ A ) )
= ( finite6739761609112101331et_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_504_finite__Pow__iff,axiom,
! [A: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_505_finite__Pow__iff,axiom,
! [A: set_set_nat] :
( ( finite6739761609112101331et_nat @ ( pow_set_nat @ A ) )
= ( finite1152437895449049373et_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_506_finite__numbers2,axiom,
! [N: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) ) ).
% finite_numbers2
thf(fact_507_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% finite_\<G>
thf(fact_508_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k2 )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k2 ) @ ( clique2971579238625216137irst_F @ k2 ) ) ) ).
% local.NEG_def
thf(fact_509_NEG___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ k2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% NEG_\<G>
thf(fact_510_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_511_Pow__top,axiom,
! [A: set_set_nat] : ( member_set_set_nat @ A @ ( pow_set_nat @ A ) ) ).
% Pow_top
thf(fact_512_Pow__top,axiom,
! [A: set_set_set_nat] : ( member2946998982187404937et_nat @ A @ ( pow_set_set_nat @ A ) ) ).
% Pow_top
thf(fact_513_Pow__top,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( pow_nat @ A ) ) ).
% Pow_top
thf(fact_514_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_515_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_516_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_517_sameprod__finite,axiom,
! [X4: set_set_nat] :
( ( finite1152437895449049373et_nat @ X4 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_518_sameprod__finite,axiom,
! [X4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ X4 )
=> ( finite3586981331298542604at_nat @ ( clique134924887794942129at_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_519_sameprod__finite,axiom,
! [X4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X4 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_520_sameprod__finite,axiom,
! [X4: set_nat] :
( ( finite_finite_nat @ X4 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) ) ) ).
% sameprod_finite
thf(fact_521_first__assumptions_Ofinite___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.finite_\<G>
thf(fact_522_first__assumptions_Ofinite__members___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite1152437895449049373et_nat @ G ) ) ) ).
% first_assumptions.finite_members_\<G>
thf(fact_523_first__assumptions_Ofinite___092_060F_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.finite_\<F>
thf(fact_524_first__assumptions_Ofinite__vG,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% first_assumptions.finite_vG
thf(fact_525_PowD,axiom,
! [A: set_nat,B: set_nat] :
( ( member_set_nat @ A @ ( pow_nat @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% PowD
thf(fact_526_PowD,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( member_set_nat_nat @ A @ ( pow_nat_nat @ B ) )
=> ( ord_le9059583361652607317at_nat @ A @ B ) ) ).
% PowD
thf(fact_527_PowD,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( member_set_set_nat @ A @ ( pow_set_nat @ B ) )
=> ( ord_le6893508408891458716et_nat @ A @ B ) ) ).
% PowD
thf(fact_528_PowD,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ A @ ( pow_set_set_set_nat @ B ) )
=> ( ord_le572741076514265352et_nat @ A @ B ) ) ).
% PowD
thf(fact_529_PowD,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( member2946998982187404937et_nat @ A @ ( pow_set_set_nat @ B ) )
=> ( ord_le9131159989063066194et_nat @ A @ B ) ) ).
% PowD
thf(fact_530_first__assumptions_O_092_060K_062___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.\<K>_\<G>
thf(fact_531_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ A2 @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_532_finite__has__maximal2,axiom,
! [A: set_set_nat_nat,A2: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( member_set_nat_nat @ A2 @ A )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A )
& ( ord_le9059583361652607317at_nat @ A2 @ X )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_533_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A2 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_534_finite__has__maximal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X: int] :
( ( member_int @ X @ A )
& ( ord_less_eq_int @ A2 @ X )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_535_finite__has__maximal2,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( member_nat_nat @ A2 @ A )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( ord_less_eq_nat_nat @ A2 @ X )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_536_finite__has__maximal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
& ( ord_le6893508408891458716et_nat @ A2 @ X )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_537_finite__has__maximal2,axiom,
! [A: set_se7970953024979822686et_nat,A2: set_set_set_set_nat] :
( ( finite5436706893199572543et_nat @ A )
=> ( ( member3774042032884853055et_nat @ A2 @ A )
=> ? [X: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ X @ A )
& ( ord_le572741076514265352et_nat @ A2 @ X )
& ! [Xa: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ Xa @ A )
=> ( ( ord_le572741076514265352et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_538_finite__has__maximal2,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( member2946998982187404937et_nat @ A2 @ A )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
& ( ord_le9131159989063066194et_nat @ A2 @ X )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_539_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_540_finite__has__minimal2,axiom,
! [A: set_set_nat_nat,A2: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( member_set_nat_nat @ A2 @ A )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A )
& ( ord_le9059583361652607317at_nat @ X @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_541_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_542_finite__has__minimal2,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ? [X: int] :
( ( member_int @ X @ A )
& ( ord_less_eq_int @ X @ A2 )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_543_finite__has__minimal2,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( member_nat_nat @ A2 @ A )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ( ord_less_eq_nat_nat @ X @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_544_finite__has__minimal2,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( member_set_set_nat @ A2 @ A )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
& ( ord_le6893508408891458716et_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_545_finite__has__minimal2,axiom,
! [A: set_se7970953024979822686et_nat,A2: set_set_set_set_nat] :
( ( finite5436706893199572543et_nat @ A )
=> ( ( member3774042032884853055et_nat @ A2 @ A )
=> ? [X: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ X @ A )
& ( ord_le572741076514265352et_nat @ X @ A2 )
& ! [Xa: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ Xa @ A )
=> ( ( ord_le572741076514265352et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_546_finite__has__minimal2,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( member2946998982187404937et_nat @ A2 @ A )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
& ( ord_le9131159989063066194et_nat @ X @ A2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_547_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_548_finite__subset,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( finite2115694454571419734at_nat @ A ) ) ) ).
% finite_subset
thf(fact_549_finite__subset,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% finite_subset
thf(fact_550_finite__subset,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( finite5926941155766903689et_nat @ B )
=> ( finite5926941155766903689et_nat @ A ) ) ) ).
% finite_subset
thf(fact_551_finite__subset,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( finite6739761609112101331et_nat @ B )
=> ( finite6739761609112101331et_nat @ A ) ) ) ).
% finite_subset
thf(fact_552_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_553_infinite__super,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S2 @ T2 )
=> ( ~ ( finite2115694454571419734at_nat @ S2 )
=> ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_554_infinite__super,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S2 @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S2 )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_555_infinite__super,axiom,
! [S2: set_set_set_set_nat,T2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ S2 @ T2 )
=> ( ~ ( finite5926941155766903689et_nat @ S2 )
=> ~ ( finite5926941155766903689et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_556_infinite__super,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S2 @ T2 )
=> ( ~ ( finite6739761609112101331et_nat @ S2 )
=> ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_557_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_558_rev__finite__subset,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( finite2115694454571419734at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_559_rev__finite__subset,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_560_rev__finite__subset,axiom,
! [B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ B )
=> ( ( ord_le572741076514265352et_nat @ A @ B )
=> ( finite5926941155766903689et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_561_rev__finite__subset,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( finite6739761609112101331et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_562_first__assumptions_Ofinite__v__gs,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ) ).
% first_assumptions.finite_v_gs
thf(fact_563_Pow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A ) @ ( pow_nat @ B ) ) ) ).
% Pow_mono
thf(fact_564_Pow__mono,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_le4954213926817602059at_nat @ ( pow_nat_nat @ A ) @ ( pow_nat_nat @ B ) ) ) ).
% Pow_mono
thf(fact_565_Pow__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_le9131159989063066194et_nat @ ( pow_set_nat @ A ) @ ( pow_set_nat @ B ) ) ) ).
% Pow_mono
thf(fact_566_Pow__mono,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ord_le3972702709577541822et_nat @ ( pow_set_set_set_nat @ A ) @ ( pow_set_set_set_nat @ B ) ) ) ).
% Pow_mono
thf(fact_567_Pow__mono,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_le572741076514265352et_nat @ ( pow_set_set_nat @ A ) @ ( pow_set_set_nat @ B ) ) ) ).
% Pow_mono
thf(fact_568_image__Pow__surj,axiom,
! [F: set_nat > nat,A: set_set_nat,B: set_nat] :
( ( ( image_set_nat_nat @ F @ A )
= B )
=> ( ( image_5842784325960735177et_nat @ ( image_set_nat_nat @ F ) @ ( pow_set_nat @ A ) )
= ( pow_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_569_image__Pow__surj,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat,B: set_set_nat] :
( ( ( image_5842784325960735177et_nat @ F @ A )
= B )
=> ( ( image_2225960715480453173et_nat @ ( image_5842784325960735177et_nat @ F ) @ ( pow_set_set_nat @ A ) )
= ( pow_set_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_570_image__Pow__surj,axiom,
! [F: ( nat > nat ) > set_set_nat,A: set_nat_nat,B: set_set_set_nat] :
( ( ( image_9186907679027735170et_nat @ F @ A )
= B )
=> ( ( image_1175979950292266990et_nat @ ( image_9186907679027735170et_nat @ F ) @ ( pow_nat_nat @ A ) )
= ( pow_set_set_nat @ B ) ) ) ).
% image_Pow_surj
thf(fact_571_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_572_all__finite__subset__image,axiom,
! [F: set_nat > nat,A: set_set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A ) )
=> ( P @ ( image_set_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_573_all__finite__subset__image,axiom,
! [F: nat > set_nat,A: set_nat,P: set_set_nat > $o] :
( ( ! [B2: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_nat_set_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_574_all__finite__subset__image,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ A ) )
=> ( P @ ( image_nat_nat_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_575_all__finite__subset__image,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B2 )
& ( ord_le9131159989063066194et_nat @ B2 @ A ) )
=> ( P @ ( image_1454916318497077779at_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_576_all__finite__subset__image,axiom,
! [F: nat > nat > nat,A: set_nat,P: set_nat_nat > $o] :
( ( ! [B2: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_577_all__finite__subset__image,axiom,
! [F: set_nat > set_nat,A: set_set_nat,P: set_set_nat > $o] :
( ( ! [B2: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A ) )
=> ( P @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_578_all__finite__subset__image,axiom,
! [F: nat > set_set_nat,A: set_nat,P: set_set_set_nat > $o] :
( ( ! [B2: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B2 )
& ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A ) )
=> ( P @ ( image_2194112158459175443et_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_579_all__finite__subset__image,axiom,
! [F: set_set_set_nat > nat,A: set_set_set_set_nat,P: set_nat > $o] :
( ( ! [B2: set_nat] :
( ( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_7011612946075707337at_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_set_set_nat] :
( ( ( finite5926941155766903689et_nat @ B2 )
& ( ord_le572741076514265352et_nat @ B2 @ A ) )
=> ( P @ ( image_7011612946075707337at_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_580_all__finite__subset__image,axiom,
! [F: set_nat > nat > nat,A: set_set_nat,P: set_nat_nat > $o] :
( ( ! [B2: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ ( image_8569768528772619084at_nat @ F @ A ) ) )
=> ( P @ B2 ) ) )
= ( ! [B2: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A ) )
=> ( P @ ( image_8569768528772619084at_nat @ F @ B2 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_581_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_nat_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_582_ex__finite__subset__image,axiom,
! [F: set_nat > nat,A: set_set_nat,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_set_nat_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A )
& ( P @ ( image_set_nat_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_583_ex__finite__subset__image,axiom,
! [F: nat > set_nat,A: set_nat,P: set_set_nat > $o] :
( ( ? [B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ ( image_nat_set_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_nat_set_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_584_ex__finite__subset__image,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ A )
& ( P @ ( image_nat_nat_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_585_ex__finite__subset__image,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
& ( ord_le9131159989063066194et_nat @ B2 @ A )
& ( P @ ( image_1454916318497077779at_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_586_ex__finite__subset__image,axiom,
! [F: nat > nat > nat,A: set_nat,P: set_nat_nat > $o] :
( ( ? [B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_nat_nat_nat2 @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_587_ex__finite__subset__image,axiom,
! [F: set_nat > set_nat,A: set_set_nat,P: set_set_nat > $o] :
( ( ? [B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A )
& ( P @ ( image_7916887816326733075et_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_588_ex__finite__subset__image,axiom,
! [F: nat > set_set_nat,A: set_nat,P: set_set_set_nat > $o] :
( ( ? [B2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B2 )
& ( ord_le9131159989063066194et_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A )
& ( P @ ( image_2194112158459175443et_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_589_ex__finite__subset__image,axiom,
! [F: set_set_set_nat > nat,A: set_set_set_set_nat,P: set_nat > $o] :
( ( ? [B2: set_nat] :
( ( finite_finite_nat @ B2 )
& ( ord_less_eq_set_nat @ B2 @ ( image_7011612946075707337at_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ B2 )
& ( ord_le572741076514265352et_nat @ B2 @ A )
& ( P @ ( image_7011612946075707337at_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_590_ex__finite__subset__image,axiom,
! [F: set_nat > nat > nat,A: set_set_nat,P: set_nat_nat > $o] :
( ( ? [B2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B2 )
& ( ord_le9059583361652607317at_nat @ B2 @ ( image_8569768528772619084at_nat @ F @ A ) )
& ( P @ B2 ) ) )
= ( ? [B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
& ( ord_le6893508408891458716et_nat @ B2 @ A )
& ( P @ ( image_8569768528772619084at_nat @ F @ B2 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_591_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_592_finite__subset__image,axiom,
! [B: set_nat,F: set_nat > nat,A: set_set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A ) )
=> ? [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A )
& ( finite1152437895449049373et_nat @ C3 )
& ( B
= ( image_set_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_593_finite__subset__image,axiom,
! [B: set_set_nat,F: nat > set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_set_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_594_finite__subset__image,axiom,
! [B: set_nat,F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A ) )
=> ? [C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C3 @ A )
& ( finite2115694454571419734at_nat @ C3 )
& ( B
= ( image_nat_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_595_finite__subset__image,axiom,
! [B: set_nat,F: set_set_nat > nat,A: set_set_set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A ) )
=> ? [C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ A )
& ( finite6739761609112101331et_nat @ C3 )
& ( B
= ( image_1454916318497077779at_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_596_finite__subset__image,axiom,
! [B: set_nat_nat,F: nat > nat > nat,A: set_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat_nat2 @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_597_finite__subset__image,axiom,
! [B: set_set_nat,F: set_nat > set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ? [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A )
& ( finite1152437895449049373et_nat @ C3 )
& ( B
= ( image_7916887816326733075et_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_598_finite__subset__image,axiom,
! [B: set_set_set_nat,F: nat > set_set_nat,A: set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_2194112158459175443et_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_599_finite__subset__image,axiom,
! [B: set_nat,F: set_set_set_nat > nat,A: set_set_set_set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_7011612946075707337at_nat @ F @ A ) )
=> ? [C3: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ C3 @ A )
& ( finite5926941155766903689et_nat @ C3 )
& ( B
= ( image_7011612946075707337at_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_600_finite__subset__image,axiom,
! [B: set_nat_nat,F: set_nat > nat > nat,A: set_set_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ ( image_8569768528772619084at_nat @ F @ A ) )
=> ? [C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ A )
& ( finite1152437895449049373et_nat @ C3 )
& ( B
= ( image_8569768528772619084at_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_601_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_602_finite__surj,axiom,
! [A: set_set_nat,B: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_603_finite__surj,axiom,
! [A: set_nat,B: set_set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A ) )
=> ( finite1152437895449049373et_nat @ B ) ) ) ).
