TPTP Problem File: SLH0631^1.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_00751_025431__16231556_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1405 ( 607 unt; 133 typ; 0 def)
% Number of atoms : 3277 (1073 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 10960 ( 361 ~; 41 |; 304 &;8866 @)
% ( 0 <=>;1388 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 777 ( 777 >; 0 *; 0 +; 0 <<)
% Number of symbols : 127 ( 126 usr; 22 con; 0-4 aty)
% Number of variables : 3231 ( 60 ^;3039 !; 132 ?;3231 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:49:02.753
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
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__Nat__Onat,type,
nat: $tType ).
% Explicit typings (126)
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions,type,
assump5453534214990993103ptions: nat > nat > nat > $o ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_OL,type,
assump1710595444109740301irst_L: nat > nat > nat ).
thf(sy_c_Assumptions__and__Approximations_Ofirst__assumptions_Om,type,
assump1710595444109740334irst_m: nat > nat ).
thf(sy_c_Assumptions__and__Approximations_Osecond__assumptions,type,
assump2881078719466019805ptions: nat > nat > nat > $o ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_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_Ofirst__assumptions_OACC,type,
clique3210737319928189260st_ACC: nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OACC__cf,type,
clique951075384711337423ACC_cf: nat > set_set_set_nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OC,type,
clique5033774636164728462irst_C: nat > ( nat > nat ) > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_OCLIQUE,type,
clique363107459185959606CLIQUE: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_ONEG,type,
clique3210737375870294875st_NEG: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060F_062,type,
clique2971579238625216137irst_F: nat > set_nat_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060G_062l,type,
clique7840962075309931874st_G_l: nat > nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060K_062,type,
clique3326749438856946062irst_K: nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_O_092_060P_062L_092_060G_062l,type,
clique2294137941332549862_L_G_l: nat > nat > nat > set_set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oaccepts,type,
clique3686358387679108662ccepts: set_set_set_nat > set_set_nat > $o ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodot,type,
clique5469973757772500719t_odot: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oodotl,type,
clique7966186356931407165_odotl: nat > nat > set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Oplucking__step,type,
clique4095374090462327202g_step: nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov,type,
clique5033774636164728513irst_v: set_set_nat > set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Ofirst__assumptions_Ov__gs,type,
clique8462013130872731469t_v_gs: set_set_set_nat > set_set_nat ).
thf(sy_c_Clique__Large__Monotone__Circuits_Onumbers,type,
clique3652268606331196573umbers: nat > set_nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(sy_c_Finite__Set_Ocard_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite_card_nat_nat: set_nat_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1149291290879098388et_nat: set_set_set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite3586981331298542604at_nat: set_set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite6739761609112101331et_nat: set_set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
finite5926941155766903689et_nat: set_set_set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_8121590178497047118at_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_inf_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
inf_in5711780100303410308et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_sup_set_nat_nat: set_nat_nat > set_nat_nat > set_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
sup_su4213647025997063966et_nat: set_set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bot_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo7376149671870096959at_nat: set_set_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
bot_bo193956671110832956et_nat: set_set_set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le152980574450754630et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_less_eq_nat_nat: ( nat > nat ) > ( nat > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le9059583361652607317at_nat: set_nat_nat > set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le4954213926817602059at_nat: set_set_nat_nat > set_set_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_J,type,
ord_le572741076514265352et_nat: set_set_set_set_nat > set_set_set_set_nat > $o ).
thf(sy_c_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_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__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_3832368097948589297at_nat: ( set_nat_nat > set_nat_nat ) > set_set_nat_nat > set_set_nat_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__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
image_7884819252390400639et_nat: ( set_set_nat > set_set_nat ) > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
image_6473237745780476395et_nat: ( set_set_set_nat > set_set_set_nat ) > set_set_set_set_nat > set_set_set_set_nat ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert_nat_nat: ( nat > nat ) > set_nat_nat > set_nat_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
insert_set_set_nat: set_set_nat > set_set_set_nat > set_set_set_nat ).
thf(sy_c_Sunflower_Osunflower_001t__Nat__Onat,type,
sunflower_nat: set_set_nat > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_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_v_G____,type,
g: nat > set_set_nat ).
thf(sy_v_Gs____,type,
gs: set_set_nat ).
thf(sy_v_S____,type,
s: set_set_nat ).
thf(sy_v_Si____,type,
si: nat > set_nat ).
thf(sy_v_U____,type,
u: set_set_set_nat ).
thf(sy_v_Us____,type,
us: set_nat ).
thf(sy_v_Vs____,type,
vs: set_nat ).
thf(sy_v_Ws____,type,
ws: 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_f____,type,
f: nat > nat ).
thf(sy_v_fstt____,type,
fstt: set_nat > nat ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_l,type,
l: nat ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_pair____,type,
pair: nat > set_nat ).
thf(sy_v_r____,type,
r: nat ).
thf(sy_v_s____,type,
s2: nat ).
thf(sy_v_si____,type,
si2: nat > nat ).
thf(sy_v_sndd____,type,
sndd: set_nat > nat ).
thf(sy_v_ti____,type,
ti: nat > nat ).
thf(sy_v_u____,type,
u2: nat > nat ).
thf(sy_v_w____,type,
w: nat > nat ).
% Relevant facts (1271)
thf(fact_0__092_060open_062w_A_092_060equiv_062_A_092_060lambda_062i_O_Asndd_A_Ipair_Ai_J_092_060close_062,axiom,
( w
= ( ^ [I: nat] : ( sndd @ ( pair @ I ) ) ) ) ).
% \<open>w \<equiv> \<lambda>i. sndd (pair i)\<close>
thf(fact_1_uwi_I4_J,axiom,
( ( f @ ( u2 @ i ) )
= ( f @ ( w @ i ) ) ) ).
% uwi(4)
thf(fact_2_uwi_I2_J,axiom,
member_nat @ ( w @ i ) @ ( clique5033774636164728513irst_v @ ( g @ i ) ) ).
% uwi(2)
thf(fact_3_i,axiom,
ord_less_nat @ i @ p ).
% i
thf(fact_4_w__def,axiom,
! [I2: nat] :
( ( w @ I2 )
= ( sndd @ ( pair @ I2 ) ) ) ).
% w_def
thf(fact_5_uwi_I6_J,axiom,
( ( pair @ i )
= ( insert_nat @ ( u2 @ i ) @ ( insert_nat @ ( w @ i ) @ bot_bot_set_nat ) ) ) ).
% uwi(6)
thf(fact_6__C_K_C_I2_J,axiom,
member_nat @ ( w @ i ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ).
% "*"(2)
thf(fact_7__C_K_C_I3_J,axiom,
member_set_nat @ ( insert_nat @ ( u2 @ i ) @ ( insert_nat @ ( w @ i ) @ bot_bot_set_nat ) ) @ ( g @ i ) ).
% "*"(3)
thf(fact_8_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_9_first__assumptions_Oplucking__step_Ocong,axiom,
clique4095374090462327202g_step = clique4095374090462327202g_step ).
% first_assumptions.plucking_step.cong
thf(fact_10_uwi_I5_J,axiom,
member_nat @ ( f @ ( w @ i ) ) @ ( clique3652268606331196573umbers @ ( minus_minus_nat @ k @ one_one_nat ) ) ).
% uwi(5)
thf(fact_11_Ws__def,axiom,
( ws
= ( minus_minus_set_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ us ) ) ).
% Ws_def
thf(fact_12_km,axiom,
ord_less_nat @ k @ ( assump1710595444109740334irst_m @ k ) ).
% km
thf(fact_13_kp,axiom,
ord_less_nat @ p @ k ).
% kp
thf(fact_14_local_Omp,axiom,
ord_less_nat @ p @ ( assump1710595444109740334irst_m @ k ) ).
% local.mp
thf(fact_15_uw_I2_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( member_nat @ ( w @ I2 ) @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) ) ) ).
% uw(2)
thf(fact_16__C_K_C_I1_J,axiom,
member_nat @ ( u2 @ i ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ).
% "*"(1)
thf(fact_17_uw_I4_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( ( f @ ( u2 @ I2 ) )
= ( f @ ( w @ I2 ) ) ) ) ).
% uw(4)
thf(fact_18_uw_I6_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( ( pair @ I2 )
= ( insert_nat @ ( u2 @ I2 ) @ ( insert_nat @ ( w @ I2 ) @ bot_bot_set_nat ) ) ) ) ).
% uw(6)
thf(fact_19_uw_I5_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( member_nat @ ( f @ ( w @ I2 ) ) @ ( clique3652268606331196573umbers @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ).
% uw(5)
thf(fact_20__092_060open_062u_A_092_060equiv_062_A_092_060lambda_062i_O_Afstt_A_Ipair_Ai_J_092_060close_062,axiom,
( u2
= ( ^ [I: nat] : ( fstt @ ( pair @ I ) ) ) ) ).
% \<open>u \<equiv> \<lambda>i. fstt (pair i)\<close>
thf(fact_21_G_I2_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( ( clique5033774636164728513irst_v @ ( g @ I2 ) )
= ( si @ I2 ) ) ) ).
% G(2)
thf(fact_22_f,axiom,
member_nat_nat @ f @ ( clique2971579238625216137irst_F @ k ) ).
% f
thf(fact_23_ex,axiom,
? [X: nat] :
( ( member_nat @ X @ ( clique3652268606331196573umbers @ ( minus_minus_nat @ k @ one_one_nat ) ) )
& ( ( image_nat_nat @ f @ ( insert_nat @ ( u2 @ i ) @ ( insert_nat @ ( w @ i ) @ bot_bot_set_nat ) ) )
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% ex
thf(fact_24_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_25_insert__Diff__single,axiom,
! [A: nat > nat,A2: set_nat_nat] :
( ( insert_nat_nat @ A @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) ) )
= ( insert_nat_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_26_insert__Diff__single,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ A @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) )
= ( insert_set_set_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_27_insert__Diff__single,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= ( insert_set_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_28_i__props_I7_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( member_set_set_nat @ ( g @ I2 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ).
% i_props(7)
thf(fact_29_uw_I1_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( member_nat @ ( u2 @ I2 ) @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) @ vs ) ) ) ).
% uw(1)
thf(fact_30_uwi_I1_J,axiom,
member_nat @ ( u2 @ i ) @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ i ) ) @ vs ) ).
% uwi(1)
thf(fact_31_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_32_Diff__insert0,axiom,
! [X2: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ X2 @ A2 )
=> ( ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ B ) )
= ( minus_8121590178497047118at_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_33_Diff__insert0,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ B ) )
= ( minus_2447799839930672331et_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_34_Diff__insert0,axiom,
! [X2: set_nat,A2: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ B ) )
= ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_35_insert__Diff1,axiom,
! [X2: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_36_insert__Diff1,axiom,
! [X2: nat > nat,B: set_nat_nat,A2: set_nat_nat] :
( ( member_nat_nat @ X2 @ B )
=> ( ( minus_8121590178497047118at_nat @ ( insert_nat_nat @ X2 @ A2 ) @ B )
= ( minus_8121590178497047118at_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_37_insert__Diff1,axiom,
! [X2: set_set_nat,B: set_set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ B )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ B )
= ( minus_2447799839930672331et_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_38_insert__Diff1,axiom,
! [X2: set_nat,B: set_set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B )
= ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_39_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_40_Diff__empty,axiom,
! [A2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A2 @ bot_bot_set_nat_nat )
= A2 ) ).
% Diff_empty
thf(fact_41_Diff__empty,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= A2 ) ).
% Diff_empty
thf(fact_42_Diff__empty,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ bot_bot_set_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_43_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_44_empty__Diff,axiom,
! [A2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ bot_bot_set_nat_nat @ A2 )
= bot_bot_set_nat_nat ) ).
% empty_Diff
thf(fact_45_empty__Diff,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ bot_bo7198184520161983622et_nat @ A2 )
= bot_bo7198184520161983622et_nat ) ).
% empty_Diff
thf(fact_46_empty__Diff,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ bot_bot_set_set_nat @ A2 )
= bot_bot_set_set_nat ) ).
% empty_Diff
thf(fact_47_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_48_Diff__cancel,axiom,
! [A2: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A2 @ A2 )
= bot_bot_set_nat_nat ) ).
% Diff_cancel
thf(fact_49_Diff__cancel,axiom,
! [A2: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ A2 )
= bot_bo7198184520161983622et_nat ) ).
% Diff_cancel
thf(fact_50_Diff__cancel,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ A2 )
= bot_bot_set_set_nat ) ).
% Diff_cancel
thf(fact_51_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_52_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_53_singletonI,axiom,
! [A: set_set_nat] : ( member_set_set_nat @ A @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ).
% singletonI
thf(fact_54_singletonI,axiom,
! [A: nat > nat] : ( member_nat_nat @ A @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) ) ).
% singletonI
thf(fact_55_image__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_56_image__eqI,axiom,
! [B2: set_nat,F: nat > set_nat,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_57_image__eqI,axiom,
! [B2: nat,F: set_nat > nat,X2: set_nat,A2: set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_58_image__eqI,axiom,
! [B2: nat > nat,F: nat > nat > nat,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_59_image__eqI,axiom,
! [B2: set_set_nat,F: nat > set_set_nat,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_set_set_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_60_image__eqI,axiom,
! [B2: set_nat,F: set_nat > set_nat,X2: set_nat,A2: set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_61_image__eqI,axiom,
! [B2: nat,F: ( nat > nat ) > nat,X2: nat > nat,A2: set_nat_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat @ B2 @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_62_image__eqI,axiom,
! [B2: nat,F: set_set_nat > nat,X2: set_set_nat,A2: set_set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( member_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_63_image__eqI,axiom,
! [B2: nat > nat,F: set_nat > nat > nat,X2: set_nat,A2: set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_nat_nat @ B2 @ ( image_8569768528772619084at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_64_image__eqI,axiom,
! [B2: set_set_nat,F: set_nat > set_set_nat,X2: set_nat,A2: set_set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ B2 @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_65_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_66_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_67_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_68_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_69_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_70_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_71_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_72_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_73_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_74_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_75_all__not__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ! [X3: set_set_nat] :
~ ( member_set_set_nat @ X3 @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_76_all__not__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ! [X3: nat > nat] :
~ ( member_nat_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% all_not_in_conv
thf(fact_77_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_78_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_79_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_80_empty__iff,axiom,
! [C: nat > nat] :
~ ( member_nat_nat @ C @ bot_bot_set_nat_nat ) ).
% empty_iff
thf(fact_81_insert__absorb2,axiom,
! [X2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A2 ) )
= ( insert_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_82_insert__absorb2,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ X2 @ ( insert_set_set_nat @ X2 @ A2 ) )
= ( insert_set_set_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_83_insert__absorb2,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ X2 @ ( insert_set_nat @ X2 @ A2 ) )
= ( insert_set_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_84_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_85_insert__iff,axiom,
! [A: set_nat,B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_86_insert__iff,axiom,
! [A: nat > nat,B2: nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ A @ ( insert_nat_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_87_insert__iff,axiom,
! [A: set_set_nat,B2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ ( insert_set_set_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_88_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_89_insertCI,axiom,
! [A: set_nat,B: set_set_nat,B2: set_nat] :
( ( ~ ( member_set_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_set_nat @ A @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_90_insertCI,axiom,
! [A: nat > nat,B: set_nat_nat,B2: nat > nat] :
( ( ~ ( member_nat_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat_nat @ A @ ( insert_nat_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_91_insertCI,axiom,
! [A: set_set_nat,B: set_set_set_nat,B2: set_set_nat] :
( ( ~ ( member_set_set_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_set_set_nat @ A @ ( insert_set_set_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_92_Diff__idemp,axiom,
! [A2: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_93_Diff__idemp,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) @ B )
= ( minus_8121590178497047118at_nat @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_94_Diff__idemp,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) @ B )
= ( minus_2447799839930672331et_nat @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_95_Diff__idemp,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) @ B )
= ( minus_2163939370556025621et_nat @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_96_Diff__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_97_Diff__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B ) )
= ( ( member_nat_nat @ C @ A2 )
& ~ ( member_nat_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_98_Diff__iff,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B ) )
= ( ( member_set_set_nat @ C @ A2 )
& ~ ( member_set_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_99_Diff__iff,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
= ( ( member_set_nat @ C @ A2 )
& ~ ( member_set_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_100_DiffI,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_101_DiffI,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( ~ ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_102_DiffI,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A2 )
=> ( ~ ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_103_DiffI,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_104_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_105_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_106_mem__Collect__eq,axiom,
! [A: nat > nat,P: ( nat > nat ) > $o] :
( ( member_nat_nat @ A @ ( collect_nat_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_107_mem__Collect__eq,axiom,
! [A: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A @ ( collect_set_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_108_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A2: set_nat_nat] :
( ( collect_nat_nat
@ ^ [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_111_Collect__mem__eq,axiom,
! [A2: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_112_empty___092_060G_062,axiom,
member_set_set_nat @ bot_bot_set_set_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% empty_\<G>
thf(fact_113_u__def,axiom,
! [I2: nat] :
( ( u2 @ I2 )
= ( fstt @ ( pair @ I2 ) ) ) ).
% u_def
thf(fact_114_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_115_image__is__empty,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( ( image_set_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_116_image__is__empty,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( ( image_nat_set_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_117_image__is__empty,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat] :
( ( ( image_1454916318497077779at_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_118_image__is__empty,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ( image_nat_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_119_image__is__empty,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( ( image_7916887816326733075et_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% image_is_empty
thf(fact_120_image__is__empty,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( ( image_2194112158459175443et_nat @ F @ A2 )
= bot_bo7198184520161983622et_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_121_image__is__empty,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( ( image_nat_nat_nat2 @ F @ A2 )
= bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_122_image__is__empty,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ( image_5842784325960735177et_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% image_is_empty
thf(fact_123_image__is__empty,axiom,
! [F: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( ( image_7432509271690132940et_nat @ F @ A2 )
= bot_bot_set_set_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% image_is_empty
thf(fact_124_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_125_empty__is__image,axiom,
! [F: set_nat > nat,A2: set_set_nat] :
( ( bot_bot_set_nat
= ( image_set_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_126_empty__is__image,axiom,
! [F: nat > set_nat,A2: set_nat] :
( ( bot_bot_set_set_nat
= ( image_nat_set_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_127_empty__is__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat] :
( ( bot_bot_set_nat
= ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_128_empty__is__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_129_empty__is__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% empty_is_image
thf(fact_130_empty__is__image,axiom,
! [F: nat > set_set_nat,A2: set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_131_empty__is__image,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( bot_bot_set_nat_nat
= ( image_nat_nat_nat2 @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_132_empty__is__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( bot_bot_set_set_nat
= ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% empty_is_image
thf(fact_133_empty__is__image,axiom,
! [F: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( bot_bot_set_set_nat
= ( image_7432509271690132940et_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% empty_is_image
thf(fact_134_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_135_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_136_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_137_image__empty,axiom,
! [F: nat > set_set_nat] :
( ( image_2194112158459175443et_nat @ F @ bot_bot_set_nat )
= bot_bo7198184520161983622et_nat ) ).
% image_empty
thf(fact_138_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_139_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_140_image__empty,axiom,
! [F: set_set_nat > nat] :
( ( image_1454916318497077779at_nat @ F @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_141_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_142_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_143_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_144_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_145_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) )
= ( image_nat_set_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_146_insert__image,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_set_nat_nat @ F @ A2 ) )
= ( image_set_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_147_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > set_set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_148_insert__image,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_149_insert__image,axiom,
! [X2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat_nat @ F @ A2 ) )
= ( image_nat_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_150_insert__image,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_151_insert__image,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( insert_set_set_nat @ ( F @ X2 ) @ ( image_6725021117256019401et_nat @ F @ A2 ) )
= ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_152_insert__image,axiom,
! [X2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_7432509271690132940et_nat @ F @ A2 ) )
= ( image_7432509271690132940et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_153_insert__image,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( image_5842784325960735177et_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_154_image__insert,axiom,
! [F: ( nat > nat ) > set_set_nat,A: nat > nat,B: set_nat_nat] :
( ( image_9186907679027735170et_nat @ F @ ( insert_nat_nat @ A @ B ) )
= ( insert_set_set_nat @ ( F @ A ) @ ( image_9186907679027735170et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_155_image__insert,axiom,
! [F: nat > nat,A: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_156_image__insert,axiom,
! [F: nat > set_set_nat,A: nat,B: set_nat] :
( ( image_2194112158459175443et_nat @ F @ ( insert_nat @ A @ B ) )
= ( insert_set_set_nat @ ( F @ A ) @ ( image_2194112158459175443et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_157_image__insert,axiom,
! [F: nat > set_nat,A: nat,B: set_nat] :
( ( image_nat_set_nat @ F @ ( insert_nat @ A @ B ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_158_image__insert,axiom,
! [F: set_set_nat > nat,A: set_set_nat,B: set_set_set_nat] :
( ( image_1454916318497077779at_nat @ F @ ( insert_set_set_nat @ A @ B ) )
= ( insert_nat @ ( F @ A ) @ ( image_1454916318497077779at_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_159_image__insert,axiom,
! [F: set_set_nat > set_set_nat,A: set_set_nat,B: set_set_set_nat] :
( ( image_7884819252390400639et_nat @ F @ ( insert_set_set_nat @ A @ B ) )
= ( insert_set_set_nat @ ( F @ A ) @ ( image_7884819252390400639et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_160_image__insert,axiom,
! [F: set_set_nat > set_nat,A: set_set_nat,B: set_set_set_nat] :
( ( image_5842784325960735177et_nat @ F @ ( insert_set_set_nat @ A @ B ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_5842784325960735177et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_161_image__insert,axiom,
! [F: set_nat > nat,A: set_nat,B: set_set_nat] :
( ( image_set_nat_nat @ F @ ( insert_set_nat @ A @ B ) )
= ( insert_nat @ ( F @ A ) @ ( image_set_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_162_image__insert,axiom,
! [F: set_nat > set_set_nat,A: set_nat,B: set_set_nat] :
( ( image_6725021117256019401et_nat @ F @ ( insert_set_nat @ A @ B ) )
= ( insert_set_set_nat @ ( F @ A ) @ ( image_6725021117256019401et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_163_image__insert,axiom,
! [F: set_nat > set_nat,A: set_nat,B: set_set_nat] :
( ( image_7916887816326733075et_nat @ F @ ( insert_set_nat @ A @ B ) )
= ( insert_set_nat @ ( F @ A ) @ ( image_7916887816326733075et_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_164_v__empty,axiom,
( ( clique5033774636164728513irst_v @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% v_empty
thf(fact_165_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_166_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_nat @ B2 @ ( image_nat_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_167_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B2: nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_set_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_168_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: nat > nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat_nat @ B2 @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_169_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: set_set_nat,F: nat > set_set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_set_nat @ B2 @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_170_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_nat @ B2 @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_171_rev__image__eqI,axiom,
! [X2: nat > nat,A2: set_nat_nat,B2: nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_nat_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_172_rev__image__eqI,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B2: nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_173_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B2: nat > nat,F: set_nat > nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat_nat @ B2 @ ( image_8569768528772619084at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_174_rev__image__eqI,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_set_nat @ B2 @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_175_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_176_ball__imageD,axiom,
! [F: ( nat > nat ) > set_set_nat,A2: set_nat_nat,P: set_set_nat > $o] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ ( image_9186907679027735170et_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_177_ball__imageD,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_178_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_179_image__cong,axiom,
! [M: set_nat_nat,N: set_nat_nat,F: ( nat > nat ) > set_set_nat,G: ( nat > nat ) > set_set_nat] :
( ( M = N )
=> ( ! [X: nat > nat] :
( ( member_nat_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_9186907679027735170et_nat @ F @ M )
= ( image_9186907679027735170et_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_180_image__cong,axiom,
! [M: set_set_set_nat,N: set_set_set_nat,F: set_set_nat > set_nat,G: set_set_nat > set_nat] :
( ( M = N )
=> ( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_5842784325960735177et_nat @ F @ M )
= ( image_5842784325960735177et_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_181_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_182_bex__imageD,axiom,
! [F: ( nat > nat ) > set_set_nat,A2: set_nat_nat,P: set_set_nat > $o] :
( ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ ( image_9186907679027735170et_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_183_bex__imageD,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_nat > $o] :
( ? [X4: set_nat] :
( ( member_set_nat @ X4 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_184_image__iff,axiom,
! [Z: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_185_image__iff,axiom,
! [Z: set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( member_set_nat @ Z @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_186_image__iff,axiom,
! [Z: set_set_nat,F: ( nat > nat ) > set_set_nat,A2: set_nat_nat] :
( ( member_set_set_nat @ Z @ ( image_9186907679027735170et_nat @ F @ A2 ) )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_187_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_188_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_nat_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_189_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_set_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_190_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_nat_nat_nat2 @ F @ A2 ) ) ) ).
% imageI
thf(fact_191_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > set_set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_set_set_nat @ ( F @ X2 ) @ ( image_2194112158459175443et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_192_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_7916887816326733075et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_193_imageI,axiom,
! [X2: nat > nat,A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_194_imageI,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,F: set_set_nat > nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_1454916318497077779at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_195_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > nat > nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_nat_nat @ ( F @ X2 ) @ ( image_8569768528772619084at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_196_imageI,axiom,
! [X2: set_nat,A2: set_set_nat,F: set_nat > set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ ( F @ X2 ) @ ( image_6725021117256019401et_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_197_first__assumptions_O_092_060F_062_Ocong,axiom,
clique2971579238625216137irst_F = clique2971579238625216137irst_F ).
% first_assumptions.\<F>.cong
thf(fact_198_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_199_ex__in__conv,axiom,
! [A2: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_200_ex__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ? [X3: set_set_nat] : ( member_set_set_nat @ X3 @ A2 ) )
= ( A2 != bot_bo7198184520161983622et_nat ) ) ).
% ex_in_conv
thf(fact_201_ex__in__conv,axiom,
! [A2: set_nat_nat] :
( ( ? [X3: nat > nat] : ( member_nat_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_nat ) ) ).
% ex_in_conv
thf(fact_202_equals0I,axiom,
! [A2: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_203_equals0I,axiom,
! [A2: set_set_nat] :
( ! [Y: set_nat] :
~ ( member_set_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_204_equals0I,axiom,
! [A2: set_set_set_nat] :
( ! [Y: set_set_nat] :
~ ( member_set_set_nat @ Y @ A2 )
=> ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% equals0I
thf(fact_205_equals0I,axiom,
! [A2: set_nat_nat] :
( ! [Y: nat > nat] :
~ ( member_nat_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat_nat ) ) ).
% equals0I
thf(fact_206_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_207_equals0D,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( A2 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_208_equals0D,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( A2 = bot_bo7198184520161983622et_nat )
=> ~ ( member_set_set_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_209_equals0D,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( A2 = bot_bot_set_nat_nat )
=> ~ ( member_nat_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_210_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_211_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_212_emptyE,axiom,
! [A: set_set_nat] :
~ ( member_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ).
% emptyE
thf(fact_213_emptyE,axiom,
! [A: nat > nat] :
~ ( member_nat_nat @ A @ bot_bot_set_nat_nat ) ).