% finite_surj
thf(fact_604_finite__surj,axiom,
! [A: set_nat_nat,B: set_nat,F: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_605_finite__surj,axiom,
! [A: set_set_set_nat,B: set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_606_finite__surj,axiom,
! [A: set_nat,B: set_nat_nat,F: nat > nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A ) )
=> ( finite2115694454571419734at_nat @ B ) ) ) ).
% finite_surj
thf(fact_607_finite__surj,axiom,
! [A: set_set_nat,B: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A ) )
=> ( finite1152437895449049373et_nat @ B ) ) ) ).
% finite_surj
thf(fact_608_finite__surj,axiom,
! [A: set_nat,B: set_set_set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A ) )
=> ( finite6739761609112101331et_nat @ B ) ) ) ).
% finite_surj
thf(fact_609_finite__surj,axiom,
! [A: set_set_nat,B: set_nat_nat,F: set_nat > nat > nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( ord_le9059583361652607317at_nat @ B @ ( image_8569768528772619084at_nat @ F @ A ) )
=> ( finite2115694454571419734at_nat @ B ) ) ) ).
% finite_surj
thf(fact_610_finite__surj,axiom,
! [A: set_nat_nat,B: set_set_nat,F: ( nat > nat ) > set_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7432509271690132940et_nat @ F @ A ) )
=> ( finite1152437895449049373et_nat @ B ) ) ) ).
% finite_surj
thf(fact_611_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_612_infinite__arbitrarily__large,axiom,
! [A: set_nat_nat,N: nat] :
( ~ ( finite2115694454571419734at_nat @ A )
=> ? [B6: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B6 )
& ( ( finite_card_nat_nat @ B6 )
= N )
& ( ord_le9059583361652607317at_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_613_infinite__arbitrarily__large,axiom,
! [A: set_set_nat,N: nat] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ? [B6: set_set_nat] :
( ( finite1152437895449049373et_nat @ B6 )
& ( ( finite_card_set_nat @ B6 )
= N )
& ( ord_le6893508408891458716et_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_614_infinite__arbitrarily__large,axiom,
! [A: set_set_set_set_nat,N: nat] :
( ~ ( finite5926941155766903689et_nat @ A )
=> ? [B6: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ B6 )
& ( ( finite8805468973633305546et_nat @ B6 )
= N )
& ( ord_le572741076514265352et_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_615_infinite__arbitrarily__large,axiom,
! [A: set_set_set_nat,N: nat] :
( ~ ( finite6739761609112101331et_nat @ A )
=> ? [B6: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B6 )
& ( ( finite1149291290879098388et_nat @ B6 )
= N )
& ( ord_le9131159989063066194et_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_616_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_617_card__subset__eq,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ( finite_card_nat_nat @ A )
= ( finite_card_nat_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_618_card__subset__eq,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ( finite_card_set_nat @ A )
= ( finite_card_set_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_619_card__subset__eq,axiom,
! [B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ B )
=> ( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( ( finite8805468973633305546et_nat @ A )
= ( finite8805468973633305546et_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_620_card__subset__eq,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ( finite1149291290879098388et_nat @ A )
= ( finite1149291290879098388et_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_621_first__assumptions_Ov___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.v_\<G>
thf(fact_622_Fpow__subset__Pow,axiom,
! [A: set_nat] : ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( pow_nat @ A ) ) ).
% Fpow_subset_Pow
thf(fact_623_Fpow__subset__Pow,axiom,
! [A: set_set_set_nat] : ( ord_le572741076514265352et_nat @ ( finite7717622420921165910et_nat @ A ) @ ( pow_set_set_nat @ A ) ) ).
% Fpow_subset_Pow
thf(fact_624_Fpow__subset__Pow,axiom,
! [A: set_set_nat] : ( ord_le9131159989063066194et_nat @ ( finite_Fpow_set_nat @ A ) @ ( pow_set_nat @ A ) ) ).
% Fpow_subset_Pow
thf(fact_625_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_626_first__assumptions_Ofinite__numbers,axiom,
! [L: nat,P2: nat,K: nat,N: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite_finite_nat @ ( clique3652268606331196573umbers @ N ) ) ) ).
% first_assumptions.finite_numbers
thf(fact_627_first__assumptions_Ov___092_060G_062__2,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ G ) ) ) ) ) ).
% first_assumptions.v_\<G>_2
thf(fact_628_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P2: nat,K: nat,F3: nat > nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_nat_nat @ F3 @ ( clique2971579238625216137irst_F @ K ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ ( clique5033774636164728462irst_C @ K @ F3 ) )
=> ( member_nat_nat @ F3 @ ( clique951075384711337423ACC_cf @ K @ X4 ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_629_card__image__le,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_630_card__image__le,axiom,
! [A: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_set_nat_nat @ F @ A ) ) @ ( finite_card_set_nat @ A ) ) ) ).
% card_image_le
thf(fact_631_card__image__le,axiom,
! [A: set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_632_card__image__le,axiom,
! [A: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F @ A ) ) @ ( finite_card_set_nat @ A ) ) ) ).
% card_image_le
thf(fact_633_card__image__le,axiom,
! [A: set_nat_nat,F: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat_nat @ F @ A ) ) @ ( finite_card_nat_nat @ A ) ) ) ).
% card_image_le
thf(fact_634_card__image__le,axiom,
! [A: set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_635_card__image__le,axiom,
! [A: set_nat,F: nat > nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ ( image_nat_nat_nat2 @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_636_card__image__le,axiom,
! [A: set_set_set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_1454916318497077779at_nat @ F @ A ) ) @ ( finite1149291290879098388et_nat @ A ) ) ) ).
% card_image_le
thf(fact_637_card__image__le,axiom,
! [A: set_set_nat,F: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( image_6725021117256019401et_nat @ F @ A ) ) @ ( finite_card_set_nat @ A ) ) ) ).
% card_image_le
thf(fact_638_card__image__le,axiom,
! [A: set_set_nat,F: set_nat > nat > nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ ( image_8569768528772619084at_nat @ F @ A ) ) @ ( finite_card_set_nat @ A ) ) ) ).
% card_image_le
thf(fact_639_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat,C2: nat] :
( ! [G5: set_nat] :
( ( ord_less_eq_set_nat @ G5 @ F3 )
=> ( ( finite_finite_nat @ G5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G5 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_640_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat_nat,C2: nat] :
( ! [G5: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ G5 @ F3 )
=> ( ( finite2115694454571419734at_nat @ G5 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ G5 ) @ C2 ) ) )
=> ( ( finite2115694454571419734at_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat_nat @ F3 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_641_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_set_nat,C2: nat] :
( ! [G5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G5 @ F3 )
=> ( ( finite1152437895449049373et_nat @ G5 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G5 ) @ C2 ) ) )
=> ( ( finite1152437895449049373et_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F3 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_642_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_set_set_set_nat,C2: nat] :
( ! [G5: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ G5 @ F3 )
=> ( ( finite5926941155766903689et_nat @ G5 )
=> ( ord_less_eq_nat @ ( finite8805468973633305546et_nat @ G5 ) @ C2 ) ) )
=> ( ( finite5926941155766903689et_nat @ F3 )
& ( ord_less_eq_nat @ ( finite8805468973633305546et_nat @ F3 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_643_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_set_set_nat,C2: nat] :
( ! [G5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ G5 @ F3 )
=> ( ( finite6739761609112101331et_nat @ G5 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ G5 ) @ C2 ) ) )
=> ( ( finite6739761609112101331et_nat @ F3 )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ F3 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_644_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_645_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ S2 ) )
=> ~ ! [T3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat_nat @ T3 )
= N )
=> ~ ( finite2115694454571419734at_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_646_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S2 ) )
=> ~ ! [T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T3 @ S2 )
=> ( ( ( finite_card_set_nat @ T3 )
= N )
=> ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_647_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_set_set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite8805468973633305546et_nat @ S2 ) )
=> ~ ! [T3: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ T3 @ S2 )
=> ( ( ( finite8805468973633305546et_nat @ T3 )
= N )
=> ~ ( finite5926941155766903689et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_648_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ S2 ) )
=> ~ ! [T3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T3 @ S2 )
=> ( ( ( finite1149291290879098388et_nat @ T3 )
= N )
=> ~ ( finite6739761609112101331et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_649_exists__subset__between,axiom,
! [A: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C2 )
& ( ( finite_card_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_650_exists__subset__between,axiom,
! [A: set_nat_nat,N: nat,C2: set_nat_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat_nat @ C2 ) )
=> ( ( ord_le9059583361652607317at_nat @ A @ C2 )
=> ( ( finite2115694454571419734at_nat @ C2 )
=> ? [B6: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A @ B6 )
& ( ord_le9059583361652607317at_nat @ B6 @ C2 )
& ( ( finite_card_nat_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_651_exists__subset__between,axiom,
! [A: set_set_nat,N: nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ C2 ) )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( finite1152437895449049373et_nat @ C2 )
=> ? [B6: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B6 )
& ( ord_le6893508408891458716et_nat @ B6 @ C2 )
& ( ( finite_card_set_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_652_exists__subset__between,axiom,
! [A: set_set_set_set_nat,N: nat,C2: set_set_set_set_nat] :
( ( ord_less_eq_nat @ ( finite8805468973633305546et_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite8805468973633305546et_nat @ C2 ) )
=> ( ( ord_le572741076514265352et_nat @ A @ C2 )
=> ( ( finite5926941155766903689et_nat @ C2 )
=> ? [B6: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B6 )
& ( ord_le572741076514265352et_nat @ B6 @ C2 )
& ( ( finite8805468973633305546et_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_653_exists__subset__between,axiom,
! [A: set_set_set_nat,N: nat,C2: set_set_set_nat] :
( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite1149291290879098388et_nat @ C2 ) )
=> ( ( ord_le9131159989063066194et_nat @ A @ C2 )
=> ( ( finite6739761609112101331et_nat @ C2 )
=> ? [B6: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ B6 )
& ( ord_le9131159989063066194et_nat @ B6 @ C2 )
& ( ( finite1149291290879098388et_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_654_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_655_card__seteq,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_nat @ B ) @ ( finite_card_nat_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_656_card__seteq,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_set_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_657_card__seteq,axiom,
! [B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ B )
=> ( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite8805468973633305546et_nat @ B ) @ ( finite8805468973633305546et_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_658_card__seteq,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B ) @ ( finite1149291290879098388et_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_659_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_660_card__mono,axiom,
! [B: set_nat_nat,A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A ) @ ( finite_card_nat_nat @ B ) ) ) ) ).
% card_mono
thf(fact_661_card__mono,axiom,
! [B: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ).
% card_mono
thf(fact_662_card__mono,axiom,
! [B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ B )
=> ( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite8805468973633305546et_nat @ A ) @ ( finite8805468973633305546et_nat @ B ) ) ) ) ).
% card_mono
thf(fact_663_card__mono,axiom,
! [B: set_set_set_nat,A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A ) @ ( finite1149291290879098388et_nat @ B ) ) ) ) ).
% card_mono
thf(fact_664_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique363107459185959606CLIQUE @ k2 ) ).
% POS_sub_CLIQUE
thf(fact_665_finK,axiom,
! [V: set_nat] : ( finite1152437895449049373et_nat @ ( k @ V ) ) ).
% finK
thf(fact_666_odot___092_060G_062,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ) ) ).
% odot_\<G>
thf(fact_667_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% empty_\<G>
thf(fact_668_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_669_card__le__if__inj__on__rel,axiom,
! [B: set_set_nat,A: set_nat,R: nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B7: set_nat] :
( ( member_set_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B5: set_nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_set_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_670_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_set_nat,R: set_nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B5: nat] :
( ( member_set_nat @ A1 @ A )
=> ( ( member_set_nat @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_671_card__le__if__inj__on__rel,axiom,
! [B: set_set_nat,A: set_set_nat,R: set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B )
=> ( ! [A5: set_nat] :
( ( member_set_nat @ A5 @ A )
=> ? [B7: set_nat] :
( ( member_set_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: set_nat,A22: set_nat,B5: set_nat] :
( ( member_set_nat @ A1 @ A )
=> ( ( member_set_nat @ A22 @ A )
=> ( ( member_set_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_672_card__le__if__inj__on__rel,axiom,
! [B: set_nat_nat,A: set_nat,R: nat > ( nat > nat ) > $o] :
( ( finite2115694454571419734at_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B7: nat > nat] :
( ( member_nat_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B5: nat > nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_673_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_set_set_nat,R: set_set_nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B5: nat] :
( ( member_set_set_nat @ A1 @ A )
=> ( ( member_set_set_nat @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_674_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat_nat,R: ( nat > nat ) > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A5: nat > nat] :
( ( member_nat_nat @ A5 @ A )
=> ? [B7: nat] :
( ( member_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: nat > nat,A22: nat > nat,B5: nat] :
( ( member_nat_nat @ A1 @ A )
=> ( ( member_nat_nat @ A22 @ A )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_675_card__le__if__inj__on__rel,axiom,
! [B: set_set_set_nat,A: set_nat,R: nat > set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B7: set_set_nat] :
( ( member_set_set_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B5: set_set_nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_set_set_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite1149291290879098388et_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_676_card__le__if__inj__on__rel,axiom,
! [B: set_set_set_set_nat,A: set_nat,R: nat > set_set_set_nat > $o] :
( ( finite5926941155766903689et_nat @ B )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A )
=> ? [B7: set_set_set_nat] :
( ( member2946998982187404937et_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: nat,A22: nat,B5: set_set_set_nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member2946998982187404937et_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite8805468973633305546et_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_677_card__le__if__inj__on__rel,axiom,
! [B: set_set_nat,A: set_set_set_nat,R: set_set_nat > set_nat > $o] :
( ( finite1152437895449049373et_nat @ B )
=> ( ! [A5: set_set_nat] :
( ( member_set_set_nat @ A5 @ A )
=> ? [B7: set_nat] :
( ( member_set_nat @ B7 @ B )
& ( R @ A5 @ B7 ) ) )
=> ( ! [A1: set_set_nat,A22: set_set_nat,B5: set_nat] :
( ( member_set_set_nat @ A1 @ A )
=> ( ( member_set_set_nat @ A22 @ A )
=> ( ( member_set_nat @ B5 @ B )
=> ( ( R @ A1 @ B5 )
=> ( ( R @ A22 @ B5 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A ) @ ( finite_card_set_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_678_first__assumptions_ONEG___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ K ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.NEG_\<G>
thf(fact_679_XY_I3_J,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% XY(3)
thf(fact_680_XY_I4_J,axiom,
ord_le9131159989063066194et_nat @ y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% XY(4)
thf(fact_681_card__POS,axiom,
( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ k2 ) )
= ( binomial @ ( assump1710595444109740334irst_m @ k2 ) @ k2 ) ) ).
% card_POS
thf(fact_682_H_I1_J,axiom,
member_set_set_nat @ h @ ( clique5469973757772500719t_odot @ x @ y ) ).
% H(1)
thf(fact_683_finV_I2_J,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ y ) ).
% finV(2)
thf(fact_684_finV_I1_J,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ x ) ).
% finV(1)
thf(fact_685_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k2 ) ) ).
% empty_CLIQUE
thf(fact_686_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_687_empty__Collect__eq,axiom,
! [P: set_set_nat > $o] :
( ( bot_bo7198184520161983622et_nat
= ( collect_set_set_nat @ P ) )
= ( ! [X3: set_set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_688_empty__Collect__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( bot_bot_set_nat_nat
= ( collect_nat_nat @ P ) )
= ( ! [X3: nat > nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_689_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_690_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_691_Collect__empty__eq,axiom,
! [P: set_set_nat > $o] :
( ( ( collect_set_set_nat @ P )
= bot_bo7198184520161983622et_nat )
= ( ! [X3: set_set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_692_Collect__empty__eq,axiom,
! [P: ( nat > nat ) > $o] :
( ( ( collect_nat_nat @ P )
= bot_bot_set_nat_nat )
= ( ! [X3: nat > nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_693_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_694_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_695_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_696_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_697_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_698_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_699_empty__iff,axiom,
! [C: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ C @ bot_bo193956671110832956et_nat ) ).
% empty_iff
thf(fact_700_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_701_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_702_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_703_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_704_finvXY,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ ( clique5469973757772500719t_odot @ x @ y ) ) ).
% finvXY
thf(fact_705_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_706_image__is__empty,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( ( image_nat_set_nat @ F @ A )
= bot_bot_set_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_707_image__is__empty,axiom,
! [F: set_nat > nat,A: set_set_nat] :
( ( ( image_set_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_708_image__is__empty,axiom,
! [F: set_nat > set_nat,A: set_set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ A )
= bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_709_image__is__empty,axiom,
! [F: nat > set_set_nat,A: set_nat] :
( ( ( image_2194112158459175443et_nat @ F @ A )
= bot_bo7198184520161983622et_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_710_image__is__empty,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( ( image_nat_nat_nat2 @ F @ A )
= bot_bot_set_nat_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_711_image__is__empty,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat] :
( ( ( image_1454916318497077779at_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_712_image__is__empty,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( ( image_nat_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_713_image__is__empty,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( ( image_5842784325960735177et_nat @ F @ A )
= bot_bot_set_set_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_714_image__is__empty,axiom,
! [F: ( nat > nat ) > set_nat,A: set_nat_nat] :
( ( ( image_7432509271690132940et_nat @ F @ A )
= bot_bot_set_set_nat )
= ( A = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_715_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_716_empty__is__image,axiom,
! [F: nat > set_nat,A: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_717_empty__is__image,axiom,
! [F: set_nat > nat,A: set_set_nat] :
( ( bot_bot_set_nat
= ( image_set_nat_nat @ F @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_718_empty__is__image,axiom,
! [F: set_nat > set_nat,A: set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_7916887816326733075et_nat @ F @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_719_empty__is__image,axiom,
! [F: nat > set_set_nat,A: set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( image_2194112158459175443et_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_720_empty__is__image,axiom,
! [F: nat > nat > nat,A: set_nat] :
( ( bot_bot_set_nat_nat
= ( image_nat_nat_nat2 @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_721_empty__is__image,axiom,
! [F: set_set_nat > nat,A: set_set_set_nat] :
( ( bot_bot_set_nat
= ( image_1454916318497077779at_nat @ F @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_722_empty__is__image,axiom,
! [F: ( nat > nat ) > nat,A: set_nat_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_723_empty__is__image,axiom,
! [F: set_set_nat > set_nat,A: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_5842784325960735177et_nat @ F @ A ) )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_724_empty__is__image,axiom,
! [F: ( nat > nat ) > set_nat,A: set_nat_nat] :
( ( bot_bot_set_set_nat
= ( image_7432509271690132940et_nat @ F @ A ) )
= ( A = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_725_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_726_image__empty,axiom,
! [F: set_nat > nat] :
( ( image_set_nat_nat @ F @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_727_image__empty,axiom,
! [F: nat > set_nat] :
( ( image_nat_set_nat @ F @ bot_bot_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_728_image__empty,axiom,
! [F: set_nat > set_nat] :
( ( image_7916887816326733075et_nat @ F @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% image_empty
thf(fact_729_image__empty,axiom,
! [F: set_set_nat > nat] :
( ( image_1454916318497077779at_nat @ F @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_730_image__empty,axiom,
! [F: ( nat > nat ) > nat] :
( ( image_nat_nat_nat @ F @ bot_bot_set_nat_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_731_image__empty,axiom,
! [F: nat > set_set_nat] :
( ( image_2194112158459175443et_nat @ F @ bot_bot_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_732_image__empty,axiom,
! [F: nat > nat > nat] :
( ( image_nat_nat_nat2 @ F @ bot_bot_set_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_733_image__empty,axiom,
! [F: set_nat > set_set_nat] :
( ( image_6725021117256019401et_nat @ F @ bot_bot_set_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_734_image__empty,axiom,
! [F: set_nat > nat > nat] :
( ( image_8569768528772619084at_nat @ F @ bot_bot_set_set_nat )
= bot_bot_set_nat_nat ) ).
% image_empty
thf(fact_735_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_736_empty__subsetI,axiom,
! [A: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A ) ).
% empty_subsetI
thf(fact_737_empty__subsetI,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% empty_subsetI
thf(fact_738_empty__subsetI,axiom,
! [A: set_set_set_set_nat] : ( ord_le572741076514265352et_nat @ bot_bo193956671110832956et_nat @ A ) ).
% empty_subsetI
thf(fact_739_empty__subsetI,axiom,
! [A: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A ) ).
% empty_subsetI
thf(fact_740_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_741_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_742_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_743_subset__empty,axiom,
! [A: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ bot_bo193956671110832956et_nat )
= ( A = bot_bo193956671110832956et_nat ) ) ).
% subset_empty
thf(fact_744_subset__empty,axiom,
! [A: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A @ bot_bo7198184520161983622et_nat )
= ( A = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_745_XYD,axiom,
ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ x @ y ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ).
% XYD
thf(fact_746_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_747_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_748_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_749_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_750_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_751_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_752_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_753_equals0I,axiom,
! [A: set_set_set_set_nat] :
( ! [Y3: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ Y3 @ A )
=> ( A = bot_bo193956671110832956et_nat ) ) ).
% equals0I
thf(fact_754_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y3: set_nat] :
~ ( member_set_nat @ Y3 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_755_equals0I,axiom,
! [A: set_set_set_nat] :
( ! [Y3: set_set_nat] :
~ ( member_set_set_nat @ Y3 @ A )
=> ( A = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_756_equals0I,axiom,
! [A: set_nat_nat] :
( ! [Y3: nat > nat] :
~ ( member_nat_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_757_equals0I,axiom,
! [A: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_758_equals0D,axiom,
! [A: set_set_set_set_nat,A2: set_set_set_nat] :
( ( A = bot_bo193956671110832956et_nat )
=> ~ ( member2946998982187404937et_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_759_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_760_equals0D,axiom,
! [A: set_set_set_nat,A2: set_set_nat] :
( ( A = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_761_equals0D,axiom,
! [A: set_nat_nat,A2: nat > nat] :
( ( A = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_762_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_763_emptyE,axiom,
! [A2: set_set_set_nat] :
~ ( member2946998982187404937et_nat @ A2 @ bot_bo193956671110832956et_nat ) ).
% emptyE
thf(fact_764_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_765_emptyE,axiom,
! [A2: set_set_nat] :
~ ( member_set_set_nat @ A2 @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_766_emptyE,axiom,
! [A2: nat > nat] :
~ ( member_nat_nat @ A2 @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_767_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_768_first__assumptions_Oempty__CLIQUE,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.empty_CLIQUE
thf(fact_769_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_770_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_771_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_772_bot_Oextremum__uniqueI,axiom,
! [A2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ bot_bot_nat_nat )
=> ( A2 = bot_bot_nat_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_773_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_774_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A2 @ bot_bo193956671110832956et_nat )
=> ( A2 = bot_bo193956671110832956et_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_775_bot_Oextremum__uniqueI,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_776_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_777_bot_Oextremum__unique,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_778_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_779_bot_Oextremum__unique,axiom,
! [A2: nat > nat] :
( ( ord_less_eq_nat_nat @ A2 @ bot_bot_nat_nat )
= ( A2 = bot_bot_nat_nat ) ) ).
% bot.extremum_unique
thf(fact_780_bot_Oextremum__unique,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_781_bot_Oextremum__unique,axiom,
! [A2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A2 @ bot_bo193956671110832956et_nat )
= ( A2 = bot_bo193956671110832956et_nat ) ) ).
% bot.extremum_unique
thf(fact_782_bot_Oextremum__unique,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% bot.extremum_unique
thf(fact_783_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_784_bot_Oextremum,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% bot.extremum
thf(fact_785_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_786_bot_Oextremum,axiom,
! [A2: nat > nat] : ( ord_less_eq_nat_nat @ bot_bot_nat_nat @ A2 ) ).
% bot.extremum
thf(fact_787_bot_Oextremum,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% bot.extremum
thf(fact_788_bot_Oextremum,axiom,
! [A2: set_set_set_set_nat] : ( ord_le572741076514265352et_nat @ bot_bo193956671110832956et_nat @ A2 ) ).
% bot.extremum
thf(fact_789_bot_Oextremum,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ).
% bot.extremum
thf(fact_790_infinite__imp__nonempty,axiom,
! [S2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S2 )
=> ( S2 != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_791_infinite__imp__nonempty,axiom,
! [S2: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S2 )
=> ( S2 != bot_bo7198184520161983622et_nat ) ) ).
% infinite_imp_nonempty
thf(fact_792_infinite__imp__nonempty,axiom,
! [S2: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ S2 )
=> ( S2 != bot_bot_set_nat_nat ) ) ).
% infinite_imp_nonempty
thf(fact_793_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_794_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_795_finite_OemptyI,axiom,
finite6739761609112101331et_nat @ bot_bo7198184520161983622et_nat ).