% emptyE
thf(fact_214_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B3: set_nat] :
( ( A2
= ( insert_nat @ A @ B3 ) )
& ~ ( member_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_215_mk__disjoint__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ? [B3: set_set_nat] :
( ( A2
= ( insert_set_nat @ A @ B3 ) )
& ~ ( member_set_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_216_mk__disjoint__insert,axiom,
! [A: nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ? [B3: set_nat_nat] :
( ( A2
= ( insert_nat_nat @ A @ B3 ) )
& ~ ( member_nat_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_217_mk__disjoint__insert,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ? [B3: set_set_set_nat] :
( ( A2
= ( insert_set_set_nat @ A @ B3 ) )
& ~ ( member_set_set_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_218_insert__commute,axiom,
! [X2: nat,Y2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y2 @ A2 ) )
= ( insert_nat @ Y2 @ ( insert_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_219_insert__commute,axiom,
! [X2: set_set_nat,Y2: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ X2 @ ( insert_set_set_nat @ Y2 @ A2 ) )
= ( insert_set_set_nat @ Y2 @ ( insert_set_set_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_220_insert__commute,axiom,
! [X2: set_nat,Y2: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ X2 @ ( insert_set_nat @ Y2 @ A2 ) )
= ( insert_set_nat @ Y2 @ ( insert_set_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_221_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_nat] :
( ( A2
= ( insert_nat @ B2 @ C2 ) )
& ~ ( member_nat @ B2 @ C2 )
& ( B
= ( insert_nat @ A @ C2 ) )
& ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_222_insert__eq__iff,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ B2 @ B )
=> ( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_set_nat] :
( ( A2
= ( insert_set_nat @ B2 @ C2 ) )
& ~ ( member_set_nat @ B2 @ C2 )
& ( B
= ( insert_set_nat @ A @ C2 ) )
& ~ ( member_set_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_223_insert__eq__iff,axiom,
! [A: nat > nat,A2: set_nat_nat,B2: nat > nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ A @ A2 )
=> ( ~ ( member_nat_nat @ B2 @ B )
=> ( ( ( insert_nat_nat @ A @ A2 )
= ( insert_nat_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_nat_nat] :
( ( A2
= ( insert_nat_nat @ B2 @ C2 ) )
& ~ ( member_nat_nat @ B2 @ C2 )
& ( B
= ( insert_nat_nat @ A @ C2 ) )
& ~ ( member_nat_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_224_insert__eq__iff,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B2: set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ A @ A2 )
=> ( ~ ( member_set_set_nat @ B2 @ B )
=> ( ( ( insert_set_set_nat @ A @ A2 )
= ( insert_set_set_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_set_set_nat] :
( ( A2
= ( insert_set_set_nat @ B2 @ C2 ) )
& ~ ( member_set_set_nat @ B2 @ C2 )
& ( B
= ( insert_set_set_nat @ A @ C2 ) )
& ~ ( member_set_set_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_225_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_226_insert__absorb,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_227_insert__absorb,axiom,
! [A: nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( insert_nat_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_228_insert__absorb,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ( ( insert_set_set_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_229_insert__ident,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ~ ( member_nat @ X2 @ B )
=> ( ( ( insert_nat @ X2 @ A2 )
= ( insert_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_230_insert__ident,axiom,
! [X2: set_nat,A2: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ~ ( member_set_nat @ X2 @ B )
=> ( ( ( insert_set_nat @ X2 @ A2 )
= ( insert_set_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_231_insert__ident,axiom,
! [X2: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ X2 @ A2 )
=> ( ~ ( member_nat_nat @ X2 @ B )
=> ( ( ( insert_nat_nat @ X2 @ A2 )
= ( insert_nat_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_232_insert__ident,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ~ ( member_set_set_nat @ X2 @ B )
=> ( ( ( insert_set_set_nat @ X2 @ A2 )
= ( insert_set_set_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_233_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ! [B3: set_nat] :
( ( A2
= ( insert_nat @ X2 @ B3 ) )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_234_Set_Oset__insert,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ~ ! [B3: set_set_nat] :
( ( A2
= ( insert_set_nat @ X2 @ B3 ) )
=> ( member_set_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_235_Set_Oset__insert,axiom,
! [X2: nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ~ ! [B3: set_nat_nat] :
( ( A2
= ( insert_nat_nat @ X2 @ B3 ) )
=> ( member_nat_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_236_Set_Oset__insert,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ~ ! [B3: set_set_set_nat] :
( ( A2
= ( insert_set_set_nat @ X2 @ B3 ) )
=> ( member_set_set_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_237_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_238_insertI2,axiom,
! [A: set_nat,B: set_set_nat,B2: set_nat] :
( ( member_set_nat @ A @ B )
=> ( member_set_nat @ A @ ( insert_set_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_239_insertI2,axiom,
! [A: nat > nat,B: set_nat_nat,B2: nat > nat] :
( ( member_nat_nat @ A @ B )
=> ( member_nat_nat @ A @ ( insert_nat_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_240_insertI2,axiom,
! [A: set_set_nat,B: set_set_set_nat,B2: set_set_nat] :
( ( member_set_set_nat @ A @ B )
=> ( member_set_set_nat @ A @ ( insert_set_set_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_241_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_242_insertI1,axiom,
! [A: set_nat,B: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B ) ) ).
% insertI1
thf(fact_243_insertI1,axiom,
! [A: nat > nat,B: set_nat_nat] : ( member_nat_nat @ A @ ( insert_nat_nat @ A @ B ) ) ).
% insertI1
thf(fact_244_insertI1,axiom,
! [A: set_set_nat,B: set_set_set_nat] : ( member_set_set_nat @ A @ ( insert_set_set_nat @ A @ B ) ) ).
% insertI1
thf(fact_245_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_246_insertE,axiom,
! [A: set_nat,B2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_247_insertE,axiom,
! [A: nat > nat,B2: nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ A @ ( insert_nat_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_248_insertE,axiom,
! [A: set_set_nat,B2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ ( insert_set_set_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_set_set_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_249_DiffD2,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_250_DiffD2,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B ) )
=> ~ ( member_nat_nat @ C @ B ) ) ).
% DiffD2
thf(fact_251_DiffD2,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B ) )
=> ~ ( member_set_set_nat @ C @ B ) ) ).
% DiffD2
thf(fact_252_DiffD2,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
=> ~ ( member_set_nat @ C @ B ) ) ).
% DiffD2
thf(fact_253_DiffD1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_254_DiffD1,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B ) )
=> ( member_nat_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_255_DiffD1,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B ) )
=> ( member_set_set_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_256_DiffD1,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
=> ( member_set_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_257_DiffE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_258_DiffE,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( minus_8121590178497047118at_nat @ A2 @ B ) )
=> ~ ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_259_DiffE,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B ) )
=> ~ ( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_260_DiffE,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
=> ~ ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_261_singleton__inject,axiom,
! [A: nat,B2: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_262_singleton__inject,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( insert_set_nat @ A @ bot_bot_set_set_nat )
= ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_263_singleton__inject,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat )
= ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_264_singleton__inject,axiom,
! [A: nat > nat,B2: nat > nat] :
( ( ( insert_nat_nat @ A @ bot_bot_set_nat_nat )
= ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_265_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_266_insert__not__empty,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ A @ A2 )
!= bot_bot_set_set_nat ) ).
% insert_not_empty
thf(fact_267_insert__not__empty,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( insert_set_set_nat @ A @ A2 )
!= bot_bo7198184520161983622et_nat ) ).
% insert_not_empty
thf(fact_268_insert__not__empty,axiom,
! [A: nat > nat,A2: set_nat_nat] :
( ( insert_nat_nat @ A @ A2 )
!= bot_bot_set_nat_nat ) ).
% insert_not_empty
thf(fact_269_doubleton__eq__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_270_doubleton__eq__iff,axiom,
! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
( ( ( insert_set_nat @ A @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) )
= ( insert_set_nat @ C @ ( insert_set_nat @ D @ bot_bot_set_set_nat ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_271_doubleton__eq__iff,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_set_nat,D: set_set_nat] :
( ( ( insert_set_set_nat @ A @ ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) )
= ( insert_set_set_nat @ C @ ( insert_set_set_nat @ D @ bot_bo7198184520161983622et_nat ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_272_doubleton__eq__iff,axiom,
! [A: nat > nat,B2: nat > nat,C: nat > nat,D: nat > nat] :
( ( ( insert_nat_nat @ A @ ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) )
= ( insert_nat_nat @ C @ ( insert_nat_nat @ D @ bot_bot_set_nat_nat ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_273_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_274_singleton__iff,axiom,
! [B2: set_nat,A: set_nat] :
( ( member_set_nat @ B2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_275_singleton__iff,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( member_set_set_nat @ B2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_276_singleton__iff,axiom,
! [B2: nat > nat,A: nat > nat] :
( ( member_nat_nat @ B2 @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_277_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_278_singletonD,axiom,
! [B2: set_nat,A: set_nat] :
( ( member_set_nat @ B2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_279_singletonD,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( member_set_set_nat @ B2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_280_singletonD,axiom,
! [B2: nat > nat,A: nat > nat] :
( ( member_nat_nat @ B2 @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_281_insert__Diff__if,axiom,
! [X2: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_282_insert__Diff__if,axiom,
! [X2: nat > nat,B: set_nat_nat,A2: set_nat_nat] :
( ( ( member_nat_nat @ X2 @ B )
=> ( ( minus_8121590178497047118at_nat @ ( insert_nat_nat @ X2 @ A2 ) @ B )
= ( minus_8121590178497047118at_nat @ A2 @ B ) ) )
& ( ~ ( member_nat_nat @ X2 @ B )
=> ( ( minus_8121590178497047118at_nat @ ( insert_nat_nat @ X2 @ A2 ) @ B )
= ( insert_nat_nat @ X2 @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_283_insert__Diff__if,axiom,
! [X2: set_set_nat,B: set_set_set_nat,A2: set_set_set_nat] :
( ( ( member_set_set_nat @ X2 @ B )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ B )
= ( minus_2447799839930672331et_nat @ A2 @ B ) ) )
& ( ~ ( member_set_set_nat @ X2 @ B )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ B )
= ( insert_set_set_nat @ X2 @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_284_insert__Diff__if,axiom,
! [X2: set_nat,B: set_set_nat,A2: set_set_nat] :
( ( ( member_set_nat @ X2 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B )
= ( minus_2163939370556025621et_nat @ A2 @ B ) ) )
& ( ~ ( member_set_nat @ X2 @ B )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B )
= ( insert_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_285_Diff__insert__absorb,axiom,
! [X2: nat,A2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_286_Diff__insert__absorb,axiom,
! [X2: nat > nat,A2: set_nat_nat] :
( ~ ( member_nat_nat @ X2 @ A2 )
=> ( ( minus_8121590178497047118at_nat @ ( insert_nat_nat @ X2 @ A2 ) @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_287_Diff__insert__absorb,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ( minus_2447799839930672331et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_288_Diff__insert__absorb,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A2 ) @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_289_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_290_Diff__insert2,axiom,
! [A2: set_nat_nat,A: nat > nat,B: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ A @ B ) )
= ( minus_8121590178497047118at_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_291_Diff__insert2,axiom,
! [A2: set_set_set_nat,A: set_set_nat,B: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B ) )
= ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_292_Diff__insert2,axiom,
! [A2: set_set_nat,A: set_nat,B: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_293_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_294_insert__Diff,axiom,
! [A: nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( insert_nat_nat @ A @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_295_insert__Diff,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ( ( insert_set_set_nat @ A @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_296_insert__Diff,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_297_Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_298_Diff__insert,axiom,
! [A2: set_nat_nat,A: nat > nat,B: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ A @ B ) )
= ( minus_8121590178497047118at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) ) ) ).
% Diff_insert
thf(fact_299_Diff__insert,axiom,
! [A2: set_set_set_nat,A: set_set_nat,B: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B ) )
= ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ) ).
% Diff_insert
thf(fact_300_Diff__insert,axiom,
! [A2: set_set_nat,A: set_nat,B: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B ) )
= ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ).
% Diff_insert
thf(fact_301_empty__CLIQUE,axiom,
~ ( member_set_set_nat @ bot_bot_set_set_nat @ ( clique363107459185959606CLIQUE @ k ) ) ).
% empty_CLIQUE
thf(fact_302_Cf,axiom,
member_set_set_nat @ ( clique5033774636164728462irst_C @ k @ f ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% Cf
thf(fact_303_i__props_I1_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( ord_less_eq_set_nat @ vs @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) ) ) ).
% i_props(1)
thf(fact_304__092_060open_062ti_A_092_060equiv_062_A_092_060lambda_062i_O_Acard_A_Iv_A_IG_Ai_J_A_N_AVs_J_092_060close_062,axiom,
( ti
= ( ^ [I: nat] : ( finite_card_nat @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) @ vs ) ) ) ) ).
% \<open>ti \<equiv> \<lambda>i. card (v (G i) - Vs)\<close>
thf(fact_305_v___092_060G_062,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% v_\<G>
thf(fact_306_Vsm,axiom,
ord_less_eq_set_nat @ vs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ).
% Vsm
thf(fact_307_finite__vG,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G2 ) ) ) ).
% finite_vG
thf(fact_308_i__props_I2_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) ) ) ).
% i_props(2)
thf(fact_309_G_I4_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( member_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) @ s ) ) ).
% G(4)
thf(fact_310_finite__members___092_060G_062,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ G2 ) ) ).
% finite_members_\<G>
thf(fact_311_GsG,axiom,
member_set_set_nat @ gs @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% GsG
thf(fact_312_fin__Vs,axiom,
finite_finite_nat @ vs ).
% fin_Vs
thf(fact_313_Snempty,axiom,
s != bot_bot_set_set_nat ).
% Snempty
thf(fact_314_finS,axiom,
finite1152437895449049373et_nat @ s ).
% finS
thf(fact_315_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_316_subset__antisym,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_317_subset__antisym,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_318_subset__antisym,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_319_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_320_subsetI,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ X @ B ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_321_subsetI,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_set_set_nat @ X @ B ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_322_subsetI,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat_nat @ X @ B ) )
=> ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_323__092_060open_062si_A_092_060equiv_062_A_092_060lambda_062i_O_Acard_A_Iv_A_IG_Ai_J_J_092_060close_062,axiom,
( si2
= ( ^ [I: nat] : ( finite_card_nat @ ( clique5033774636164728513irst_v @ ( g @ I ) ) ) ) ) ).
% \<open>si \<equiv> \<lambda>i. card (v (G i))\<close>
thf(fact_324_i__props_I6_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( finite1152437895449049373et_nat @ ( g @ I2 ) ) ) ).
% i_props(6)
thf(fact_325_vGs,axiom,
ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ vs ).
% vGs
thf(fact_326_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_327_subset__empty,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_328_subset__empty,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ bot_bo7198184520161983622et_nat )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% subset_empty
thf(fact_329_subset__empty,axiom,
! [A2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ bot_bot_set_nat_nat )
= ( A2 = bot_bot_set_nat_nat ) ) ).
% subset_empty
thf(fact_330_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_331_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_332_empty__subsetI,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ bot_bo7198184520161983622et_nat @ A2 ) ).
% empty_subsetI
thf(fact_333_empty__subsetI,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ bot_bot_set_nat_nat @ A2 ) ).