% finite.emptyI
thf(fact_796_finite_OemptyI,axiom,
finite2115694454571419734at_nat @ bot_bot_set_nat_nat ).
% finite.emptyI
thf(fact_797_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_798_Pow__bottom,axiom,
! [B: set_set_nat] : ( member_set_set_nat @ bot_bot_set_set_nat @ ( pow_set_nat @ B ) ) ).
% Pow_bottom
thf(fact_799_Pow__bottom,axiom,
! [B: set_set_set_nat] : ( member2946998982187404937et_nat @ bot_bo7198184520161983622et_nat @ ( pow_set_set_nat @ B ) ) ).
% Pow_bottom
thf(fact_800_Pow__bottom,axiom,
! [B: set_nat_nat] : ( member_set_nat_nat @ bot_bot_set_nat_nat @ ( pow_nat_nat @ B ) ) ).
% Pow_bottom
thf(fact_801_Pow__bottom,axiom,
! [B: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B ) ) ).
% Pow_bottom
thf(fact_802_Pow__not__empty,axiom,
! [A: set_nat] :
( ( pow_nat @ A )
!= bot_bot_set_set_nat ) ).
% Pow_not_empty
thf(fact_803_Pow__not__empty,axiom,
! [A: set_set_nat] :
( ( pow_set_nat @ A )
!= bot_bo7198184520161983622et_nat ) ).
% Pow_not_empty
thf(fact_804_empty__in__Fpow,axiom,
! [A: set_set_nat] : ( member_set_set_nat @ bot_bot_set_set_nat @ ( finite_Fpow_set_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_805_empty__in__Fpow,axiom,
! [A: set_set_set_nat] : ( member2946998982187404937et_nat @ bot_bo7198184520161983622et_nat @ ( finite7717622420921165910et_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_806_empty__in__Fpow,axiom,
! [A: set_nat_nat] : ( member_set_nat_nat @ bot_bot_set_nat_nat @ ( finite_Fpow_nat_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_807_empty__in__Fpow,axiom,
! [A: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_808_Fpow__not__empty,axiom,
! [A: set_nat] :
( ( finite_Fpow_nat @ A )
!= bot_bot_set_set_nat ) ).
% Fpow_not_empty
thf(fact_809_Fpow__not__empty,axiom,
! [A: set_set_nat] :
( ( finite_Fpow_set_nat @ A )
!= bot_bo7198184520161983622et_nat ) ).
% Fpow_not_empty
thf(fact_810_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_811_finite__has__minimal,axiom,
! [A: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( A != bot_bo7376149671870096959at_nat )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_812_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_813_finite__has__minimal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X: int] :
( ( member_int @ X @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_814_finite__has__minimal,axiom,
! [A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( A != bot_bot_set_nat_nat )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_815_finite__has__minimal,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_816_finite__has__minimal,axiom,
! [A: set_se7970953024979822686et_nat] :
( ( finite5436706893199572543et_nat @ A )
=> ( ( A != bot_bo7308840002416255730et_nat )
=> ? [X: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ X @ A )
& ! [Xa: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ Xa @ A )
=> ( ( ord_le572741076514265352et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_817_finite__has__minimal,axiom,
! [A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( A != bot_bo193956671110832956et_nat )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_818_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_819_finite__has__maximal,axiom,
! [A: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A )
=> ( ( A != bot_bo7376149671870096959at_nat )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A )
=> ( ( ord_le9059583361652607317at_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_820_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_821_finite__has__maximal,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ? [X: int] :
( ( member_int @ X @ A )
& ! [Xa: int] :
( ( member_int @ Xa @ A )
=> ( ( ord_less_eq_int @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_822_finite__has__maximal,axiom,
! [A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( A != bot_bot_set_nat_nat )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A )
=> ( ( ord_less_eq_nat_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_823_finite__has__maximal,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_824_finite__has__maximal,axiom,
! [A: set_se7970953024979822686et_nat] :
( ( finite5436706893199572543et_nat @ A )
=> ( ( A != bot_bo7308840002416255730et_nat )
=> ? [X: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ X @ A )
& ! [Xa: set_set_set_set_nat] :
( ( member3774042032884853055et_nat @ Xa @ A )
=> ( ( ord_le572741076514265352et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_825_finite__has__maximal,axiom,
! [A: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A )
=> ( ( A != bot_bo193956671110832956et_nat )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A )
=> ( ( ord_le9131159989063066194et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_826_first__assumptions_Ocard__POS,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ K ) )
= ( binomial @ ( assump1710595444109740334irst_m @ K ) @ K ) ) ) ).
% first_assumptions.card_POS
thf(fact_827_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M2: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_828_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_829_first__assumptions_Oempty___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.empty_\<G>
thf(fact_830_first__assumptions_OPOS__sub__CLIQUE,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_sub_CLIQUE
thf(fact_831_first__assumptions_Oodot___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.odot_\<G>
thf(fact_832_first__assumptions_ONEG__def,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737375870294875st_NEG @ K )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ K ) @ ( clique2971579238625216137irst_F @ K ) ) ) ) ).
% first_assumptions.NEG_def
thf(fact_833_G_I2_J,axiom,
member_set_set_nat @ g @ ( clique3210737319928189260st_ACC @ k2 @ ( clique5469973757772500719t_odot @ x @ y ) ) ).
% G(2)
thf(fact_834_card__v__gs__join,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z3 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ) ) ) ).
% card_v_gs_join
thf(fact_835__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_836_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique3210737375870294875st_NEG @ k2 ) ) ).
% finite_POS_NEG
thf(fact_837_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique363107459185959606CLIQUE @ k2 ) ).
% POS_CLIQUE
thf(fact_838_ACC__cf__odot,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k2 @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k2 @ X4 ) @ ( clique951075384711337423ACC_cf @ k2 @ Y ) ) ) ).
% ACC_cf_odot
thf(fact_839_first__assumptions_Ocard__v__gs__join,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat,Z3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z3 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ) ) ) ) ).
% first_assumptions.card_v_gs_join
thf(fact_840_ACC__I,axiom,
! [G: set_set_nat,X4: set_set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ k2 @ X4 ) ) ) ) ).
% ACC_I
thf(fact_841_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k2 @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_842_ACC__union,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k2 @ X4 ) @ ( clique3210737319928189260st_ACC @ k2 @ Y ) ) ) ).
% ACC_union
thf(fact_843_IntI,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ A )
=> ( ( member2946998982187404937et_nat @ C @ B )
=> ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_844_IntI,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_845_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_846_IntI,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A )
=> ( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_847_IntI,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A )
=> ( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_848_Int__iff,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( inf_in2396666505901392698et_nat @ A @ B ) )
= ( ( member2946998982187404937et_nat @ C @ A )
& ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_849_Int__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
& ( member_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_850_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_851_Int__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
& ( member_nat_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_852_Int__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
& ( member_set_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_853_UnCI,axiom,
! [C: set_set_set_nat,B: set_set_set_set_nat,A: set_set_set_set_nat] :
( ( ~ ( member2946998982187404937et_nat @ C @ B )
=> ( member2946998982187404937et_nat @ C @ A ) )
=> ( member2946998982187404937et_nat @ C @ ( sup_su3906748206781935060et_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_854_UnCI,axiom,
! [C: set_set_nat,B: set_set_set_nat,A: set_set_set_nat] :
( ( ~ ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ A ) )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_855_UnCI,axiom,
! [C: set_nat,B: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ A ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_856_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_857_UnCI,axiom,
! [C: nat > nat,B: set_nat_nat,A: set_nat_nat] :
( ( ~ ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ A ) )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_858_Un__iff,axiom,
! [C: set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( member2946998982187404937et_nat @ C @ ( sup_su3906748206781935060et_nat @ A @ B ) )
= ( ( member2946998982187404937et_nat @ C @ A )
| ( member2946998982187404937et_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_859_Un__iff,axiom,
! [C: set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A @ B ) )
= ( ( member_set_set_nat @ C @ A )
| ( member_set_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_860_Un__iff,axiom,
! [C: set_nat,A: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B ) )
= ( ( member_set_nat @ C @ A )
| ( member_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_861_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_862_Un__iff,axiom,
! [C: nat > nat,A: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A @ B ) )
= ( ( member_nat_nat @ C @ A )
| ( member_nat_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_863_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_864_Int__subset__iff,axiom,
! [C2: set_nat_nat,A: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C2 @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C2 @ A )
& ( ord_le9059583361652607317at_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_865_Int__subset__iff,axiom,
! [C2: set_set_nat,A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C2 @ ( inf_inf_set_set_nat @ A @ B ) )
= ( ( ord_le6893508408891458716et_nat @ C2 @ A )
& ( ord_le6893508408891458716et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_866_Int__subset__iff,axiom,
! [C2: set_set_set_set_nat,A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ C2 @ ( inf_in2396666505901392698et_nat @ A @ B ) )
= ( ( ord_le572741076514265352et_nat @ C2 @ A )
& ( ord_le572741076514265352et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_867_Int__subset__iff,axiom,
! [C2: set_set_set_nat,A: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C2 @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( ( ord_le9131159989063066194et_nat @ C2 @ A )
& ( ord_le9131159989063066194et_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_868_finite__Int,axiom,
! [F3: set_set_nat,G: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ F3 )
| ( finite1152437895449049373et_nat @ G ) )
=> ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_869_finite__Int,axiom,
! [F3: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F3 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_870_finite__Int,axiom,
! [F3: set_nat_nat,G: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ F3 )
| ( finite2115694454571419734at_nat @ G ) )
=> ( finite2115694454571419734at_nat @ ( inf_inf_set_nat_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_871_finite__Int,axiom,
! [F3: set_set_set_nat,G: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ F3 )
| ( finite6739761609112101331et_nat @ G ) )
=> ( finite6739761609112101331et_nat @ ( inf_in5711780100303410308et_nat @ F3 @ G ) ) ) ).
% finite_Int
thf(fact_872_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_873_Un__subset__iff,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A @ B ) @ C2 )
= ( ( ord_le9059583361652607317at_nat @ A @ C2 )
& ( ord_le9059583361652607317at_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_874_Un__subset__iff,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A @ B ) @ C2 )
= ( ( ord_le6893508408891458716et_nat @ A @ C2 )
& ( ord_le6893508408891458716et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_875_Un__subset__iff,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,C2: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ ( sup_su3906748206781935060et_nat @ A @ B ) @ C2 )
= ( ( ord_le572741076514265352et_nat @ A @ C2 )
& ( ord_le572741076514265352et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_876_Un__subset__iff,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A @ B ) @ C2 )
= ( ( ord_le9131159989063066194et_nat @ A @ C2 )
& ( ord_le9131159989063066194et_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_877_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_878_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_879_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_880_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_881_finite__Un,axiom,
! [F3: set_set_set_nat,G: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F3 @ G ) )
= ( ( finite6739761609112101331et_nat @ F3 )
& ( finite6739761609112101331et_nat @ G ) ) ) ).
% finite_Un
thf(fact_882_finite__Un,axiom,
! [F3: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F3 @ G ) )
= ( ( finite1152437895449049373et_nat @ F3 )
& ( finite1152437895449049373et_nat @ G ) ) ) ).
% finite_Un
thf(fact_883_finite__Un,axiom,
! [F3: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F3 @ G ) )
= ( ( finite_finite_nat @ F3 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_884_finite__Un,axiom,
! [F3: set_nat_nat,G: set_nat_nat] :
( ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F3 @ G ) )
= ( ( finite2115694454571419734at_nat @ F3 )
& ( finite2115694454571419734at_nat @ G ) ) ) ).
% finite_Un
thf(fact_885_Int__Un__eq_I4_J,axiom,
! [T2: set_set_set_nat,S2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ T2 @ ( inf_in5711780100303410308et_nat @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_886_Int__Un__eq_I4_J,axiom,
! [T2: set_set_nat,S2: set_set_nat] :
( ( sup_sup_set_set_nat @ T2 @ ( inf_inf_set_set_nat @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_887_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S2: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_888_Int__Un__eq_I4_J,axiom,
! [T2: set_nat_nat,S2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T2 @ ( inf_inf_set_nat_nat @ S2 @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_889_Int__Un__eq_I3_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ S2 @ ( inf_in5711780100303410308et_nat @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_890_Int__Un__eq_I3_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ S2 @ ( inf_inf_set_set_nat @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_891_Int__Un__eq_I3_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S2 @ ( inf_inf_set_nat @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_892_Int__Un__eq_I3_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S2 @ ( inf_inf_set_nat_nat @ S2 @ T2 ) )
= S2 ) ).
% Int_Un_eq(3)
thf(fact_893_Int__Un__eq_I2_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_894_Int__Un__eq_I2_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_895_Int__Un__eq_I2_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_896_Int__Un__eq_I2_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_897_Int__Un__eq_I1_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_898_Int__Un__eq_I1_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_899_Int__Un__eq_I1_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_900_Int__Un__eq_I1_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Int_Un_eq(1)
thf(fact_901_Un__Int__eq_I4_J,axiom,
! [T2: set_set_set_nat,S2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ T2 @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_902_Un__Int__eq_I4_J,axiom,
! [T2: set_set_nat,S2: set_set_nat] :
( ( inf_inf_set_set_nat @ T2 @ ( sup_sup_set_set_nat @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_903_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S2: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_904_Un__Int__eq_I4_J,axiom,
! [T2: set_nat_nat,S2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T2 @ ( sup_sup_set_nat_nat @ S2 @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_905_Un__Int__eq_I3_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ S2 @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_906_Un__Int__eq_I3_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ S2 @ ( sup_sup_set_set_nat @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_907_Un__Int__eq_I3_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S2 @ ( sup_sup_set_nat @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_908_Un__Int__eq_I3_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S2 @ ( sup_sup_set_nat_nat @ S2 @ T2 ) )
= S2 ) ).