% empty_subsetI
thf(fact_334_insert__subset,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
= ( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_335_insert__subset,axiom,
! [X2: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ A2 ) @ B )
= ( ( member_set_nat @ X2 @ B )
& ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_336_insert__subset,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ X2 @ A2 ) @ B )
= ( ( member_set_set_nat @ X2 @ B )
& ( ord_le9131159989063066194et_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_337_insert__subset,axiom,
! [X2: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( insert_nat_nat @ X2 @ A2 ) @ B )
= ( ( member_nat_nat @ X2 @ B )
& ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_338_finite__numbers,axiom,
! [N2: nat] : ( finite_finite_nat @ ( clique3652268606331196573umbers @ N2 ) ) ).
% finite_numbers
thf(fact_339_card__numbers,axiom,
! [N2: nat] :
( ( finite_card_nat @ ( clique3652268606331196573umbers @ N2 ) )
= N2 ) ).
% card_numbers
thf(fact_340_ti__def,axiom,
! [I2: nat] :
( ( ti @ I2 )
= ( finite_card_nat @ ( minus_minus_set_nat @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) @ vs ) ) ) ).
% ti_def
thf(fact_341_singleton__insert__inj__eq,axiom,
! [B2: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_342_singleton__insert__inj__eq,axiom,
! [B2: set_nat,A: set_nat,A2: set_set_nat] :
( ( ( insert_set_nat @ B2 @ bot_bot_set_set_nat )
= ( insert_set_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_343_singleton__insert__inj__eq,axiom,
! [B2: set_set_nat,A: set_set_nat,A2: set_set_set_nat] :
( ( ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat )
= ( insert_set_set_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_344_singleton__insert__inj__eq,axiom,
! [B2: nat > nat,A: nat > nat,A2: set_nat_nat] :
( ( ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat )
= ( insert_nat_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le9059583361652607317at_nat @ A2 @ ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_345_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B2: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_346_singleton__insert__inj__eq_H,axiom,
! [A: set_nat,A2: set_set_nat,B2: set_nat] :
( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) )
= ( ( A = B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B2 @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_347_singleton__insert__inj__eq_H,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B2: set_set_nat] :
( ( ( insert_set_set_nat @ A @ A2 )
= ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) )
= ( ( A = B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ B2 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_348_singleton__insert__inj__eq_H,axiom,
! [A: nat > nat,A2: set_nat_nat,B2: nat > nat] :
( ( ( insert_nat_nat @ A @ A2 )
= ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) )
= ( ( A = B2 )
& ( ord_le9059583361652607317at_nat @ A2 @ ( insert_nat_nat @ B2 @ bot_bot_set_nat_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_349_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_350_Diff__eq__empty__iff,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ( minus_2163939370556025621et_nat @ A2 @ B )
= bot_bot_set_set_nat )
= ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_351_Diff__eq__empty__iff,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( ( minus_2447799839930672331et_nat @ A2 @ B )
= bot_bo7198184520161983622et_nat )
= ( ord_le9131159989063066194et_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_352_Diff__eq__empty__iff,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ( minus_8121590178497047118at_nat @ A2 @ B )
= bot_bot_set_nat_nat )
= ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_353_s__def,axiom,
( s2
= ( finite_card_nat @ vs ) ) ).
% s_def
thf(fact_354_local_ONEG__def,axiom,
( ( clique3210737375870294875st_NEG @ k )
= ( image_9186907679027735170et_nat @ ( clique5033774636164728462irst_C @ k ) @ ( clique2971579238625216137irst_F @ k ) ) ) ).
% local.NEG_def
thf(fact_355__092_060open_062C_Af_A_092_060in_062_ANEG_092_060close_062,axiom,
member_set_set_nat @ ( clique5033774636164728462irst_C @ k @ f ) @ ( clique3210737375870294875st_NEG @ k ) ).
% \<open>C f \<in> NEG\<close>
thf(fact_356_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_357_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_358_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_359_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_360_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_361_set__eq__subset,axiom,
( ( ^ [Y3: set_set_nat,Z2: set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A3 @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_362_set__eq__subset,axiom,
( ( ^ [Y3: set_set_set_nat,Z2: set_set_set_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_set_set_nat,B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A3 @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_363_set__eq__subset,axiom,
( ( ^ [Y3: set_nat_nat,Z2: set_nat_nat] : ( Y3 = Z2 ) )
= ( ^ [A3: set_nat_nat,B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A3 @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_364_subset__trans,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_365_subset__trans,axiom,
! [A2: set_set_nat,B: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C3 )
=> ( ord_le6893508408891458716et_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_366_subset__trans,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C3 )
=> ( ord_le9131159989063066194et_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_367_subset__trans,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ord_le9059583361652607317at_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_368_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_369_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_370_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_371_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_372_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_373_subset__refl,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_374_subset__refl,axiom,
! [A2: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_375_subset__refl,axiom,
! [A2: set_nat_nat] : ( ord_le9059583361652607317at_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_376_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_377_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A3 )
=> ( member_set_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_378_subset__iff,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B4: set_set_set_nat] :
! [T: set_set_nat] :
( ( member_set_set_nat @ T @ A3 )
=> ( member_set_set_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_379_subset__iff,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B4: set_nat_nat] :
! [T: nat > nat] :
( ( member_nat_nat @ T @ A3 )
=> ( member_nat_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_380_equalityD2,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_381_equalityD2,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2 = B )
=> ( ord_le6893508408891458716et_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_382_equalityD2,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( A2 = B )
=> ( ord_le9131159989063066194et_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_383_equalityD2,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( A2 = B )
=> ( ord_le9059583361652607317at_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_384_equalityD1,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_385_equalityD1,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2 = B )
=> ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_386_equalityD1,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( A2 = B )
=> ( ord_le9131159989063066194et_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_387_equalityD1,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( A2 = B )
=> ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_388_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_389_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A3 )
=> ( member_set_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_390_subset__eq,axiom,
( ord_le9131159989063066194et_nat
= ( ^ [A3: set_set_set_nat,B4: set_set_set_nat] :
! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A3 )
=> ( member_set_set_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_391_subset__eq,axiom,
( ord_le9059583361652607317at_nat
= ( ^ [A3: set_nat_nat,B4: set_nat_nat] :
! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A3 )
=> ( member_nat_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_392_equalityE,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_393_equalityE,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( A2 = B )
=> ~ ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ~ ( ord_le6893508408891458716et_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_394_equalityE,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( A2 = B )
=> ~ ( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ~ ( ord_le9131159989063066194et_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_395_equalityE,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( A2 = B )
=> ~ ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ~ ( ord_le9059583361652607317at_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_396_subsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_397_subsetD,axiom,
! [A2: set_set_nat,B: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_398_subsetD,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_399_subsetD,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_400_in__mono,axiom,
! [A2: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_401_in__mono,axiom,
! [A2: set_set_nat,B: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_402_in__mono,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,X2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_403_in__mono,axiom,
! [A2: set_nat_nat,B: set_nat_nat,X2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( member_nat_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_404_first__assumptions_OC_Ocong,axiom,
clique5033774636164728462irst_C = clique5033774636164728462irst_C ).
% first_assumptions.C.cong
thf(fact_405_first__assumptions_OCLIQUE_Ocong,axiom,
clique363107459185959606CLIQUE = clique363107459185959606CLIQUE ).
% first_assumptions.CLIQUE.cong
thf(fact_406_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_407_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_408_image__mono,axiom,
! [A2: set_set_nat,B: set_set_nat,F: set_nat > nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_409_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > set_set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_410_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B ) ) ) ).
% image_mono
thf(fact_411_image__mono,axiom,
! [A2: set_set_nat,B: set_set_nat,F: set_nat > set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_412_image__mono,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,F: set_set_nat > nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ ( image_1454916318497077779at_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_413_image__mono,axiom,
! [A2: set_nat_nat,B: set_nat_nat,F: ( nat > nat ) > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ ( image_nat_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_414_image__mono,axiom,
! [A2: set_set_nat,B: set_set_nat,F: set_nat > set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_le9131159989063066194et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) @ ( image_6725021117256019401et_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_415_image__mono,axiom,
! [A2: set_set_nat,B: set_set_nat,F: set_nat > nat > nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_le9059583361652607317at_nat @ ( image_8569768528772619084at_nat @ F @ A2 ) @ ( image_8569768528772619084at_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_416_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_417_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > nat,B: set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_418_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_nat,B: set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_nat_set_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_419_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat,B: set_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_420_image__subsetI,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat,B: set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_421_image__subsetI,axiom,
! [A2: set_set_nat,F: set_nat > set_nat,B: set_set_nat] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_422_image__subsetI,axiom,
! [A2: set_nat,F: nat > set_set_nat,B: set_set_set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_set_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le9131159989063066194et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_423_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > nat,B: set_nat_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat_nat @ ( F @ X ) @ B ) )
=> ( ord_le9059583361652607317at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_424_image__subsetI,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat,B: set_set_nat] :
( ! [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_425_image__subsetI,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_nat,B: set_set_nat] :
( ! [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_426_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_427_subset__imageE,axiom,
! [B: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C4 @ A2 )
=> ( B
!= ( image_set_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_428_subset__imageE,axiom,
! [B: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B
!= ( image_nat_set_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_429_subset__imageE,axiom,
! [B: set_nat,F: set_set_nat > nat,A2: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ~ ! [C4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C4 @ A2 )
=> ( B
!= ( image_1454916318497077779at_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_430_subset__imageE,axiom,
! [B: set_nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C4 @ A2 )
=> ( B
!= ( image_nat_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_431_subset__imageE,axiom,
! [B: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ~ ! [C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C4 @ A2 )
=> ( B
!= ( image_7916887816326733075et_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_432_subset__imageE,axiom,
! [B: set_set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B
!= ( image_2194112158459175443et_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_433_subset__imageE,axiom,
! [B: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B
!= ( image_nat_nat_nat2 @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_434_subset__imageE,axiom,
! [B: set_set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ~ ! [C4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C4 @ A2 )
=> ( B
!= ( image_5842784325960735177et_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_435_subset__imageE,axiom,
! [B: set_set_nat,F: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_7432509271690132940et_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C4 @ A2 )
=> ( B
!= ( image_7432509271690132940et_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_436_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_437_image__subset__iff,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ B )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( member_set_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_438_image__subset__iff,axiom,
! [F: ( nat > nat ) > set_set_nat,A2: set_nat_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( image_9186907679027735170et_nat @ F @ A2 ) @ B )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( member_set_set_nat @ ( F @ X3 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_439_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_440_subset__image__iff,axiom,
! [B: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B
= ( image_set_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_441_subset__image__iff,axiom,
! [B: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_442_subset__image__iff,axiom,
! [B: set_nat,F: set_set_nat > nat,A2: set_set_set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B
= ( image_1454916318497077779at_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_443_subset__image__iff,axiom,
! [B: set_nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B
= ( image_nat_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_444_subset__image__iff,axiom,
! [B: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) )
= ( ? [AA: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ AA @ A2 )
& ( B
= ( image_7916887816326733075et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_445_subset__image__iff,axiom,
! [B: set_set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_2194112158459175443et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_446_subset__image__iff,axiom,
! [B: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat_nat2 @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_447_subset__image__iff,axiom,
! [B: set_set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) )
= ( ? [AA: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ AA @ A2 )
& ( B
= ( image_5842784325960735177et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_448_subset__image__iff,axiom,
! [B: set_set_nat,F: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_7432509271690132940et_nat @ F @ A2 ) )
= ( ? [AA: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ AA @ A2 )
& ( B
= ( image_7432509271690132940et_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_449_insert__mono,axiom,
! [C3: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C3 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C3 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_450_insert__mono,axiom,
! [C3: set_set_nat,D2: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ D2 )
=> ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ A @ C3 ) @ ( insert_set_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_451_insert__mono,axiom,
! [C3: set_set_set_nat,D2: set_set_set_nat,A: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ D2 )
=> ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ A @ C3 ) @ ( insert_set_set_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_452_insert__mono,axiom,
! [C3: set_nat_nat,D2: set_nat_nat,A: nat > nat] :
( ( ord_le9059583361652607317at_nat @ C3 @ D2 )
=> ( ord_le9059583361652607317at_nat @ ( insert_nat_nat @ A @ C3 ) @ ( insert_nat_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_453_subset__insert,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_454_subset__insert,axiom,
! [X2: set_nat,A2: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ B ) )
= ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_455_subset__insert,axiom,
! [X2: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X2 @ B ) )
= ( ord_le9131159989063066194et_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_456_subset__insert,axiom,
! [X2: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ X2 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ ( insert_nat_nat @ X2 @ B ) )
= ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_457_subset__insertI,axiom,
! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_458_subset__insertI,axiom,
! [B: set_set_nat,A: set_nat] : ( ord_le6893508408891458716et_nat @ B @ ( insert_set_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_459_subset__insertI,axiom,
! [B: set_set_set_nat,A: set_set_nat] : ( ord_le9131159989063066194et_nat @ B @ ( insert_set_set_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_460_subset__insertI,axiom,
! [B: set_nat_nat,A: nat > nat] : ( ord_le9059583361652607317at_nat @ B @ ( insert_nat_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_461_subset__insertI2,axiom,
! [A2: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_462_subset__insertI2,axiom,
! [A2: set_set_nat,B: set_set_nat,B2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_463_subset__insertI2,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,B2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_464_subset__insertI2,axiom,
! [A2: set_nat_nat,B: set_nat_nat,B2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( insert_nat_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_465_Diff__mono,axiom,
! [A2: set_nat,C3: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_466_Diff__mono,axiom,
! [A2: set_set_nat,C3: set_set_nat,D2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ C3 )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) @ ( minus_2163939370556025621et_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_467_Diff__mono,axiom,
! [A2: set_set_set_nat,C3: set_set_set_nat,D2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ C3 )
=> ( ( ord_le9131159989063066194et_nat @ D2 @ B )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) @ ( minus_2447799839930672331et_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_468_Diff__mono,axiom,
! [A2: set_nat_nat,C3: set_nat_nat,D2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ C3 )
=> ( ( ord_le9059583361652607317at_nat @ D2 @ B )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) @ ( minus_8121590178497047118at_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_469_Diff__subset,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_470_Diff__subset,axiom,
! [A2: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_471_Diff__subset,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_472_Diff__subset,axiom,
! [A2: set_nat_nat,B: set_nat_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_473_double__diff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_474_double__diff,axiom,
! [A2: set_set_nat,B: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ C3 )
=> ( ( minus_2163939370556025621et_nat @ B @ ( minus_2163939370556025621et_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_475_double__diff,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ C3 )
=> ( ( minus_2447799839930672331et_nat @ B @ ( minus_2447799839930672331et_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_476_double__diff,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ C3 )
=> ( ( minus_8121590178497047118at_nat @ B @ ( minus_8121590178497047118at_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_477_subset__singletonD,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_478_subset__singletonD,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
=> ( ( A2 = bot_bot_set_set_nat )
| ( A2
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_479_subset__singletonD,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) )
=> ( ( A2 = bot_bo7198184520161983622et_nat )
| ( A2
= ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% subset_singletonD
thf(fact_480_subset__singletonD,axiom,
! [A2: set_nat_nat,X2: nat > nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) )
=> ( ( A2 = bot_bot_set_nat_nat )
| ( A2
= ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) ) ) ).
% subset_singletonD
thf(fact_481_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_482_subset__singleton__iff,axiom,
! [X5: set_set_nat,A: set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
= ( ( X5 = bot_bot_set_set_nat )
| ( X5
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_483_subset__singleton__iff,axiom,
! [X5: set_set_set_nat,A: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) )
= ( ( X5 = bot_bo7198184520161983622et_nat )
| ( X5
= ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_484_subset__singleton__iff,axiom,
! [X5: set_nat_nat,A: nat > nat] :
( ( ord_le9059583361652607317at_nat @ X5 @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) )
= ( ( X5 = bot_bot_set_nat_nat )
| ( X5
= ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_485_image__diff__subset,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_486_image__diff__subset,axiom,
! [F: set_nat > nat,A2: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_set_nat_nat @ F @ A2 ) @ ( image_set_nat_nat @ F @ B ) ) @ ( image_set_nat_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_487_image__diff__subset,axiom,
! [F: nat > set_nat,A2: set_nat,B: set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_nat_set_nat @ F @ A2 ) @ ( image_nat_set_nat @ F @ B ) ) @ ( image_nat_set_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_488_image__diff__subset,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,B: set_nat_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat_nat @ F @ A2 ) @ ( image_nat_nat_nat @ F @ B ) ) @ ( image_nat_nat_nat @ F @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_489_image__diff__subset,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,B: set_set_set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) @ ( image_1454916318497077779at_nat @ F @ B ) ) @ ( image_1454916318497077779at_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_490_image__diff__subset,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,B: set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) @ ( image_7916887816326733075et_nat @ F @ B ) ) @ ( image_7916887816326733075et_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_491_image__diff__subset,axiom,
! [F: nat > set_set_nat,A2: set_nat,B: set_nat] : ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) @ ( image_2194112158459175443et_nat @ F @ B ) ) @ ( image_2194112158459175443et_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_492_image__diff__subset,axiom,
! [F: nat > nat > nat,A2: set_nat,B: set_nat] : ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ ( image_nat_nat_nat2 @ F @ A2 ) @ ( image_nat_nat_nat2 @ F @ B ) ) @ ( image_nat_nat_nat2 @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_493_image__diff__subset,axiom,
! [F: ( nat > nat ) > set_nat,A2: set_nat_nat,B: set_nat_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) @ ( image_7432509271690132940et_nat @ F @ B ) ) @ ( image_7432509271690132940et_nat @ F @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_494_image__diff__subset,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,B: set_set_set_nat] : ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) @ ( image_5842784325960735177et_nat @ F @ B ) ) @ ( image_5842784325960735177et_nat @ F @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_495_subset__Diff__insert,axiom,
! [A2: set_nat,B: set_nat,X2: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat @ X2 @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C3 ) )
& ~ ( member_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_496_subset__Diff__insert,axiom,
! [A2: set_set_nat,B: set_set_nat,X2: set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B @ ( insert_set_nat @ X2 @ C3 ) ) )
= ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B @ C3 ) )
& ~ ( member_set_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_497_subset__Diff__insert,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,X2: set_set_nat,C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( minus_2447799839930672331et_nat @ B @ ( insert_set_set_nat @ X2 @ C3 ) ) )
= ( ( ord_le9131159989063066194et_nat @ A2 @ ( minus_2447799839930672331et_nat @ B @ C3 ) )
& ~ ( member_set_set_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_498_subset__Diff__insert,axiom,
! [A2: set_nat_nat,B: set_nat_nat,X2: nat > nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( minus_8121590178497047118at_nat @ B @ ( insert_nat_nat @ X2 @ C3 ) ) )
= ( ( ord_le9059583361652607317at_nat @ A2 @ ( minus_8121590178497047118at_nat @ B @ C3 ) )
& ~ ( member_nat_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_499_in__image__insert__iff,axiom,
! [B: set_set_nat,X2: nat,A2: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B )
=> ~ ( member_nat @ X2 @ C4 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X2 ) @ B ) )
= ( ( member_nat @ X2 @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_500_in__image__insert__iff,axiom,
! [B: set_set_nat_nat,X2: nat > nat,A2: set_nat_nat] :
( ! [C4: set_nat_nat] :
( ( member_set_nat_nat @ C4 @ B )
=> ~ ( member_nat_nat @ X2 @ C4 ) )
=> ( ( member_set_nat_nat @ A2 @ ( image_3832368097948589297at_nat @ ( insert_nat_nat @ X2 ) @ B ) )
= ( ( member_nat_nat @ X2 @ A2 )
& ( member_set_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_501_in__image__insert__iff,axiom,
! [B: set_set_set_set_nat,X2: set_set_nat,A2: set_set_set_nat] :
( ! [C4: set_set_set_nat] :
( ( member2946998982187404937et_nat @ C4 @ B )
=> ~ ( member_set_set_nat @ X2 @ C4 ) )
=> ( ( member2946998982187404937et_nat @ A2 @ ( image_6473237745780476395et_nat @ ( insert_set_set_nat @ X2 ) @ B ) )
= ( ( member_set_set_nat @ X2 @ A2 )
& ( member2946998982187404937et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_502_in__image__insert__iff,axiom,
! [B: set_set_set_nat,X2: set_nat,A2: set_set_nat] :
( ! [C4: set_set_nat] :
( ( member_set_set_nat @ C4 @ B )
=> ~ ( member_set_nat @ X2 @ C4 ) )
=> ( ( member_set_set_nat @ A2 @ ( image_7884819252390400639et_nat @ ( insert_set_nat @ X2 ) @ B ) )
= ( ( member_set_nat @ X2 @ A2 )
& ( member_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_503_subset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
= ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_504_subset__insert__iff,axiom,
! [A2: set_set_nat,X2: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ B ) )
= ( ( ( member_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B ) )
& ( ~ ( member_set_nat @ X2 @ A2 )
=> ( ord_le6893508408891458716et_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_505_subset__insert__iff,axiom,
! [A2: set_set_set_nat,X2: set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X2 @ B ) )
= ( ( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) @ B ) )
& ( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ord_le9131159989063066194et_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_506_subset__insert__iff,axiom,
! [A2: set_nat_nat,X2: nat > nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ ( insert_nat_nat @ X2 @ B ) )
= ( ( ( member_nat_nat @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) @ B ) )
& ( ~ ( member_nat_nat @ X2 @ A2 )
=> ( ord_le9059583361652607317at_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_507_Diff__single__insert,axiom,
! [A2: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_508_Diff__single__insert,axiom,
! [A2: set_set_nat,X2: set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B )
=> ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_509_Diff__single__insert,axiom,
! [A2: set_set_set_nat,X2: set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) @ B )
=> ( ord_le9131159989063066194et_nat @ A2 @ ( insert_set_set_nat @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_510_Diff__single__insert,axiom,
! [A2: set_nat_nat,X2: nat > nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) @ B )
=> ( ord_le9059583361652607317at_nat @ A2 @ ( insert_nat_nat @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_511_si__def,axiom,
! [I2: nat] :
( ( si2 @ I2 )
= ( finite_card_nat @ ( clique5033774636164728513irst_v @ ( g @ I2 ) ) ) ) ).
% si_def
thf(fact_512_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_513_card__Diff__insert,axiom,
! [A: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ~ ( member_nat_nat @ A @ B )
=> ( ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_514_card__Diff__insert,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ( ~ ( member_set_set_nat @ A @ B )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B ) ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_515_card__Diff__insert,axiom,
! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ~ ( member_set_nat @ A @ B )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_516_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_517_finite__Diff__insert,axiom,
! [A2: set_nat_nat,A: nat > nat,B: set_nat_nat] :
( ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ A @ B ) ) )
= ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_518_finite__Diff__insert,axiom,
! [A2: set_set_set_nat,A: set_set_nat,B: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ A @ B ) ) )
= ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_519_finite__Diff__insert,axiom,
! [A2: set_set_nat,A: set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ B ) ) )
= ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_520__092_060open_062f_A_092_060in_062_AACC__cf_A_123Gs_125_092_060close_062,axiom,
member_nat_nat @ f @ ( clique951075384711337423ACC_cf @ k @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ).
% \<open>f \<in> ACC_cf {Gs}\<close>
thf(fact_521_card__Diff__singleton,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_522_card__Diff__singleton,axiom,
! [X2: nat > nat,A2: set_nat_nat] :
( ( member_nat_nat @ X2 @ A2 )
=> ( ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_523_card__Diff__singleton,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_524_card__Diff__singleton,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_525_card__Diff__singleton__if,axiom,
! [X2: nat,A2: set_nat] :
( ( ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_526_card__Diff__singleton__if,axiom,
! [X2: nat > nat,A2: set_nat_nat] :
( ( ( member_nat_nat @ X2 @ A2 )
=> ( ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat_nat @ X2 @ A2 )
=> ( ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) )
= ( finite_card_nat_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_527_card__Diff__singleton__if,axiom,
! [X2: set_set_nat,A2: set_set_set_nat] :
( ( ( member_set_set_nat @ X2 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_set_nat @ X2 @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) )
= ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_528_card__Diff__singleton__if,axiom,
! [X2: set_nat,A2: set_set_nat] :
( ( ( member_set_nat @ X2 @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_nat @ X2 @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
= ( finite_card_set_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_529_card__Diff1__less,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_530_card__Diff1__less,axiom,
! [A2: set_nat_nat,X2: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_531_card__Diff1__less,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_532_card__Diff1__less,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_533_card__Diff2__less,axiom,
! [A2: set_nat,X2: nat,Y2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_534_card__Diff2__less,axiom,
! [A2: set_nat_nat,X2: nat > nat,Y2: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ X2 @ A2 )
=> ( ( member_nat_nat @ Y2 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) @ ( insert_nat_nat @ Y2 @ bot_bot_set_nat_nat ) ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_535_card__Diff2__less,axiom,
! [A2: set_set_set_nat,X2: set_set_nat,Y2: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ X2 @ A2 )
=> ( ( member_set_set_nat @ Y2 @ A2 )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) @ ( insert_set_set_nat @ Y2 @ bot_bo7198184520161983622et_nat ) ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_536_card__Diff2__less,axiom,
! [A2: set_set_nat,X2: set_nat,Y2: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X2 @ A2 )
=> ( ( member_set_nat @ Y2 @ A2 )
=> ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_537_card__Diff1__less__iff,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X2 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_538_card__Diff1__less__iff,axiom,
! [A2: set_nat_nat,X2: nat > nat] :
( ( ord_less_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) ) @ ( finite_card_nat_nat @ A2 ) )
= ( ( finite2115694454571419734at_nat @ A2 )
& ( member_nat_nat @ X2 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_539_card__Diff1__less__iff,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] :
( ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) ) @ ( finite1149291290879098388et_nat @ A2 ) )
= ( ( finite6739761609112101331et_nat @ A2 )
& ( member_set_set_nat @ X2 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_540_card__Diff1__less__iff,axiom,
! [A2: set_set_nat,X2: set_nat] :
( ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) )
= ( ( finite1152437895449049373et_nat @ A2 )
& ( member_set_nat @ X2 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_541_GsCf,axiom,
ord_le6893508408891458716et_nat @ gs @ ( clique5033774636164728462irst_C @ k @ f ) ).
% GsCf
thf(fact_542_ACC__cf__mono,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique951075384711337423ACC_cf @ k @ Y4 ) ) ) ).
% ACC_cf_mono
thf(fact_543_finite___092_060F_062,axiom,
finite2115694454571419734at_nat @ ( clique2971579238625216137irst_F @ k ) ).
% finite_\<F>
thf(fact_544_v__mono,axiom,
! [G2: set_set_nat,H: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G2 @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_mono
thf(fact_545_S_I3_J,axiom,
( ( finite_card_set_nat @ s )
= p ) ).
% S(3)
thf(fact_546_ACC__cf__empty,axiom,
( ( clique951075384711337423ACC_cf @ k @ bot_bo7198184520161983622et_nat )
= bot_bot_set_nat_nat ) ).
% ACC_cf_empty
thf(fact_547_ACC__cf___092_060F_062,axiom,
! [X5: set_set_set_nat] : ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique2971579238625216137irst_F @ k ) ) ).
% ACC_cf_\<F>
thf(fact_548_NEG___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3210737375870294875st_NEG @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% NEG_\<G>
thf(fact_549_finite__imageI,axiom,
! [F2: set_nat,H2: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_550_finite__imageI,axiom,
! [F2: set_nat,H2: nat > set_nat] :
( ( finite_finite_nat @ F2 )
=> ( finite1152437895449049373et_nat @ ( image_nat_set_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_551_finite__imageI,axiom,
! [F2: set_set_nat,H2: set_nat > nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( finite_finite_nat @ ( image_set_nat_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_552_finite__imageI,axiom,
! [F2: set_nat,H2: nat > nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite2115694454571419734at_nat @ ( image_nat_nat_nat2 @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_553_finite__imageI,axiom,
! [F2: set_nat,H2: nat > set_set_nat] :
( ( finite_finite_nat @ F2 )
=> ( finite6739761609112101331et_nat @ ( image_2194112158459175443et_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_554_finite__imageI,axiom,
! [F2: set_set_nat,H2: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( finite1152437895449049373et_nat @ ( image_7916887816326733075et_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_555_finite__imageI,axiom,
! [F2: set_nat_nat,H2: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_556_finite__imageI,axiom,
! [F2: set_set_set_nat,H2: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( finite_finite_nat @ ( image_1454916318497077779at_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_557_finite__imageI,axiom,
! [F2: set_set_nat,H2: set_nat > nat > nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( finite2115694454571419734at_nat @ ( image_8569768528772619084at_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_558_finite__imageI,axiom,
! [F2: set_set_nat,H2: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( finite6739761609112101331et_nat @ ( image_6725021117256019401et_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_559_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_560_finite__insert,axiom,
! [A: set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( insert_set_nat @ A @ A2 ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ).