% Un_Int_eq(3)
thf(fact_909_Un__Int__eq_I2_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_910_Un__Int__eq_I2_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_911_Un__Int__eq_I2_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_912_Un__Int__eq_I2_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S2 @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_913_Un__Int__eq_I1_J,axiom,
! [S2: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_914_Un__Int__eq_I1_J,axiom,
! [S2: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_915_Un__Int__eq_I1_J,axiom,
! [S2: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_916_Un__Int__eq_I1_J,axiom,
! [S2: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S2 @ T2 ) @ S2 )
= S2 ) ).
% Un_Int_eq(1)
thf(fact_917_psubsetI,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_918_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_919_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_920_psubsetI,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat] :
( ( ord_le572741076514265352et_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le52856854838348540et_nat @ A @ B ) ) ) ).
% psubsetI
thf(fact_921_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_922_Pow__Int__eq,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( pow_set_nat @ ( inf_inf_set_set_nat @ A @ B ) )
= ( inf_in5711780100303410308et_nat @ ( pow_set_nat @ A ) @ ( pow_set_nat @ B ) ) ) ).
% Pow_Int_eq
thf(fact_923_Pow__Int__eq,axiom,
! [A: set_nat_nat,B: set_nat_nat] :
( ( pow_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) )
= ( inf_in710756014367367485at_nat @ ( pow_nat_nat @ A ) @ ( pow_nat_nat @ B ) ) ) ).
% Pow_Int_eq
thf(fact_924_Pow__Int__eq,axiom,
! [A: set_set_set_nat,B: set_set_set_nat] :
( ( pow_set_set_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) )
= ( inf_in2396666505901392698et_nat @ ( pow_set_set_nat @ A ) @ ( pow_set_set_nat @ B ) ) ) ).
% Pow_Int_eq
thf(fact_925_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_926_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_927_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k2 @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_928_first__assumptions_OACC__union,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K @ X4 ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_union
thf(fact_929_first__assumptions_OACC__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.ACC_empty
thf(fact_930_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_931_Un__Int__assoc__eq,axiom,
! [A: set_nat_nat,B: set_nat_nat,C2: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C2 ) ) )
= ( ord_le9059583361652607317at_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_932_Un__Int__assoc__eq,axiom,
! [A: set_set_nat,B: set_set_nat,C2: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C2 ) ) )
= ( ord_le6893508408891458716et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_933_Un__Int__assoc__eq,axiom,
! [A: set_set_set_set_nat,B: set_set_set_set_nat,C2: set_set_set_set_nat] :
( ( ( sup_su3906748206781935060et_nat @ ( inf_in2396666505901392698et_nat @ A @ B ) @ C2 )
= ( inf_in2396666505901392698et_nat @ A @ ( sup_su3906748206781935060et_nat @ B @ C2 ) ) )
= ( ord_le572741076514265352et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_934_Un__Int__assoc__eq,axiom,
! [A: set_set_set_nat,B: set_set_set_nat,C2: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A @ B ) @ C2 )
= ( inf_in5711780100303410308et_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C2 ) ) )
= ( ord_le9131159989063066194et_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_935_lt__ex,axiom,
! [X2: real] :
? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).
% lt_ex
thf(fact_936_lt__ex,axiom,
! [X2: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X2 ) ).
% lt_ex
thf(fact_937_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_938_gt__ex,axiom,
! [X2: real] :
? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% gt_ex
thf(fact_939_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_940_dense,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ? [Z4: real] :
( ( ord_less_real @ X2 @ Z4 )
& ( ord_less_real @ Z4 @ Y2 ) ) ) ).
% dense
thf(fact_941_less__imp__neq,axiom,
! [X2: set_set_set_nat,Y2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_942_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_943_less__imp__neq,axiom,
! [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_944_less__imp__neq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_945_order_Oasym,axiom,
! [A2: set_set_set_nat,B3: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B3 )
=> ~ ( ord_le152980574450754630et_nat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_946_order_Oasym,axiom,
! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% order.asym
thf(fact_947_order_Oasym,axiom,
! [A2: real,B3: real] :
( ( ord_less_real @ A2 @ B3 )
=> ~ ( ord_less_real @ B3 @ A2 ) ) ).
% order.asym
thf(fact_948_order_Oasym,axiom,
! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ~ ( ord_less_int @ B3 @ A2 ) ) ).
% order.asym
thf(fact_949_ord__eq__less__trans,axiom,
! [A2: set_set_set_nat,B3: set_set_set_nat,C: set_set_set_nat] :
( ( A2 = B3 )
=> ( ( ord_le152980574450754630et_nat @ B3 @ C )
=> ( ord_le152980574450754630et_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_950_ord__eq__less__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( A2 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_951_ord__eq__less__trans,axiom,
! [A2: real,B3: real,C: real] :
( ( A2 = B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_952_ord__eq__less__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( A2 = B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_953_ord__less__eq__trans,axiom,
! [A2: set_set_set_nat,B3: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_le152980574450754630et_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_954_ord__less__eq__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_955_ord__less__eq__trans,axiom,
! [A2: real,B3: real,C: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_956_ord__less__eq__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_957_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X )
=> ( P @ Y6 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_958_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_959_antisym__conv3,axiom,
! [Y2: real,X2: real] :
( ~ ( ord_less_real @ Y2 @ X2 )
=> ( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_960_antisym__conv3,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_int @ Y2 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_961_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_962_linorder__cases,axiom,
! [X2: real,Y2: real] :
( ~ ( ord_less_real @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_real @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_963_linorder__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_964_dual__order_Oasym,axiom,
! [B3: set_set_set_nat,A2: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ B3 @ A2 )
=> ~ ( ord_le152980574450754630et_nat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_965_dual__order_Oasym,axiom,
! [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_966_dual__order_Oasym,axiom,
! [B3: real,A2: real] :
( ( ord_less_real @ B3 @ A2 )
=> ~ ( ord_less_real @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_967_dual__order_Oasym,axiom,
! [B3: int,A2: int] :
( ( ord_less_int @ B3 @ A2 )
=> ~ ( ord_less_int @ A2 @ B3 ) ) ).
% dual_order.asym
thf(fact_968_dual__order_Oirrefl,axiom,
! [A2: set_set_set_nat] :
~ ( ord_le152980574450754630et_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_969_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_970_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_971_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_972_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ~ ( P5 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_973_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_974_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B3: real] :
( ! [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_975_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B3: int] :
( ! [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B5: int] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_976_order_Ostrict__trans,axiom,
! [A2: set_set_set_nat,B3: set_set_set_nat,C: set_set_set_nat] :
( ( ord_le152980574450754630et_nat @ A2 @ B3 )
=> ( ( ord_le152980574450754630et_nat @ B3 @ C )
=> ( ord_le152980574450754630et_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_977_order_Ostrict__trans,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_978_order_Ostrict__trans,axiom,
! [A2: real,B3: real,C: real] :
( ( ord_less_real @ A2 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_979_order_Ostrict__trans,axiom,
! [A2: int,B3: int,C: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( ( ord_less_int @ B3 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_980_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_981_not__less__iff__gr__or__eq,axiom,
! [X2: real,Y2: real] :
( ( ~ ( ord_less_real @ X2 @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_982_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_983_dual__order_Ostrict__trans,axiom,
! [B3: real,A2: real,C: real] :
( ( ord_less_real @ B3 @ A2 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_984_dual__order_Ostrict__trans,axiom,
! [B3: int,A2: int,C: int] :
( ( ord_less_int @ B3 @ A2 )
=> ( ( ord_less_int @ C @ B3 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_985_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_986_first__assumptions_OACC__cf__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ) ).
% first_assumptions.ACC_cf_empty
thf(fact_987_first__assumptions_OACC__cf__odot,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique951075384711337423ACC_cf @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_odot
thf(fact_988_first__assumptions_OPOS__CLIQUE,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique363107459185959606CLIQUE @ K ) ) ) ).
% first_assumptions.POS_CLIQUE
thf(fact_989_first__assumptions_Ov__gs__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ) ).
% first_assumptions.v_gs_empty
thf(fact_990_first__assumptions_Ov__empty,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ) ).
% first_assumptions.v_empty
thf(fact_991_first__assumptions_Ofinite__POS__NEG,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737375870294875st_NEG @ K ) ) ) ) ).
% first_assumptions.finite_POS_NEG
thf(fact_992_first__assumptions_OACC__I,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X4 @ G )
=> ( member_set_set_nat @ G @ ( clique3210737319928189260st_ACC @ K @ X4 ) ) ) ) ) ).
% first_assumptions.ACC_I
thf(fact_993_v__union,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_union
thf(fact_994_v__gs__union,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ).
% v_gs_union
thf(fact_995_km,axiom,
ord_less_nat @ k2 @ ( assump1710595444109740334irst_m @ k2 ) ).
% km
thf(fact_996_ACC__odot,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k2 @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k2 @ X4 ) @ ( clique3210737319928189260st_ACC @ k2 @ Y ) ) ) ).
% ACC_odot
thf(fact_997_ACC__cf__union,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k2 @ X4 ) @ ( clique951075384711337423ACC_cf @ k2 @ Y ) ) ) ).
% ACC_cf_union
thf(fact_998_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k2 ) @ ( clique3210737375870294875st_NEG @ k2 ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_999_local_Omerge__def,axiom,
! [C2: set_nat,V: set_nat] :
( ( merge @ C2 @ V )
= ( clique6722202388162463298od_nat @ ( sup_sup_set_nat @ C2 @ V ) @ ( sup_sup_set_nat @ C2 @ V ) ) ) ).
% local.merge_def
thf(fact_1000_union___092_060G_062,axiom,
! [G: set_set_nat,H: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ) ) ).
% union_\<G>
thf(fact_1001_first__assumptions_Ov__union,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_union
thf(fact_1002_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N3 )
=> ( ord_less_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1003_bounded__nat__set__is__finite,axiom,
! [N2: set_nat,N: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N2 )
=> ( ord_less_nat @ X @ N ) )
=> ( finite_finite_nat @ N2 ) ) ).
% bounded_nat_set_is_finite
thf(fact_1004_first__assumptions_Ok,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_1005_first__assumptions_Okp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ K ) ) ).
% first_assumptions.kp
thf(fact_1006_first__assumptions_Opl,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ P2 ) ) ).
% first_assumptions.pl
thf(fact_1007_first__assumptions_Omp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.mp
thf(fact_1008_first__assumptions_Okm,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ K @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.km
thf(fact_1009_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X4 ) @ ( clique951075384711337423ACC_cf @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_1010_first__assumptions_Ov__gs__union,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) @ ( clique8462013130872731469t_v_gs @ Y ) ) ) ) ).
% first_assumptions.v_gs_union
thf(fact_1011_first__assumptions_OACC__odot,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X4 @ Y ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K @ X4 ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ).
% first_assumptions.ACC_odot
thf(fact_1012_first__assumptions_OCLIQUE__NEG,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ K ) @ ( clique3210737375870294875st_NEG @ K ) )
= bot_bo7198184520161983622et_nat ) ) ).
% first_assumptions.CLIQUE_NEG
thf(fact_1013_first__assumptions_Ounion___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.union_\<G>
thf(fact_1014__092_060open_062G_A_092_060notin_062_AACC_A_IX_A_092_060odot_062l_AY_J_A_092_060union_062_AGS_092_060close_062,axiom,
~ ( member_set_set_nat @ g @ ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) @ gs ) ) ).
% \<open>G \<notin> ACC (X \<odot>l Y) \<union> GS\<close>
thf(fact_1015_k,axiom,
ord_less_nat @ l @ k2 ).
% k
thf(fact_1016_H_I2_J,axiom,
~ ( member_set_set_nat @ h @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ).
% H(2)
thf(fact_1017_joinl__join,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k2 @ X4 @ Y ) @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ).
% joinl_join
thf(fact_1018_G_I3_J,axiom,
~ ( member_set_set_nat @ g @ ( clique3210737319928189260st_ACC @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ) ).
% G(3)
thf(fact_1019__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062H_O_A_092_060lbrakk_062H_A_092_060in_062_AX_A_092_060odot_062_AY_059_AH_A_092_060notin_062_AX_A_092_060odot_062l_AY_059_AH_A_092_060subseteq_062_AG_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [H3: set_set_nat] :
( ( member_set_set_nat @ H3 @ ( clique5469973757772500719t_odot @ x @ y ) )
=> ( ~ ( member_set_set_nat @ H3 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) )
=> ~ ( ord_le6893508408891458716et_nat @ H3 @ g ) ) ) ).
% \<open>\<And>thesis. (\<And>H. \<lbrakk>H \<in> X \<odot> Y; H \<notin> X \<odot>l Y; H \<subseteq> G\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1020__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 @ k2 @ 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_1021_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_1022_first__assumptions_Ojoinl__join,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K @ X4 @ Y ) @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ) ).
% first_assumptions.joinl_join
thf(fact_1023_sub,axiom,
ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) @ ( clique7840962075309931874st_G_l @ l @ k2 ) ).
% sub
thf(fact_1024_odotl__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k2 @ X4 @ Y )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y ) @ ( clique7840962075309931874st_G_l @ l @ k2 ) ) ) ).