% finite_insert
thf(fact_561_finite__insert,axiom,
! [A: nat > nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ ( insert_nat_nat @ A @ A2 ) )
= ( finite2115694454571419734at_nat @ A2 ) ) ).
% finite_insert
thf(fact_562_finite__insert,axiom,
! [A: set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( insert_set_set_nat @ A @ A2 ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ).
% finite_insert
thf(fact_563_finite__Diff2,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_564_finite__Diff2,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) )
= ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_565_finite__Diff2,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) )
= ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_566_finite__Diff2,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
= ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_567_finite__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_568_finite__Diff,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_569_finite__Diff,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_570_finite__Diff,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_571_finite__ACC,axiom,
! [X5: set_set_set_nat] : ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) ) ).
% finite_ACC
thf(fact_572_i__props_I3_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( ( ti @ I2 )
= ( minus_minus_nat @ ( si2 @ I2 ) @ s2 ) ) ) ).
% i_props(3)
thf(fact_573_finite___092_060G_062,axiom,
finite6739761609112101331et_nat @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% finite_\<G>
thf(fact_574_first__assumptions_ONEG_Ocong,axiom,
clique3210737375870294875st_NEG = clique3210737375870294875st_NEG ).
% first_assumptions.NEG.cong
thf(fact_575_first__assumptions_OACC__cf_Ocong,axiom,
clique951075384711337423ACC_cf = clique951075384711337423ACC_cf ).
% first_assumptions.ACC_cf.cong
thf(fact_576_finite__has__minimal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ( ord_less_eq_nat_nat @ X @ A )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_577_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ X @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_578_finite__has__minimal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( ord_le6893508408891458716et_nat @ X @ A )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_579_finite__has__minimal2,axiom,
! [A2: set_set_set_set_nat,A: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( member2946998982187404937et_nat @ A @ A2 )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
& ( ord_le9131159989063066194et_nat @ X @ A )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_580_finite__has__minimal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
& ( ord_le9059583361652607317at_nat @ X @ A )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_581_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ X @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_582_finite__has__maximal2,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( member_nat_nat @ A @ A2 )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ( ord_less_eq_nat_nat @ A @ X )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_583_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ( ord_less_eq_set_nat @ A @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_584_finite__has__maximal2,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( member_set_set_nat @ A @ A2 )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ( ord_le6893508408891458716et_nat @ A @ X )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_585_finite__has__maximal2,axiom,
! [A2: set_set_set_set_nat,A: set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( member2946998982187404937et_nat @ A @ A2 )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
& ( ord_le9131159989063066194et_nat @ A @ X )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_586_finite__has__maximal2,axiom,
! [A2: set_set_nat_nat,A: set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( member_set_nat_nat @ A @ A2 )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
& ( ord_le9059583361652607317at_nat @ A @ X )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_587_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ A @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_588_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_589_all__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_set_nat_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A2 )
=> ( P @ ( image_set_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_590_all__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_nat_set_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_591_all__subset__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B4 @ A2 )
=> ( P @ ( image_1454916318497077779at_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_592_all__subset__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A2 )
=> ( P @ ( image_nat_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_593_all__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ A2 )
=> ( P @ ( image_7916887816326733075et_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_594_all__subset__image,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ! [B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B4 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_2194112158459175443et_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_595_all__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( P @ ( image_nat_nat_nat2 @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_596_all__subset__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ B4 @ A2 )
=> ( P @ ( image_5842784325960735177et_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_597_all__subset__image,axiom,
! [F: ( nat > nat ) > set_nat,A2: set_nat_nat,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ B4 @ ( image_7432509271690132940et_nat @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ B4 @ A2 )
=> ( P @ ( image_7432509271690132940et_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_598_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_599_rev__finite__subset,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_600_rev__finite__subset,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_601_rev__finite__subset,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_602_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_603_infinite__super,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_604_infinite__super,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ S @ T2 )
=> ( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_605_infinite__super,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ S @ T2 )
=> ( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_606_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_607_finite__subset,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_608_finite__subset,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( finite6739761609112101331et_nat @ B )
=> ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_609_finite__subset,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_610_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_611_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_612_infinite__imp__nonempty,axiom,
! [S: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S )
=> ( S != bot_bo7198184520161983622et_nat ) ) ).
% infinite_imp_nonempty
thf(fact_613_infinite__imp__nonempty,axiom,
! [S: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ S )
=> ( S != bot_bot_set_nat_nat ) ) ).
% infinite_imp_nonempty
thf(fact_614_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_615_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_616_finite_OemptyI,axiom,
finite6739761609112101331et_nat @ bot_bo7198184520161983622et_nat ).
% finite.emptyI
thf(fact_617_finite_OemptyI,axiom,
finite2115694454571419734at_nat @ bot_bot_set_nat_nat ).
% finite.emptyI
thf(fact_618_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_619_finite_OinsertI,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( finite1152437895449049373et_nat @ ( insert_set_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_620_finite_OinsertI,axiom,
! [A2: set_nat_nat,A: nat > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( finite2115694454571419734at_nat @ ( insert_nat_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_621_finite_OinsertI,axiom,
! [A2: set_set_set_nat,A: set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( finite6739761609112101331et_nat @ ( insert_set_set_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_622_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_623_Diff__infinite__finite,axiom,
! [T2: set_nat_nat,S: set_nat_nat] :
( ( finite2115694454571419734at_nat @ T2 )
=> ( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_624_Diff__infinite__finite,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ T2 )
=> ( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_625_Diff__infinite__finite,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( finite1152437895449049373et_nat @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_626_finite__has__minimal,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat_nat )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_627_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_628_finite__has__minimal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_629_finite__has__minimal,axiom,
! [A2: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( A2 != bot_bo193956671110832956et_nat )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_630_finite__has__minimal,axiom,
! [A2: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_631_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_632_finite__has__maximal,axiom,
! [A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat_nat )
=> ? [X: nat > nat] :
( ( member_nat_nat @ X @ A2 )
& ! [Xa: nat > nat] :
( ( member_nat_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_633_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_634_finite__has__maximal,axiom,
! [A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( A2 != bot_bo7198184520161983622et_nat )
=> ? [X: set_set_nat] :
( ( member_set_set_nat @ X @ A2 )
& ! [Xa: set_set_nat] :
( ( member_set_set_nat @ Xa @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_635_finite__has__maximal,axiom,
! [A2: set_set_set_set_nat] :
( ( finite5926941155766903689et_nat @ A2 )
=> ( ( A2 != bot_bo193956671110832956et_nat )
=> ? [X: set_set_set_nat] :
( ( member2946998982187404937et_nat @ X @ A2 )
& ! [Xa: set_set_set_nat] :
( ( member2946998982187404937et_nat @ Xa @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_636_finite__has__maximal,axiom,
! [A2: set_set_nat_nat] :
( ( finite3586981331298542604at_nat @ A2 )
=> ( ( A2 != bot_bo7376149671870096959at_nat )
=> ? [X: set_nat_nat] :
( ( member_set_nat_nat @ X @ A2 )
& ! [Xa: set_nat_nat] :
( ( member_set_nat_nat @ Xa @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_637_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_638_finite__surj,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_639_finite__surj,axiom,
! [A2: set_set_nat,B: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_640_finite__surj,axiom,
! [A2: set_nat,B: set_set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ( finite1152437895449049373et_nat @ B ) ) ) ).
% finite_surj
thf(fact_641_finite__surj,axiom,
! [A2: set_nat_nat,B: set_nat,F: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_642_finite__surj,axiom,
! [A2: set_set_set_nat,B: set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_643_finite__surj,axiom,
! [A2: set_set_nat,B: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( finite1152437895449049373et_nat @ B ) ) ) ).
% finite_surj
thf(fact_644_finite__surj,axiom,
! [A2: set_nat,B: set_set_set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( finite6739761609112101331et_nat @ B ) ) ) ).
% finite_surj
thf(fact_645_finite__surj,axiom,
! [A2: set_nat,B: set_nat_nat,F: nat > nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( finite2115694454571419734at_nat @ B ) ) ) ).
% finite_surj
thf(fact_646_finite__surj,axiom,
! [A2: set_nat_nat,B: set_set_nat,F: ( nat > nat ) > set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7432509271690132940et_nat @ F @ A2 ) )
=> ( finite1152437895449049373et_nat @ B ) ) ) ).
% finite_surj
thf(fact_647_finite__surj,axiom,
! [A2: set_set_set_nat,B: set_set_nat,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( finite1152437895449049373et_nat @ B ) ) ) ).
% finite_surj
thf(fact_648_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_649_finite__subset__image,axiom,
! [B: set_nat,F: set_nat > nat,A2: set_set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A2 ) )
=> ? [C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C4 @ A2 )
& ( finite1152437895449049373et_nat @ C4 )
& ( B
= ( image_set_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_650_finite__subset__image,axiom,
! [B: set_set_nat,F: nat > set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_set_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_651_finite__subset__image,axiom,
! [B: set_nat,F: set_set_nat > nat,A2: set_set_set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ? [C4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C4 @ A2 )
& ( finite6739761609112101331et_nat @ C4 )
& ( B
= ( image_1454916318497077779at_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_652_finite__subset__image,axiom,
! [B: set_nat,F: ( nat > nat ) > nat,A2: set_nat_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ? [C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C4 @ A2 )
& ( finite2115694454571419734at_nat @ C4 )
& ( B
= ( image_nat_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_653_finite__subset__image,axiom,
! [B: set_set_nat,F: set_nat > set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ? [C4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C4 @ A2 )
& ( finite1152437895449049373et_nat @ C4 )
& ( B
= ( image_7916887816326733075et_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_654_finite__subset__image,axiom,
! [B: set_set_set_nat,F: nat > set_set_nat,A2: set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_2194112158459175443et_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_655_finite__subset__image,axiom,
! [B: set_nat_nat,F: nat > nat > nat,A2: set_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat_nat2 @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_656_finite__subset__image,axiom,
! [B: set_set_nat,F: set_set_nat > set_nat,A2: set_set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ? [C4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C4 @ A2 )
& ( finite6739761609112101331et_nat @ C4 )
& ( B
= ( image_5842784325960735177et_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_657_finite__subset__image,axiom,
! [B: set_set_nat,F: ( nat > nat ) > set_nat,A2: set_nat_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7432509271690132940et_nat @ F @ A2 ) )
=> ? [C4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C4 @ A2 )
& ( finite2115694454571419734at_nat @ C4 )
& ( B
= ( image_7432509271690132940et_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_658_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 )
& ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_659_ex__finite__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_set_nat_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A2 )
& ( P @ ( image_set_nat_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_660_ex__finite__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ? [B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 )
& ( P @ ( image_nat_set_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_661_ex__finite__subset__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_1454916318497077779at_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ A2 )
& ( P @ ( image_1454916318497077779at_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_662_ex__finite__subset__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A2 )
& ( P @ ( image_nat_nat_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_663_ex__finite__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ? [B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ ( image_7916887816326733075et_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A2 )
& ( P @ ( image_7916887816326733075et_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_664_ex__finite__subset__image,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ? [B4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ ( image_2194112158459175443et_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 )
& ( P @ ( image_2194112158459175443et_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_665_ex__finite__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ? [B4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ ( image_nat_nat_nat2 @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 )
& ( P @ ( image_nat_nat_nat2 @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_666_ex__finite__subset__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ? [B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ ( image_5842784325960735177et_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ A2 )
& ( P @ ( image_5842784325960735177et_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_667_ex__finite__subset__image,axiom,
! [F: ( nat > nat ) > set_nat,A2: set_nat_nat,P: set_set_nat > $o] :
( ( ? [B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ ( image_7432509271690132940et_nat @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A2 )
& ( P @ ( image_7432509271690132940et_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_668_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_669_all__finite__subset__image,axiom,
! [F: set_nat > nat,A2: set_set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_set_nat_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A2 ) )
=> ( P @ ( image_set_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_670_all__finite__subset__image,axiom,
! [F: nat > set_nat,A2: set_nat,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ ( image_nat_set_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) )
=> ( P @ ( image_nat_set_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_671_all__finite__subset__image,axiom,
! [F: set_set_nat > nat,A2: set_set_set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_1454916318497077779at_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ A2 ) )
=> ( P @ ( image_1454916318497077779at_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_672_all__finite__subset__image,axiom,
! [F: ( nat > nat ) > nat,A2: set_nat_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A2 ) )
=> ( P @ ( image_nat_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_673_all__finite__subset__image,axiom,
! [F: set_nat > set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ ( image_7916887816326733075et_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ A2 ) )
=> ( P @ ( image_7916887816326733075et_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_674_all__finite__subset__image,axiom,
! [F: nat > set_set_nat,A2: set_nat,P: set_set_set_nat > $o] :
( ( ! [B4: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ ( image_2194112158459175443et_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) )
=> ( P @ ( image_2194112158459175443et_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_675_all__finite__subset__image,axiom,
! [F: nat > nat > nat,A2: set_nat,P: set_nat_nat > $o] :
( ( ! [B4: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ ( image_nat_nat_nat2 @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A2 ) )
=> ( P @ ( image_nat_nat_nat2 @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_676_all__finite__subset__image,axiom,
! [F: set_set_nat > set_nat,A2: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ ( image_5842784325960735177et_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ B4 )
& ( ord_le9131159989063066194et_nat @ B4 @ A2 ) )
=> ( P @ ( image_5842784325960735177et_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_677_all__finite__subset__image,axiom,
! [F: ( nat > nat ) > set_nat,A2: set_nat_nat,P: set_set_nat > $o] :
( ( ! [B4: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ B4 )
& ( ord_le6893508408891458716et_nat @ B4 @ ( image_7432509271690132940et_nat @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ B4 )
& ( ord_le9059583361652607317at_nat @ B4 @ A2 ) )
=> ( P @ ( image_7432509271690132940et_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_678_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_679_infinite__finite__induct,axiom,
! [P: set_set_nat > $o,A2: set_set_nat] :
( ! [A4: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [X: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ~ ( member_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_680_infinite__finite__induct,axiom,
! [P: set_set_set_nat > $o,A2: set_set_set_nat] :
( ! [A4: set_set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ~ ( member_set_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_681_infinite__finite__induct,axiom,
! [P: set_nat_nat > $o,A2: set_nat_nat] :
( ! [A4: set_nat_nat] :
( ~ ( finite2115694454571419734at_nat @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_nat_nat )
=> ( ! [X: nat > nat,F3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ F3 )
=> ( ~ ( member_nat_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_nat @ X @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_682_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X: nat] : ( P @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_683_finite__ne__induct,axiom,
! [F2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( F2 != bot_bot_set_set_nat )
=> ( ! [X: set_nat] : ( P @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
=> ( ! [X: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( F3 != bot_bot_set_set_nat )
=> ( ~ ( member_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_684_finite__ne__induct,axiom,
! [F2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( ( F2 != bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat] : ( P @ ( insert_set_set_nat @ X @ bot_bo7198184520161983622et_nat ) )
=> ( ! [X: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ( F3 != bot_bo7198184520161983622et_nat )
=> ( ~ ( member_set_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_685_finite__ne__induct,axiom,
! [F2: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat_nat )
=> ( ! [X: nat > nat] : ( P @ ( insert_nat_nat @ X @ bot_bot_set_nat_nat ) )
=> ( ! [X: nat > nat,F3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat_nat )
=> ( ~ ( member_nat_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_nat @ X @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_686_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_687_finite__induct,axiom,
! [F2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [X: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ~ ( member_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_688_finite__induct,axiom,
! [F2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [X: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ~ ( member_set_set_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_689_finite__induct,axiom,
! [F2: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ F2 )
=> ( ( P @ bot_bot_set_nat_nat )
=> ( ! [X: nat > nat,F3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ F3 )
=> ( ~ ( member_nat_nat @ X @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_nat @ X @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_690_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A5
= ( insert_nat @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_691_finite_Osimps,axiom,
( finite1152437895449049373et_nat
= ( ^ [A5: set_set_nat] :
( ( A5 = bot_bot_set_set_nat )
| ? [A3: set_set_nat,B5: set_nat] :
( ( A5
= ( insert_set_nat @ B5 @ A3 ) )
& ( finite1152437895449049373et_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_692_finite_Osimps,axiom,
( finite6739761609112101331et_nat
= ( ^ [A5: set_set_set_nat] :
( ( A5 = bot_bo7198184520161983622et_nat )
| ? [A3: set_set_set_nat,B5: set_set_nat] :
( ( A5
= ( insert_set_set_nat @ B5 @ A3 ) )
& ( finite6739761609112101331et_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_693_finite_Osimps,axiom,
( finite2115694454571419734at_nat
= ( ^ [A5: set_nat_nat] :
( ( A5 = bot_bot_set_nat_nat )
| ? [A3: set_nat_nat,B5: nat > nat] :
( ( A5
= ( insert_nat_nat @ B5 @ A3 ) )
& ( finite2115694454571419734at_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_694_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A4: set_nat] :
( ? [A6: nat] :
( A
= ( insert_nat @ A6 @ A4 ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ) ).