% odotl_def
thf(fact_1025_finite__v__gs__Gl,axiom,
! [X4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ).
% finite_v_gs_Gl
thf(fact_1026_XY_I1_J,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique7840962075309931874st_G_l @ l @ k2 ) ).
% XY(1)
thf(fact_1027_XY_I2_J,axiom,
ord_le9131159989063066194et_nat @ y @ ( clique7840962075309931874st_G_l @ l @ k2 ) ).
% XY(2)
thf(fact_1028_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_1029_first__assumptions_Ofinite__v__gs__Gl,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ) ).
% first_assumptions.finite_v_gs_Gl
thf(fact_1030_first__assumptions_Oodotl__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique7966186356931407165_odotl @ L @ K @ X4 @ Y )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X4 @ Y ) @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ).
% first_assumptions.odotl_def
thf(fact_1031_kml,axiom,
ord_less_eq_nat @ k2 @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k2 ) @ l ) ).
% kml
thf(fact_1032_second__assumptions__axioms,axiom,
assump2881078719466019805ptions @ l @ p @ k2 ).
% second_assumptions_axioms
thf(fact_1033_pointwise__minimal__pointwise__maximal_I2_J,axiom,
! [S3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat_nat )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ S3 )
=> ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S3 )
=> ( ( ord_less_eq_nat_nat @ X @ Xa2 )
| ( ord_less_eq_nat_nat @ Xa2 @ X ) ) ) )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ S3 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S3 )
=> ( ord_less_eq_nat_nat @ Xa @ X ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(2)
thf(fact_1034_pointwise__minimal__pointwise__maximal_I1_J,axiom,
! [S3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ S3 )
=> ( ( S3 != bot_bot_set_nat_nat )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ S3 )
=> ! [Xa2: nat > nat] :
( ( member_nat_nat @ Xa2 @ S3 )
=> ( ( ord_less_eq_nat_nat @ X @ Xa2 )
| ( ord_less_eq_nat_nat @ Xa2 @ X ) ) ) )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ S3 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ S3 )
=> ( ord_less_eq_nat_nat @ X @ Xa ) ) ) ) ) ) ).
% pointwise_minimal_pointwise_maximal(1)
thf(fact_1035_kp,axiom,
ord_less_nat @ p @ k2 ).
% kp
thf(fact_1036_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_1037_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k2 ) ).
% local.mp
thf(fact_1038_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k2 ).
% first_assumptions_axioms
thf(fact_1039_Y,axiom,
member2946998982187404937et_nat @ y @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) ).
% Y
thf(fact_1040_X,axiom,
member2946998982187404937et_nat @ x @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) ).
% X
thf(fact_1041_first__assumptions_Okml,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ K @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ K ) @ L ) ) ) ).
% first_assumptions.kml
thf(fact_1042__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_1043_Lm,axiom,
ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k2 ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lm
thf(fact_1044_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1045__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 @ k2 @ x @ y ) ) ) ).
% \<open>n * (p - 1) \<le> card (v_gs (X \<odot>l Y))\<close>
thf(fact_1046_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_1047_deviate__pos__cup,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( ( clique3314026705536850673os_cup @ l @ p @ k2 @ X4 @ Y )
= bot_bo7198184520161983622et_nat ) ) ) ).
% deviate_pos_cup
thf(fact_1048__092_060open_062X_A_092_060sqinter_062_AY_A_092_060in_062_A_092_060P_062L_092_060G_062l_092_060close_062,axiom,
member2946998982187404937et_nat @ ( clique2586627118206219037_sqcap @ l @ p @ k2 @ x @ y ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) ).
% \<open>X \<sqinter> Y \<in> \<P>L\<G>l\<close>
thf(fact_1049_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_1050_first__assumptions_OL_Ocong,axiom,
assump1710595444109740301irst_L = assump1710595444109740301irst_L ).
% first_assumptions.L.cong
thf(fact_1051_second__assumptions_OLp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ ( assump1710595444109740301irst_L @ L @ P2 ) ) ) ).
% second_assumptions.Lp
thf(fact_1052_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1053_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_1054_eq__imp__le,axiom,
! [M5: nat,N: nat] :
( ( M5 = N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% eq_imp_le
thf(fact_1055_le__antisym,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( ord_less_eq_nat @ N @ M5 )
=> ( M5 = N ) ) ) ).
% le_antisym
thf(fact_1056_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_1057_nat__le__linear,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
| ( ord_less_eq_nat @ N @ M5 ) ) ).
% nat_le_linear
thf(fact_1058_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_1059_GreatestI__ex__nat,axiom,
! [P: nat > $o,B3: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_1060_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1061_second__assumptions_OLm,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ ( assump1710595444109740334irst_m @ K ) @ ( assump1710595444109740301irst_L @ L @ P2 ) ) ) ).
% second_assumptions.Lm
thf(fact_1062_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1063_less__imp__le__nat,axiom,
! [M5: nat,N: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% less_imp_le_nat
thf(fact_1064_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1065_less__or__eq__imp__le,axiom,
! [M5: nat,N: nat] :
( ( ( ord_less_nat @ M5 @ N )
| ( M5 = N ) )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1066_le__neq__implies__less,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( M5 != N )
=> ( ord_less_nat @ M5 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1067_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_1068_diff__le__mono2,axiom,
! [M5: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M5 ) ) ) ).
% diff_le_mono2
thf(fact_1069_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1070_diff__le__self,axiom,
! [M5: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ N ) @ M5 ) ).
% diff_le_self
thf(fact_1071_diff__le__mono,axiom,
! [M5: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1072_Nat_Odiff__diff__eq,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M5 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1073_le__diff__iff,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1074_eq__diff__iff,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M5 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M5 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1075_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_1076_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_1077_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_1078_le__square,axiom,
! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ).
% le_square
thf(fact_1079_le__cube,axiom,
! [M5: nat] : ( ord_less_eq_nat @ M5 @ ( times_times_nat @ M5 @ ( times_times_nat @ M5 @ M5 ) ) ) ).
% le_cube
thf(fact_1080_less__diff__iff,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M5 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M5 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1081_diff__less__mono,axiom,
! [A2: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1082_sqcup__sub,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique3210737319928189260st_ACC @ k2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k2 @ ( clique2586627118207531017_sqcup @ l @ p @ k2 @ X4 @ Y ) ) ) ) ) ).
% sqcup_sub
thf(fact_1083__092_060open_062PLU_A_IX_A_092_060odot_062l_AY_J_A_092_060in_062_A_092_060P_062L_092_060G_062l_092_060close_062,axiom,
member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) ).
% \<open>PLU (X \<odot>l Y) \<in> \<P>L\<G>l\<close>
thf(fact_1084_PLU__union,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ l @ p @ k2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) ) ) ) ).
% PLU_union
thf(fact_1085_sqcup,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ l @ p @ k2 @ X4 @ Y ) @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) ) ) ) ).
% sqcup
thf(fact_1086_sqcup__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118207531017_sqcup @ l @ p @ k2 @ X4 @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) ).
% sqcup_def
thf(fact_1087_sqcap__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique2586627118206219037_sqcap @ l @ p @ k2 @ X4 @ Y )
= ( clique2699557479641037314nd_PLU @ l @ p @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ X4 @ Y ) ) ) ).
% sqcap_def
thf(fact_1088_deviate__pos__cup__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705536850673os_cup @ l @ p @ k2 @ X4 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique3210737319928189260st_ACC @ k2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k2 @ ( clique2586627118207531017_sqcup @ l @ p @ k2 @ X4 @ Y ) ) ) ) ).
% deviate_pos_cup_def
thf(fact_1089_deviate__pos__cap__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique3314026705535538693os_cap @ l @ p @ k2 @ X4 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique3210737319928189260st_ACC @ k2 @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ k2 @ ( clique2586627118206219037_sqcap @ l @ p @ k2 @ X4 @ Y ) ) ) ) ).
% deviate_pos_cap_def
thf(fact_1090_deviate__neg__cup__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987439376245eg_cup @ l @ p @ k2 @ X4 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k2 @ ( clique2586627118207531017_sqcup @ l @ p @ k2 @ X4 @ Y ) ) @ ( clique951075384711337423ACC_cf @ k2 @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) ) ).
% deviate_neg_cup_def
thf(fact_1091_deviate__neg__cap__def,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( clique1591571987438064265eg_cap @ l @ p @ k2 @ X4 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k2 @ ( clique2586627118206219037_sqcap @ l @ p @ k2 @ X4 @ Y ) ) @ ( clique951075384711337423ACC_cf @ k2 @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ) ) ).
% deviate_neg_cap_def
thf(fact_1092_second__assumptions_Odeviate__pos__cup_Ocong,axiom,
clique3314026705536850673os_cup = clique3314026705536850673os_cup ).
% second_assumptions.deviate_pos_cup.cong
thf(fact_1093_second__assumptions_Osqcap_Ocong,axiom,
clique2586627118206219037_sqcap = clique2586627118206219037_sqcap ).
% second_assumptions.sqcap.cong
thf(fact_1094_second__assumptions_Osqcup_Ocong,axiom,
clique2586627118207531017_sqcup = clique2586627118207531017_sqcup ).
% second_assumptions.sqcup.cong
thf(fact_1095_second__assumptions_OPLU_Ocong,axiom,
clique2699557479641037314nd_PLU = clique2699557479641037314nd_PLU ).
% second_assumptions.PLU.cong
thf(fact_1096_second__assumptions_Osqcup__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X4 @ Y )
= ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) ) ).
% second_assumptions.sqcup_def
thf(fact_1097_second__assumptions_Osqcap__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X4 @ Y )
= ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( clique7966186356931407165_odotl @ L @ K @ X4 @ Y ) ) ) ) ).
% second_assumptions.sqcap_def
thf(fact_1098_second__assumptions_Osqcup,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X4 @ Y ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).
% second_assumptions.sqcup
thf(fact_1099_second__assumptions_Odeviate__pos__cup__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique3314026705536850673os_cup @ L @ P2 @ K @ X4 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X4 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_pos_cup_def
thf(fact_1100_second__assumptions_OPLU__union,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( member2946998982187404937et_nat @ ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) ) ) ) ) ).
% second_assumptions.PLU_union
thf(fact_1101_second__assumptions_Odeviate__pos__cup,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( clique3314026705536850673os_cup @ L @ P2 @ K @ X4 @ Y )
= bot_bo7198184520161983622et_nat ) ) ) ) ).
% second_assumptions.deviate_pos_cup
thf(fact_1102_second__assumptions_Osqcup__sub,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X4 @ Y ) ) ) ) ) ) ).
% second_assumptions.sqcup_sub
thf(fact_1103_plucking__step_I3_J,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X4 ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique3210737319928189260st_ACC @ k2 @ X4 ) ) @ ( clique3210737319928189260st_ACC @ k2 @ Y ) ) ) ) ) ).
% plucking_step(3)
thf(fact_1104_PLU,axiom,
( ( clique2699557479641037314nd_PLU @ l @ p @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) )
= z ) ).
% PLU
thf(fact_1105_plucking__step_I5_J,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X4 ) )
=> ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ).
% plucking_step(5)
thf(fact_1106_plucking__step_I2_J,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X4 ) )
=> ( ord_le9131159989063066194et_nat @ Y @ ( clique7840962075309931874st_G_l @ l @ k2 ) ) ) ) ) ).
% plucking_step(2)
thf(fact_1107_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_1108_second__assumptions_Odeviate__neg__cap_Ocong,axiom,
clique1591571987438064265eg_cap = clique1591571987438064265eg_cap ).
% second_assumptions.deviate_neg_cap.cong
thf(fact_1109_second__assumptions_Odeviate__neg__cup_Ocong,axiom,
clique1591571987439376245eg_cup = clique1591571987439376245eg_cup ).
% second_assumptions.deviate_neg_cup.cong
thf(fact_1110_second__assumptions_Odeviate__pos__cap_Ocong,axiom,
clique3314026705535538693os_cap = clique3314026705535538693os_cap ).
% second_assumptions.deviate_pos_cap.cong
thf(fact_1111_second__assumptions_Odeviate__neg__cap__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique1591571987438064265eg_cap @ L @ P2 @ K @ X4 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X4 @ Y ) ) @ ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_neg_cap_def
thf(fact_1112_second__assumptions_Oplucking__step_I2_J,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P2 @ X4 ) )
=> ( ord_le9131159989063066194et_nat @ Y @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ) ) ).
% second_assumptions.plucking_step(2)
thf(fact_1113_second__assumptions_Odeviate__neg__cup__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique1591571987439376245eg_cup @ L @ P2 @ K @ X4 @ Y )
= ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ ( clique2586627118207531017_sqcup @ L @ P2 @ K @ X4 @ Y ) ) @ ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X4 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_neg_cup_def
thf(fact_1114_second__assumptions_Oplucking__step_I5_J,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P2 @ X4 ) )
=> ( Y != bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% second_assumptions.plucking_step(5)
thf(fact_1115_second__assumptions_Oplucking__step_I3_J,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P2 @ X4 ) )
=> ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ X4 ) ) @ ( clique3210737319928189260st_ACC @ K @ Y ) ) ) ) ) ) ).
% second_assumptions.plucking_step(3)
thf(fact_1116_second__assumptions_Odeviate__pos__cap__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique3314026705535538693os_cap @ L @ P2 @ K @ X4 @ Y )
= ( minus_2447799839930672331et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X4 @ Y ) ) ) @ ( clique3210737319928189260st_ACC @ K @ ( clique2586627118206219037_sqcap @ L @ P2 @ K @ X4 @ Y ) ) ) ) ) ).
% second_assumptions.deviate_pos_cap_def
thf(fact_1117_PLU__main_Opinduct,axiom,
! [A0: set_set_set_nat,P: set_set_set_nat > $o] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k2 ) @ A0 )
=> ( ! [X8: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ l @ p @ k2 ) @ X8 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X8 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X8 ) ) ) )
=> ( P @ ( clique4095374090462327202g_step @ p @ X8 ) ) )
=> ( P @ X8 ) ) )
=> ( P @ A0 ) ) ) ).
% PLU_main.pinduct
thf(fact_1118_plucking__step_I1_J,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X4 ) )
=> ( 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 @ X4 ) ) @ p ) @ one_one_nat ) ) ) ) ) ).
% plucking_step(1)
thf(fact_1119_res,axiom,
( ( clique429652266423863867U_main @ l @ p @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) )
= ( produc2803780273060847707at_nat @ z @ n ) ) ).
% res
thf(fact_1120_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ).
% lm
thf(fact_1121_nat__add__left__cancel__le,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M5 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1122_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_1123_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_1124_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_1125__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,
~ ! [Z5: set_set_set_nat,N5: nat] :
( ( clique429652266423863867U_main @ l @ p @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) )
!= ( produc2803780273060847707at_nat @ Z5 @ N5 ) ) ).
% \<open>\<And>thesis. (\<And>Z n. PLU_main (X \<odot>l Y) = (Z, n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1126_PLU__main__n,axiom,
! [X4: set_set_set_nat,Z3: set_set_set_nat,N: nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
=> ( ( ( clique429652266423863867U_main @ l @ p @ k2 @ X4 )
= ( produc2803780273060847707at_nat @ Z3 @ N ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ p @ one_one_nat ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ) ) ).
% PLU_main_n
thf(fact_1127_add__leE,axiom,
! [M5: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M5 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1128_le__add1,axiom,
! [N: nat,M5: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M5 ) ) ).
% le_add1
thf(fact_1129_le__add2,axiom,
! [N: nat,M5: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M5 @ N ) ) ).
% le_add2
thf(fact_1130_add__leD1,axiom,
! [M5: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% add_leD1
thf(fact_1131_add__leD2,axiom,
! [M5: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M5 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1132_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N5: nat] :
( L
= ( plus_plus_nat @ K @ N5 ) ) ) ).
% le_Suc_ex
thf(fact_1133_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_1134_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_1135_trans__le__add1,axiom,
! [I: nat,J: nat,M5: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M5 ) ) ) ).
% trans_le_add1
thf(fact_1136_trans__le__add2,axiom,
! [I: nat,J: nat,M5: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M5 @ J ) ) ) ).
% trans_le_add2
thf(fact_1137_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1138_second__assumptions_OPLU__main_Ocong,axiom,
clique429652266423863867U_main = clique429652266423863867U_main ).
% second_assumptions.PLU_main.cong
thf(fact_1139_mono__nat__linear__lb,axiom,
! [F: nat > nat,M5: nat,K: nat] :
( ! [M3: nat,N5: nat] :
( ( ord_less_nat @ M3 @ N5 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N5 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M5 ) @ K ) @ ( F @ ( plus_plus_nat @ M5 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1140_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_1141_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_1142_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_1143_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_1144_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_1145_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_1146_first__assumptions_Olm,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( plus_plus_nat @ L @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.lm
thf(fact_1147_second__assumptions_OPLU__main__n,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Z3: set_set_set_nat,N: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ( clique429652266423863867U_main @ L @ P2 @ K @ X4 )
= ( produc2803780273060847707at_nat @ Z3 @ N ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ P2 @ one_one_nat ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) ) ) ) ) ).
% second_assumptions.PLU_main_n
thf(fact_1148_second__assumptions_OPLU__main_Opinduct,axiom,
! [L: nat,P2: nat,K: nat,A0: set_set_set_nat,P: set_set_set_nat > $o] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P2 @ K ) @ A0 )
=> ( ! [X8: set_set_set_nat] :
( ( accp_set_set_set_nat @ ( clique8954521387433384062in_rel @ L @ P2 @ K ) @ X8 )
=> ( ( ( ( ord_le9131159989063066194et_nat @ X8 @ ( clique7840962075309931874st_G_l @ L @ K ) )
& ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X8 ) ) ) )
=> ( P @ ( clique4095374090462327202g_step @ P2 @ X8 ) ) )
=> ( P @ X8 ) ) )
=> ( P @ A0 ) ) ) ) ).
% second_assumptions.PLU_main.pinduct
thf(fact_1149_second__assumptions_Oplucking__step_I1_J,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P2 @ X4 ) )
=> ( 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 @ X4 ) ) @ P2 ) @ one_one_nat ) ) ) ) ) ) ).
% second_assumptions.plucking_step(1)
thf(fact_1150_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_1151_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_1152_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_1153_choose__mult__lemma,axiom,
! [M5: nat,R: nat,K: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M5 @ R ) @ K ) @ ( plus_plus_nat @ M5 @ K ) ) @ ( binomial @ ( plus_plus_nat @ M5 @ K ) @ K ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M5 @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M5 @ R ) @ M5 ) ) ) ).
% choose_mult_lemma
thf(fact_1154_choose__mult,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M5 )
=> ( ( ord_less_eq_nat @ M5 @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M5 ) @ ( binomial @ M5 @ K ) )
= ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M5 @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_1155_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_1156_PLU__def,axiom,
! [X4: set_set_set_nat] :
( ( clique2699557479641037314nd_PLU @ l @ p @ k2 @ X4 )
= ( produc6523417423482510407at_nat @ ( clique429652266423863867U_main @ l @ p @ k2 @ X4 ) ) ) ).
% PLU_def
thf(fact_1157_second__assumptions_OPLU__def,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( clique2699557479641037314nd_PLU @ L @ P2 @ K @ X4 )
= ( produc6523417423482510407at_nat @ ( clique429652266423863867U_main @ L @ P2 @ K @ X4 ) ) ) ) ).
% second_assumptions.PLU_def
thf(fact_1158_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M5 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M5 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1159_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M5 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M5 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1160_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M5 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M5 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1161_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M5 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M5
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1162_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M5 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M5 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1163_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M5 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M5 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1164_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M5 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M5 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1165_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M5 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M5 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1166_main,axiom,
( ( member2946998982187404937et_nat @ z @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
& ( ( z = bot_bo7198184520161983622et_nat )
= ( ( clique7966186356931407165_odotl @ l @ k2 @ x @ y )
= bot_bo7198184520161983622et_nat ) )
& ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique3210737319928189260st_ACC @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ) @ ( clique3210737319928189260st_ACC @ k2 @ 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 @ k2 @ z ) @ ( clique951075384711337423ACC_cf @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ n ) ) ) ).
% main
thf(fact_1167_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1168_k2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ k2 ).
% k2
thf(fact_1169_l8,axiom,
ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ l ).
% l8
thf(fact_1170_l2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ l ).
% l2
thf(fact_1171_p,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ).
% p
thf(fact_1172_v__mem__sub,axiom,
! [E: set_nat,G: set_set_nat] :
( ( ( finite_card_nat @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( member_set_nat @ E @ G )
=> ( ord_less_eq_set_nat @ E @ ( clique5033774636164728513irst_v @ G ) ) ) ) ).
% v_mem_sub
thf(fact_1173_m2,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( assump1710595444109740334irst_m @ k2 ) ).
% m2
thf(fact_1174_m__def,axiom,
( ( assump1710595444109740334irst_m @ k2 )
= ( power_power_nat @ k2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% m_def
thf(fact_1175_kl2,axiom,
( k2
= ( power_power_nat @ l @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% kl2
thf(fact_1176__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_1177_sameprod___092_060G_062,axiom,
! [X4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ X4 ) )
=> ( member_set_set_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ) ) ).
% sameprod_\<G>
thf(fact_1178_v__card2,axiom,
! [G: set_set_nat] :
( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k2 ) ) ) )
=> ( ( G != bot_bot_set_set_nat )
=> ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ) ) ).
% v_card2
thf(fact_1179_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_1180__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_1181_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_1182__092_060open_062_Ik_A_N_A1_J_A_094_Am_A_K_An_A_060_A_Ik_A_N_A1_J_A_094_Am_A_K_A_I2_A_K_AL_092_060_094sup_0622_J_092_060close_062,axiom,
ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ n ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% \<open>(k - 1) ^ m * n < (k - 1) ^ m * (2 * L\<^sup>2)\<close>
thf(fact_1183_card,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% card
thf(fact_1184__092_060open_0622_A_K_A_I2_A_094_A_Ip_A_N_A1_J_A_K_Acard_A_I_092_060partial_062_092_060sqinter_062Neg_AX_AY_J_J_A_060_A2_A_K_A_I_Ik_A_N_A1_J_A_094_Am_A_K_AL_092_060_094sup_0622_J_092_060close_062,axiom,
ord_less_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ p @ one_one_nat ) ) @ ( finite_card_nat_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k2 @ x @ y ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% \<open>2 * (2 ^ (p - 1) * card (\<partial>\<sqinter>Neg X Y)) < 2 * ((k - 1) ^ m * L\<^sup>2)\<close>
thf(fact_1185__092_060open_0622_A_094_A_Ip_A_N_A1_J_A_K_Acard_A_I_092_060partial_062_092_060sqinter_062Neg_AX_AY_J_A_060_A_Ik_A_N_A1_J_A_094_Am_A_K_AL_092_060_094sup_0622_092_060close_062,axiom,
ord_less_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ p @ one_one_nat ) ) @ ( finite_card_nat_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k2 @ x @ y ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% \<open>2 ^ (p - 1) * card (\<partial>\<sqinter>Neg X Y) < (k - 1) ^ m * L\<^sup>2\<close>
thf(fact_1186__092_060open_0622_A_094_Ap_A_K_Acard_A_I_092_060partial_062_092_060sqinter_062Neg_AX_AY_J_A_092_060le_062_A2_A_094_Ap_A_K_Acard_A_IACC__cf_AZ_A_N_AACC__cf_A_IX_A_092_060odot_062l_AY_J_J_092_060close_062,axiom,
ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ) @ ( finite_card_nat_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k2 @ x @ y ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ p ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k2 @ z ) @ ( clique951075384711337423ACC_cf @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ) ) ) ).
% \<open>2 ^ p * card (\<partial>\<sqinter>Neg X Y) \<le> 2 ^ p * card (ACC_cf Z - ACC_cf (X \<odot>l Y))\<close>
thf(fact_1187_plucking__step_I4_J,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ p @ X4 ) )
=> ( 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 @ k2 @ Y ) @ ( clique951075384711337423ACC_cf @ k2 @ X4 ) ) ) ) @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) ) ) ) ) ).
% plucking_step(4)
thf(fact_1188__092_060open_0622_A_094_Ap_A_K_Acard_A_IACC__cf_AZ_A_N_AACC__cf_A_IX_A_092_060odot_062l_AY_J_J_A_092_060le_062_A_Ik_A_N_A1_J_A_094_Am_A_K_An_092_060close_062,axiom,
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 @ k2 @ z ) @ ( clique951075384711337423ACC_cf @ k2 @ ( clique7966186356931407165_odotl @ l @ k2 @ x @ y ) ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ n ) ).
% \<open>2 ^ p * card (ACC_cf Z - ACC_cf (X \<odot>l Y)) \<le> (k - 1) ^ m * n\<close>
thf(fact_1189_v__numbers2,axiom,
! [X2: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
=> ( ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X2 ) @ ( clique3652268606331196573umbers @ X2 ) ) )
= ( clique3652268606331196573umbers @ X2 ) ) ) ).
% v_numbers2
thf(fact_1190_v__sameprod,axiom,
! [X4: set_nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ X4 ) )
=> ( ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ X4 @ X4 ) )
= X4 ) ) ).
% v_sameprod
thf(fact_1191_card__numbers2,axiom,
! [N: nat] :
( ( finite_card_set_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N ) @ ( clique3652268606331196573umbers @ N ) ) )
= ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% card_numbers2
thf(fact_1192_PLU__main,axiom,
! [X4: set_set_set_nat,Z3: set_set_set_nat,N: nat] :
( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ l @ k2 ) )
=> ( ( ( clique429652266423863867U_main @ l @ p @ k2 @ X4 )
= ( produc2803780273060847707at_nat @ Z3 @ N ) )
=> ( ( member2946998982187404937et_nat @ Z3 @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
& ( ( Z3 = bot_bo7198184520161983622et_nat )
= ( X4 = bot_bo7198184520161983622et_nat ) )
& ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k2 ) @ ( clique3210737319928189260st_ACC @ k2 @ X4 ) ) @ ( clique3210737319928189260st_ACC @ k2 @ Z3 ) )
& ( 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 @ k2 @ Z3 ) @ ( clique951075384711337423ACC_cf @ k2 @ X4 ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ N ) ) ) ) ) ).
% PLU_main
thf(fact_1193_binomial__le__pow,axiom,
! [R: nat,N: nat] :
( ( ord_less_eq_nat @ R @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).
% binomial_le_pow
thf(fact_1194_second__assumptions_Okl2,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( K
= ( power_power_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% second_assumptions.kl2
thf(fact_1195_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1196_first__assumptions_Om__def,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( assump1710595444109740334irst_m @ K )
= ( power_power_nat @ K @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% first_assumptions.m_def
thf(fact_1197_first__assumptions_Om2,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( assump1710595444109740334irst_m @ K ) ) ) ).