% finite.cases
thf(fact_695_finite_Ocases,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ~ ! [A4: set_set_nat] :
( ? [A6: set_nat] :
( A
= ( insert_set_nat @ A6 @ A4 ) )
=> ~ ( finite1152437895449049373et_nat @ A4 ) ) ) ) ).
% finite.cases
thf(fact_696_finite_Ocases,axiom,
! [A: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A )
=> ( ( A != bot_bo7198184520161983622et_nat )
=> ~ ! [A4: set_set_set_nat] :
( ? [A6: set_set_nat] :
( A
= ( insert_set_set_nat @ A6 @ A4 ) )
=> ~ ( finite6739761609112101331et_nat @ A4 ) ) ) ) ).
% finite.cases
thf(fact_697_finite_Ocases,axiom,
! [A: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A )
=> ( ( A != bot_bot_set_nat_nat )
=> ~ ! [A4: set_nat_nat] :
( ? [A6: nat > nat] :
( A
= ( insert_nat_nat @ A6 @ A4 ) )
=> ~ ( finite2115694454571419734at_nat @ A4 ) ) ) ) ).
% finite.cases
thf(fact_698_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ( finite_card_nat @ B3 )
= N2 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_699_infinite__arbitrarily__large,axiom,
! [A2: set_set_nat,N2: nat] :
( ~ ( finite1152437895449049373et_nat @ A2 )
=> ? [B3: set_set_nat] :
( ( finite1152437895449049373et_nat @ B3 )
& ( ( finite_card_set_nat @ B3 )
= N2 )
& ( ord_le6893508408891458716et_nat @ B3 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_700_infinite__arbitrarily__large,axiom,
! [A2: set_set_set_nat,N2: nat] :
( ~ ( finite6739761609112101331et_nat @ A2 )
=> ? [B3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B3 )
& ( ( finite1149291290879098388et_nat @ B3 )
= N2 )
& ( ord_le9131159989063066194et_nat @ B3 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_701_infinite__arbitrarily__large,axiom,
! [A2: set_nat_nat,N2: nat] :
( ~ ( finite2115694454571419734at_nat @ A2 )
=> ? [B3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B3 )
& ( ( finite_card_nat_nat @ B3 )
= N2 )
& ( ord_le9059583361652607317at_nat @ B3 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_702_card__subset__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_703_card__subset__eq,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ( finite_card_set_nat @ A2 )
= ( finite_card_set_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_704_card__subset__eq,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( ( finite1149291290879098388et_nat @ A2 )
= ( finite1149291290879098388et_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_705_card__subset__eq,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ( finite_card_nat_nat @ A2 )
= ( finite_card_nat_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_706_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_707_finite__subset__induct_H,axiom,
! [F2: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( ord_le6893508408891458716et_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [A6: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( member_set_nat @ A6 @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
=> ( ~ ( member_set_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_708_finite__subset__induct_H,axiom,
! [F2: set_set_set_nat,A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( ( ord_le9131159989063066194et_nat @ F2 @ A2 )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [A6: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ( member_set_set_nat @ A6 @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ F3 @ A2 )
=> ( ~ ( member_set_set_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_709_finite__subset__induct_H,axiom,
! [F2: set_nat_nat,A2: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ F2 )
=> ( ( ord_le9059583361652607317at_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_nat )
=> ( ! [A6: nat > nat,F3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ F3 )
=> ( ( member_nat_nat @ A6 @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ F3 @ A2 )
=> ( ~ ( member_nat_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_nat @ A6 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_710_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A6 @ A2 )
=> ( ~ ( member_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_711_finite__subset__induct,axiom,
! [F2: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( ord_le6893508408891458716et_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [A6: set_nat,F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( member_set_nat @ A6 @ A2 )
=> ( ~ ( member_set_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_712_finite__subset__induct,axiom,
! [F2: set_set_set_nat,A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ F2 )
=> ( ( ord_le9131159989063066194et_nat @ F2 @ A2 )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [A6: set_set_nat,F3: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ F3 )
=> ( ( member_set_set_nat @ A6 @ A2 )
=> ( ~ ( member_set_set_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_set_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_713_finite__subset__induct,axiom,
! [F2: set_nat_nat,A2: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ F2 )
=> ( ( ord_le9059583361652607317at_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_nat )
=> ( ! [A6: nat > nat,F3: set_nat_nat] :
( ( finite2115694454571419734at_nat @ F3 )
=> ( ( member_nat_nat @ A6 @ A2 )
=> ( ~ ( member_nat_nat @ A6 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_nat @ A6 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_714_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: nat,A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A6 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_715_finite__empty__induct,axiom,
! [A2: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: nat > nat,A4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A4 )
=> ( ( member_nat_nat @ A6 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_8121590178497047118at_nat @ A4 @ ( insert_nat_nat @ A6 @ bot_bot_set_nat_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_716_finite__empty__induct,axiom,
! [A2: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: set_set_nat,A4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A4 )
=> ( ( member_set_set_nat @ A6 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_2447799839930672331et_nat @ A4 @ ( insert_set_set_nat @ A6 @ bot_bo7198184520161983622et_nat ) ) ) ) ) )
=> ( P @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_717_finite__empty__induct,axiom,
! [A2: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A6: set_nat,A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A6 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ A6 @ bot_bot_set_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_718_infinite__coinduct,axiom,
! [X5: set_nat > $o,A2: set_nat] :
( ( X5 @ A2 )
=> ( ! [A4: set_nat] :
( ( X5 @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( X5 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_719_infinite__coinduct,axiom,
! [X5: set_nat_nat > $o,A2: set_nat_nat] :
( ( X5 @ A2 )
=> ( ! [A4: set_nat_nat] :
( ( X5 @ A4 )
=> ? [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A4 )
& ( ( X5 @ ( minus_8121590178497047118at_nat @ A4 @ ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) )
| ~ ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ A4 @ ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) ) ) ) )
=> ~ ( finite2115694454571419734at_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_720_infinite__coinduct,axiom,
! [X5: set_set_set_nat > $o,A2: set_set_set_nat] :
( ( X5 @ A2 )
=> ( ! [A4: set_set_set_nat] :
( ( X5 @ A4 )
=> ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A4 )
& ( ( X5 @ ( minus_2447799839930672331et_nat @ A4 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) )
| ~ ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ A4 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ) ) ) )
=> ~ ( finite6739761609112101331et_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_721_infinite__coinduct,axiom,
! [X5: set_set_nat > $o,A2: set_set_nat] :
( ( X5 @ A2 )
=> ( ! [A4: set_set_nat] :
( ( X5 @ A4 )
=> ? [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
& ( ( X5 @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) )
| ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ) ) )
=> ~ ( finite1152437895449049373et_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_722_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_723_infinite__remove,axiom,
! [S: set_nat_nat,A: nat > nat] :
( ~ ( finite2115694454571419734at_nat @ S )
=> ~ ( finite2115694454571419734at_nat @ ( minus_8121590178497047118at_nat @ S @ ( insert_nat_nat @ A @ bot_bot_set_nat_nat ) ) ) ) ).
% infinite_remove
thf(fact_724_infinite__remove,axiom,
! [S: set_set_set_nat,A: set_set_nat] :
( ~ ( finite6739761609112101331et_nat @ S )
=> ~ ( finite6739761609112101331et_nat @ ( minus_2447799839930672331et_nat @ S @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% infinite_remove
thf(fact_725_infinite__remove,axiom,
! [S: set_set_nat,A: set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_726_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X: nat] :
( A2
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_727_card__1__singletonE,axiom,
! [A2: set_set_nat] :
( ( ( finite_card_set_nat @ A2 )
= one_one_nat )
=> ~ ! [X: set_nat] :
( A2
!= ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_728_card__1__singletonE,axiom,
! [A2: set_set_set_nat] :
( ( ( finite1149291290879098388et_nat @ A2 )
= one_one_nat )
=> ~ ! [X: set_set_nat] :
( A2
!= ( insert_set_set_nat @ X @ bot_bo7198184520161983622et_nat ) ) ) ).
% card_1_singletonE
thf(fact_729_card__1__singletonE,axiom,
! [A2: set_nat_nat] :
( ( ( finite_card_nat_nat @ A2 )
= one_one_nat )
=> ~ ! [X: nat > nat] :
( A2
!= ( insert_nat_nat @ X @ bot_bot_set_nat_nat ) ) ) ).
% card_1_singletonE
thf(fact_730_card__less__sym__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_731_card__less__sym__Diff,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_less_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B ) )
=> ( ord_less_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_732_card__less__sym__Diff,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_less_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B ) )
=> ( ord_less_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_733_card__less__sym__Diff,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B ) )
=> ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_734_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( P @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_735_finite__remove__induct,axiom,
! [B: set_set_nat,P: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( P @ bot_bot_set_set_nat )
=> ( ! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( P @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_736_finite__remove__induct,axiom,
! [B: set_set_set_nat,P: set_set_set_nat > $o] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( P @ bot_bo7198184520161983622et_nat )
=> ( ! [A4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A4 )
=> ( ( A4 != bot_bo7198184520161983622et_nat )
=> ( ( ord_le9131159989063066194et_nat @ A4 @ B )
=> ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A4 )
=> ( P @ ( minus_2447799839930672331et_nat @ A4 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_737_finite__remove__induct,axiom,
! [B: set_nat_nat,P: set_nat_nat > $o] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( P @ bot_bot_set_nat_nat )
=> ( ! [A4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat_nat )
=> ( ( ord_le9059583361652607317at_nat @ A4 @ B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A4 )
=> ( P @ ( minus_8121590178497047118at_nat @ A4 @ ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_738_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( P @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_739_remove__induct,axiom,
! [P: set_set_nat > $o,B: set_set_nat] :
( ( P @ bot_bot_set_set_nat )
=> ( ( ~ ( finite1152437895449049373et_nat @ B )
=> ( P @ B ) )
=> ( ! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A4 )
=> ( P @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_740_remove__induct,axiom,
! [P: set_set_set_nat > $o,B: set_set_set_nat] :
( ( P @ bot_bo7198184520161983622et_nat )
=> ( ( ~ ( finite6739761609112101331et_nat @ B )
=> ( P @ B ) )
=> ( ! [A4: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A4 )
=> ( ( A4 != bot_bo7198184520161983622et_nat )
=> ( ( ord_le9131159989063066194et_nat @ A4 @ B )
=> ( ! [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ A4 )
=> ( P @ ( minus_2447799839930672331et_nat @ A4 @ ( insert_set_set_nat @ X4 @ bot_bo7198184520161983622et_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_741_remove__induct,axiom,
! [P: set_nat_nat > $o,B: set_nat_nat] :
( ( P @ bot_bot_set_nat_nat )
=> ( ( ~ ( finite2115694454571419734at_nat @ B )
=> ( P @ B ) )
=> ( ! [A4: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat_nat )
=> ( ( ord_le9059583361652607317at_nat @ A4 @ B )
=> ( ! [X4: nat > nat] :
( ( member_nat_nat @ X4 @ A4 )
=> ( P @ ( minus_8121590178497047118at_nat @ A4 @ ( insert_nat_nat @ X4 @ bot_bot_set_nat_nat ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_742_card__Diff__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_743_card__Diff__subset,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_744_card__Diff__subset,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ B @ A2 )
=> ( ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_745_card__Diff__subset,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ B @ A2 )
=> ( ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_746_odot___092_060G_062,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% odot_\<G>
thf(fact_747__092_060K_062___092_060G_062,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% \<K>_\<G>
thf(fact_748_accepts__def,axiom,
( clique3686358387679108662ccepts
= ( ^ [X6: set_set_set_nat,G3: set_set_nat] :
? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X6 )
& ( ord_le6893508408891458716et_nat @ X3 @ G3 ) ) ) ) ).