% first_assumptions.m2
thf(fact_1198_first__assumptions_Op,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P2 ) ) ).
% first_assumptions.p
thf(fact_1199_first__assumptions_Ok2,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) ) ).
% first_assumptions.k2
thf(fact_1200_first__assumptions_Ol2,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L ) ) ).
% first_assumptions.l2
thf(fact_1201_first__assumptions_Ointro,axiom,
! [L: nat,P2: nat,K: nat] :
( ( ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ L )
=> ( ( ord_less_nat @ L @ P2 )
=> ( ( ord_less_nat @ P2 @ K )
=> ( assump5453534214990993103ptions @ L @ P2 @ K ) ) ) ) ).
% first_assumptions.intro
thf(fact_1202_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_1203_first__assumptions_Ov__sameprod,axiom,
! [L: nat,P2: nat,K: nat,X4: set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ X4 ) )
=> ( ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ X4 @ X4 ) )
= X4 ) ) ) ).
% first_assumptions.v_sameprod
thf(fact_1204_second__assumptions__axioms_Ointro,axiom,
! [K: nat,L: nat] :
( ( K
= ( power_power_nat @ L @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L )
=> ( assump8934899134041091456axioms @ L @ K ) ) ) ).
% second_assumptions_axioms.intro
thf(fact_1205_second__assumptions__axioms__def,axiom,
( assump8934899134041091456axioms
= ( ^ [L2: nat,K2: nat] :
( ( K2
= ( power_power_nat @ L2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
& ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L2 ) ) ) ) ).
% second_assumptions_axioms_def
thf(fact_1206_second__assumptions_Ol8,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) @ L ) ) ).
% second_assumptions.l8
thf(fact_1207_first__assumptions_Ov__mem__sub,axiom,
! [L: nat,P2: nat,K: nat,E: set_nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ( finite_card_nat @ E )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( member_set_nat @ E @ G )
=> ( ord_less_eq_set_nat @ E @ ( clique5033774636164728513irst_v @ G ) ) ) ) ) ).
% first_assumptions.v_mem_sub
thf(fact_1208_first__assumptions_Ov__numbers2,axiom,
! [L: nat,P2: nat,K: nat,X2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 )
=> ( ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ X2 ) @ ( clique3652268606331196573umbers @ X2 ) ) )
= ( clique3652268606331196573umbers @ X2 ) ) ) ) ).
% first_assumptions.v_numbers2
thf(fact_1209_binomial__le__pow2,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_1210_first__assumptions_Osameprod___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,X4: set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_less_eq_set_nat @ X4 @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ X4 ) )
=> ( member_set_set_nat @ ( clique6722202388162463298od_nat @ X4 @ X4 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.sameprod_\<G>
thf(fact_1211_first__assumptions_Ov__card2,axiom,
! [L: nat,P2: nat,K: nat,G: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( G != bot_bot_set_set_nat )
=> ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( finite_card_nat @ ( clique5033774636164728513irst_v @ G ) ) ) ) ) ) ).
% first_assumptions.v_card2
thf(fact_1212_second__assumptions_Oplucking__step_I4_J,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ord_less_nat @ ( assump1710595444109740301irst_L @ L @ P2 ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X4 ) ) )
=> ( ( Y
= ( clique4095374090462327202g_step @ P2 @ X4 ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P2 ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ Y ) @ ( clique951075384711337423ACC_cf @ K @ X4 ) ) ) ) @ ( power_power_nat @ ( minus_minus_nat @ K @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% second_assumptions.plucking_step(4)
thf(fact_1213_second__assumptions_OPLU__main,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Z3: set_set_set_nat,N: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X4 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( ( ( clique429652266423863867U_main @ L @ P2 @ K @ X4 )
= ( produc2803780273060847707at_nat @ Z3 @ N ) )
=> ( ( member2946998982187404937et_nat @ Z3 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
& ( ( Z3 = bot_bo7198184520161983622et_nat )
= ( X4 = bot_bo7198184520161983622et_nat ) )
& ( ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ K ) @ ( clique3210737319928189260st_ACC @ K @ X4 ) ) @ ( clique3210737319928189260st_ACC @ K @ Z3 ) )
& ( ord_less_eq_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P2 ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ K @ Z3 ) @ ( clique951075384711337423ACC_cf @ K @ X4 ) ) ) ) @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ K @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) @ N ) ) ) ) ) ) ).
% second_assumptions.PLU_main
thf(fact_1214__092_060open_062real_A_I2_A_094_A_Ip_A_N_A1_J_A_K_Acard_A_I_092_060partial_062_092_060sqinter_062Neg_AX_AY_J_J_A_060_Areal_A_I_Ik_A_N_A1_J_A_094_Am_A_K_AL_092_060_094sup_0622_J_092_060close_062,axiom,
ord_less_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ p @ one_one_nat ) ) @ ( finite_card_nat_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k2 @ x @ y ) ) ) ) @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% \<open>real (2 ^ (p - 1) * card (\<partial>\<sqinter>Neg X Y)) < real ((k - 1) ^ m * L\<^sup>2)\<close>
thf(fact_1215_power2__nat__le__imp__le,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M5 @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_1216_power2__nat__le__eq__le,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_1217_self__le__ge2__pow,axiom,
! [K: nat,M5: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M5 @ ( power_power_nat @ K @ M5 ) ) ) ).
% self_le_ge2_pow
thf(fact_1218_diff__le__diff__pow,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M5 @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M5 ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1219_ex__power__ivl1,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
=> ( ( ord_less_eq_nat @ one_one_nat @ K )
=> ? [N5: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N5 ) @ K )
& ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1220_ex__power__ivl2,axiom,
! [B3: nat,K: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ? [N5: nat] :
( ( ord_less_nat @ ( power_power_nat @ B3 @ N5 ) @ K )
& ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1221_int__ops_I5_J,axiom,
! [A2: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(5)
thf(fact_1222_int__plus,axiom,
! [N: nat,M5: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M5 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M5 ) ) ) ).
% int_plus
thf(fact_1223_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_1224_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1225_int__ops_I7_J,axiom,
! [A2: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B3 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(7)
thf(fact_1226_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1227_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1228__092_060open_062real_A_Icard_A_I_092_060partial_062_092_060sqinter_062Neg_AX_AY_J_J_A_060_Areal_A_I_Ik_A_N_A1_J_A_094_Am_A_K_AL_092_060_094sup_0622_J_A_P_A2_A_094_A_Ip_A_N_A1_J_092_060close_062,axiom,
ord_less_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat_nat @ ( clique1591571987438064265eg_cap @ l @ p @ k2 @ x @ y ) ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ ( power_power_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_nat @ p @ one_one_nat ) ) ) ).
% \<open>real (card (\<partial>\<sqinter>Neg X Y)) < real ((k - 1) ^ m * L\<^sup>2) / 2 ^ (p - 1)\<close>
thf(fact_1229_deviate__neg__cup,axiom,
! [X4: set_set_set_nat,Y: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ l @ p @ k2 ) )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat_nat @ ( clique1591571987439376245eg_cup @ l @ p @ k2 @ X4 @ Y ) ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) @ ( assump1710595444109740301irst_L @ l @ p ) ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_nat @ p @ one_one_nat ) ) ) ) ) ) ).
% deviate_neg_cup
thf(fact_1230_int__if,axiom,
! [P: $o,A2: nat,B3: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B3 ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B3 ) )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% int_if
thf(fact_1231_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_1232_second__assumptions_Odeviate__neg__cup,axiom,
! [L: nat,P2: nat,K: nat,X4: set_set_set_nat,Y: set_set_set_nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ( member2946998982187404937et_nat @ X4 @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ( member2946998982187404937et_nat @ Y @ ( clique2294137941332549862_L_G_l @ L @ P2 @ K ) )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat_nat @ ( clique1591571987439376245eg_cup @ L @ P2 @ K @ X4 @ Y ) ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( times_times_nat @ ( power_power_nat @ ( minus_minus_nat @ K @ one_one_nat ) @ ( assump1710595444109740334irst_m @ K ) ) @ ( assump1710595444109740301irst_L @ L @ P2 ) ) ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( minus_minus_nat @ P2 @ one_one_nat ) ) ) ) ) ) ) ).
% second_assumptions.deviate_neg_cup
thf(fact_1233_numeral__le__real__of__nat__iff,axiom,
! [N: num,M5: nat] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M5 ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M5 ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_1234_binomial__strict__mono,axiom,
! [K: nat,K3: nat,N: nat] :
( ( ord_less_nat @ K @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) @ N )
=> ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K3 ) ) ) ) ).
% binomial_strict_mono
thf(fact_1235_int__ops_I8_J,axiom,
! [A2: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A2 @ B3 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(8)
thf(fact_1236_binomial__antimono,axiom,
! [K: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ K3 )
=> ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K3 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_1237_binomial__maximum,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% binomial_maximum
thf(fact_1238_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% choose_two
thf(fact_1239_nat__le__real__less,axiom,
( ord_less_eq_nat
= ( ^ [N4: nat,M4: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M4 ) @ one_one_real ) ) ) ) ).
% nat_le_real_less
thf(fact_1240_binomial__maximum_H,axiom,
! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_1241_binomial__mono,axiom,
! [K: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K3 ) ) ) ) ).
% binomial_mono
thf(fact_1242_binomial__strict__antimono,axiom,
! [K: nat,K3: nat,N: nat] :
( ( ord_less_nat @ K @ K3 )
=> ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ord_less_nat @ ( binomial @ N @ K3 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_1243__C_K_C,axiom,
ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( minus_minus_nat @ k2 @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k2 ) ) ).
% "*"
thf(fact_1244_L3,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% L3
thf(fact_1245_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1246_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1247_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1248_diff__is__0__eq_H,axiom,
! [M5: nat,N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( ( minus_minus_nat @ M5 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1249_diff__is__0__eq,axiom,
! [M5: nat,N: nat] :
( ( ( minus_minus_nat @ M5 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M5 @ N ) ) ).
% diff_is_0_eq
thf(fact_1250_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_1251_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_1252_nat__mult__le__cancel__disj,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M5 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1253_mult__le__cancel2,axiom,
! [M5: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M5 @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M5 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1254_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_1255_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ~ ( P @ I3 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1256_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1257_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_1258_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_1259_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1260_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1261_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1262_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1263_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_1264_nat__mult__le__cancel1,axiom,
! [K: nat,M5: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M5 ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M5 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1265_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_1266_kuhn__lemma,axiom,
! [P2: nat,N: nat,Label: ( nat > nat ) > nat > nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ! [X: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_nat @ ( X @ I3 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ( Label @ X @ I2 )
= zero_zero_nat )
| ( ( Label @ X @ I2 )
= one_one_nat ) ) ) )
=> ( ! [X: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_nat @ ( X @ I3 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ( X @ I2 )
= zero_zero_nat )
=> ( ( Label @ X @ I2 )
= zero_zero_nat ) ) ) )
=> ( ! [X: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_nat @ ( X @ I3 ) @ P2 ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ( X @ I2 )
= P2 )
=> ( ( Label @ X @ I2 )
= one_one_nat ) ) ) )
=> ~ ! [Q2: nat > nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_nat @ ( Q2 @ I3 ) @ P2 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [R2: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J3 ) @ ( R2 @ J3 ) )
& ( ord_less_eq_nat @ ( R2 @ J3 ) @ ( plus_plus_nat @ ( Q2 @ J3 ) @ one_one_nat ) ) ) )
& ? [S4: nat > nat] :
( ! [J3: nat] :
( ( ord_less_nat @ J3 @ N )
=> ( ( ord_less_eq_nat @ ( Q2 @ J3 ) @ ( S4 @ J3 ) )
& ( ord_less_eq_nat @ ( S4 @ J3 ) @ ( plus_plus_nat @ ( Q2 @ J3 ) @ one_one_nat ) ) ) )
& ( ( Label @ R2 @ I3 )
!= ( Label @ S4 @ I3 ) ) ) ) ) ) ) ) ) ) ).
% kuhn_lemma
thf(fact_1267_second__assumptions_OL3,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump2881078719466019805ptions @ L @ P2 @ K )
=> ( ord_less_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( assump1710595444109740301irst_L @ L @ P2 ) ) ) ).
% second_assumptions.L3
thf(fact_1268_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_1269_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_1270_central__binomial__lower__bound,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_1271_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1272_int__ops_I6_J,axiom,
! [A2: nat,B3: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1273_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X: nat > real] :
( ( P @ X )
=> ( P @ ( F @ X ) ) )
=> ( ! [X: nat > real] :
( ( P @ X )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) )
& ( ord_less_eq_real @ ( X @ I2 ) @ one_one_real ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X6: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L3 @ X6 @ I3 ) @ one_one_nat )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q @ I3 )
& ( ( X6 @ I3 )
= zero_zero_real ) )
=> ( ( L3 @ X6 @ I3 )
= zero_zero_nat ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q @ I3 )
& ( ( X6 @ I3 )
= one_one_real ) )
=> ( ( L3 @ X6 @ I3 )
= one_one_nat ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q @ I3 )
& ( ( L3 @ X6 @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X6 @ I3 ) @ ( F @ X6 @ I3 ) ) )
& ! [X6: nat > real,I3: nat] :
( ( ( P @ X6 )
& ( Q @ I3 )
& ( ( L3 @ X6 @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X6 @ I3 ) @ ( X6 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ h ) @ ( clique5033774636164728513irst_v @ g ) ).
%------------------------------------------------------------------------------