% accepts_def
thf(fact_749_v___092_060G_062__2,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ G2 ) ) ) ) ).
% v_\<G>_2
thf(fact_750_POS__sub__CLIQUE,axiom,
ord_le9131159989063066194et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_sub_CLIQUE
thf(fact_751_i__props_I4_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( ord_less_eq_nat @ s2 @ ( si2 @ I2 ) ) ) ).
% i_props(4)
thf(fact_752__092_060open_062v__gs_A_123Gs_125_A_061_A_123v_AGs_125_092_060close_062,axiom,
( ( clique8462013130872731469t_v_gs @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) )
= ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ).
% \<open>v_gs {Gs} = {v Gs}\<close>
thf(fact_753_finite__v__gs,axiom,
! [X5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ).
% finite_v_gs
thf(fact_754_v__gs__def,axiom,
( clique8462013130872731469t_v_gs
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v ) ) ).
% v_gs_def
thf(fact_755_v__sameprod__subset,axiom,
! [Vs: set_nat] : ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ ( clique6722202388162463298od_nat @ Vs @ Vs ) ) @ Vs ) ).
% v_sameprod_subset
thf(fact_756_v__gs__mono,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y4 ) ) ) ).
% v_gs_mono
thf(fact_757_Gs__def,axiom,
( gs
= ( clique6722202388162463298od_nat @ vs @ vs ) ) ).
% Gs_def
thf(fact_758_rq,axiom,
ord_less_eq_nat @ p @ r ).
% rq
thf(fact_759_acceptsI,axiom,
! [D2: set_set_nat,G2: set_set_nat,X5: set_set_set_nat] :
( ( ord_le6893508408891458716et_nat @ D2 @ G2 )
=> ( ( member_set_set_nat @ D2 @ X5 )
=> ( clique3686358387679108662ccepts @ X5 @ G2 ) ) ) ).
% acceptsI
thf(fact_760_finite__numbers2,axiom,
! [N2: nat] : ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N2 ) @ ( clique3652268606331196573umbers @ N2 ) ) ) ).
% finite_numbers2
thf(fact_761_v__gs__empty,axiom,
( ( clique8462013130872731469t_v_gs @ bot_bo7198184520161983622et_nat )
= bot_bot_set_set_nat ) ).
% v_gs_empty
thf(fact_762_ACC__cf__I,axiom,
! [F2: nat > nat,X5: set_set_set_nat] :
( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ k ) )
=> ( ( clique3686358387679108662ccepts @ X5 @ ( clique5033774636164728462irst_C @ k @ F2 ) )
=> ( member_nat_nat @ F2 @ ( clique951075384711337423ACC_cf @ k @ X5 ) ) ) ) ).
% ACC_cf_I
thf(fact_763_first__assumptions_O_092_060K_062_Ocong,axiom,
clique3326749438856946062irst_K = clique3326749438856946062irst_K ).
% first_assumptions.\<K>.cong
thf(fact_764_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_765_card__insert__le,axiom,
! [A2: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_766_card__insert__le,axiom,
! [A2: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_767_card__insert__le,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] : ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ ( insert_set_set_nat @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_768_card__image__le,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_769_card__image__le,axiom,
! [A2: set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_nat_set_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_770_card__image__le,axiom,
! [A2: set_set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_set_nat_nat @ F @ A2 ) ) @ ( finite_card_set_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_771_card__image__le,axiom,
! [A2: set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( image_2194112158459175443et_nat @ F @ A2 ) ) @ ( finite_card_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_772_card__image__le,axiom,
! [A2: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_7916887816326733075et_nat @ F @ A2 ) ) @ ( finite_card_set_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_773_card__image__le,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat_nat @ F @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_774_card__image__le,axiom,
! [A2: set_set_set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_1454916318497077779at_nat @ F @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_775_card__image__le,axiom,
! [A2: set_set_nat,F: set_nat > set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( image_6725021117256019401et_nat @ F @ A2 ) ) @ ( finite_card_set_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_776_card__image__le,axiom,
! [A2: set_nat_nat,F: ( nat > nat ) > set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_7432509271690132940et_nat @ F @ A2 ) ) @ ( finite_card_nat_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_777_card__image__le,axiom,
! [A2: set_set_set_nat,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( image_5842784325960735177et_nat @ F @ A2 ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ).
% card_image_le
thf(fact_778_card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_779_card__mono,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B ) ) ) ) ).
% card_mono
thf(fact_780_card__mono,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B ) ) ) ) ).
% card_mono
thf(fact_781_card__mono,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B ) ) ) ) ).
% card_mono
thf(fact_782_card__seteq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_783_card__seteq,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_set_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_784_card__seteq,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B ) @ ( finite1149291290879098388et_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_785_card__seteq,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_nat @ B ) @ ( finite_card_nat_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_786_exists__subset__between,axiom,
! [A2: set_nat,N2: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ C3 )
& ( ( finite_card_nat @ B3 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_787_exists__subset__between,axiom,
! [A2: set_set_nat,N2: nat,C3: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_set_nat @ C3 ) )
=> ( ( ord_le6893508408891458716et_nat @ A2 @ C3 )
=> ( ( finite1152437895449049373et_nat @ C3 )
=> ? [B3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
& ( ord_le6893508408891458716et_nat @ B3 @ C3 )
& ( ( finite_card_set_nat @ B3 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_788_exists__subset__between,axiom,
! [A2: set_set_set_nat,N2: nat,C3: set_set_set_nat] :
( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite1149291290879098388et_nat @ C3 ) )
=> ( ( ord_le9131159989063066194et_nat @ A2 @ C3 )
=> ( ( finite6739761609112101331et_nat @ C3 )
=> ? [B3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ B3 )
& ( ord_le9131159989063066194et_nat @ B3 @ C3 )
& ( ( finite1149291290879098388et_nat @ B3 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_789_exists__subset__between,axiom,
! [A2: set_nat_nat,N2: nat,C3: set_nat_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat_nat @ C3 ) )
=> ( ( ord_le9059583361652607317at_nat @ A2 @ C3 )
=> ( ( finite2115694454571419734at_nat @ C3 )
=> ? [B3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ A2 @ B3 )
& ( ord_le9059583361652607317at_nat @ B3 @ C3 )
& ( ( finite_card_nat_nat @ B3 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_790_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_791_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_set_nat @ S ) )
=> ~ ! [T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T3 @ S )
=> ( ( ( finite_card_set_nat @ T3 )
= N2 )
=> ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_792_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_set_set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite1149291290879098388et_nat @ S ) )
=> ~ ! [T3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ T3 @ S )
=> ( ( ( finite1149291290879098388et_nat @ T3 )
= N2 )
=> ~ ( finite6739761609112101331et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_793_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat_nat @ S ) )
=> ~ ! [T3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ T3 @ S )
=> ( ( ( finite_card_nat_nat @ T3 )
= N2 )
=> ~ ( finite2115694454571419734at_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_794_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C3: nat] :
( ! [G4: set_nat] :
( ( ord_less_eq_set_nat @ G4 @ F2 )
=> ( ( finite_finite_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G4 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_795_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_nat,C3: nat] :
( ! [G4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G4 @ F2 )
=> ( ( finite1152437895449049373et_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G4 ) @ C3 ) ) )
=> ( ( finite1152437895449049373et_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_796_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_set_nat,C3: nat] :
( ! [G4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ G4 @ F2 )
=> ( ( finite6739761609112101331et_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ G4 ) @ C3 ) ) )
=> ( ( finite6739761609112101331et_nat @ F2 )
& ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_797_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat_nat,C3: nat] :
( ! [G4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ G4 @ F2 )
=> ( ( finite2115694454571419734at_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ G4 ) @ C3 ) ) )
=> ( ( finite2115694454571419734at_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_798_card__le__sym__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_799_card__le__sym__Diff,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( finite2115694454571419734at_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_800_card__le__sym__Diff,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( finite6739761609112101331et_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_801_card__le__sym__Diff,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_802_surj__card__le,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_803_surj__card__le,axiom,
! [A2: set_set_nat,B: set_nat,F: set_nat > nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_set_nat_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_804_surj__card__le,axiom,
! [A2: set_nat,B: set_set_nat,F: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_nat_set_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_805_surj__card__le,axiom,
! [A2: set_nat_nat,B: set_nat,F: ( nat > nat ) > nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_806_surj__card__le,axiom,
! [A2: set_set_set_nat,B: set_nat,F: set_set_nat > nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_1454916318497077779at_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_807_surj__card__le,axiom,
! [A2: set_set_nat,B: set_set_nat,F: set_nat > set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7916887816326733075et_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_set_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_808_surj__card__le,axiom,
! [A2: set_nat,B: set_set_set_nat,F: nat > set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9131159989063066194et_nat @ B @ ( image_2194112158459175443et_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ B ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_809_surj__card__le,axiom,
! [A2: set_nat,B: set_nat_nat,F: nat > nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_le9059583361652607317at_nat @ B @ ( image_nat_nat_nat2 @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat_nat @ B ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_810_surj__card__le,axiom,
! [A2: set_nat_nat,B: set_set_nat,F: ( nat > nat ) > set_nat] :
( ( finite2115694454571419734at_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_7432509271690132940et_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite_card_nat_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_811_surj__card__le,axiom,
! [A2: set_set_set_nat,B: set_set_nat,F: set_set_nat > set_nat] :
( ( finite6739761609112101331et_nat @ A2 )
=> ( ( ord_le6893508408891458716et_nat @ B @ ( image_5842784325960735177et_nat @ F @ A2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ B ) @ ( finite1149291290879098388et_nat @ A2 ) ) ) ) ).
% surj_card_le
thf(fact_812_card__Diff1__le,axiom,
! [A2: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_813_card__Diff1__le,axiom,
! [A2: set_nat_nat,X2: nat > nat] : ( ord_less_eq_nat @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ ( insert_nat_nat @ X2 @ bot_bot_set_nat_nat ) ) ) @ ( finite_card_nat_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_814_card__Diff1__le,axiom,
! [A2: set_set_set_nat,X2: set_set_nat] : ( ord_less_eq_nat @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ ( insert_set_set_nat @ X2 @ bot_bo7198184520161983622et_nat ) ) ) @ ( finite1149291290879098388et_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_815_card__Diff1__le,axiom,
! [A2: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_816_diff__card__le__card__Diff,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_817_diff__card__le__card__Diff,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat_nat @ A2 ) @ ( finite_card_nat_nat @ B ) ) @ ( finite_card_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_818_diff__card__le__card__Diff,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite1149291290879098388et_nat @ A2 ) @ ( finite1149291290879098388et_nat @ B ) ) @ ( finite1149291290879098388et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_819_diff__card__le__card__Diff,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A2 ) @ ( finite_card_set_nat @ B ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_820_sameprod__mono,axiom,
! [X5: set_set_nat,Y4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ X5 @ Y4 )
=> ( ord_le9131159989063066194et_nat @ ( clique8906516429304539640et_nat @ X5 @ X5 ) @ ( clique8906516429304539640et_nat @ Y4 @ Y4 ) ) ) ).
% sameprod_mono
thf(fact_821_sameprod__mono,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ Y4 )
=> ( ord_le572741076514265352et_nat @ ( clique1181040904276305582et_nat @ X5 @ X5 ) @ ( clique1181040904276305582et_nat @ Y4 @ Y4 ) ) ) ).
% sameprod_mono
thf(fact_822_sameprod__mono,axiom,
! [X5: set_nat_nat,Y4: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ X5 @ Y4 )
=> ( ord_le4954213926817602059at_nat @ ( clique134924887794942129at_nat @ X5 @ X5 ) @ ( clique134924887794942129at_nat @ Y4 @ Y4 ) ) ) ).
% sameprod_mono
thf(fact_823_sameprod__mono,axiom,
! [X5: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( clique6722202388162463298od_nat @ X5 @ X5 ) @ ( clique6722202388162463298od_nat @ Y4 @ Y4 ) ) ) ).
% sameprod_mono
thf(fact_824_sameprod__finite,axiom,
! [X5: set_set_nat] :
( ( finite1152437895449049373et_nat @ X5 )
=> ( finite6739761609112101331et_nat @ ( clique8906516429304539640et_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_825_sameprod__finite,axiom,
! [X5: set_nat_nat] :
( ( finite2115694454571419734at_nat @ X5 )
=> ( finite3586981331298542604at_nat @ ( clique134924887794942129at_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_826_sameprod__finite,axiom,
! [X5: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ X5 )
=> ( finite5926941155766903689et_nat @ ( clique1181040904276305582et_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_827_sameprod__finite,axiom,
! [X5: set_nat] :
( ( finite_finite_nat @ X5 )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ X5 @ X5 ) ) ) ).
% sameprod_finite
thf(fact_828_uw_I3_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( member_set_nat @ ( pair @ I2 ) @ ( inf_inf_set_set_nat @ ( g @ I2 ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% uw(3)
thf(fact_829_card__v__gs__join,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat,Z3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z3 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y4 ) ) ) ) ) ) ) ).
% card_v_gs_join
thf(fact_830_uwi_I3_J,axiom,
member_set_nat @ ( pair @ i ) @ ( inf_inf_set_set_nat @ ( g @ i ) @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ).
% uwi(3)
thf(fact_831_diff__diff__cancel,axiom,
! [I2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_832_fake,axiom,
member_nat_nat @ f @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) @ ( clique951075384711337423ACC_cf @ k @ u ) ) ).
% fake
thf(fact_833_CLIQUE__NEG,axiom,
( ( inf_in5711780100303410308et_nat @ ( clique363107459185959606CLIQUE @ k ) @ ( clique3210737375870294875st_NEG @ k ) )
= bot_bo7198184520161983622et_nat ) ).
% CLIQUE_NEG
thf(fact_834_finite__POS__NEG,axiom,
finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737375870294875st_NEG @ k ) ) ).
% finite_POS_NEG
thf(fact_835_Unempty,axiom,
u != bot_bo7198184520161983622et_nat ).
% Unempty
thf(fact_836_finU,axiom,
finite6739761609112101331et_nat @ u ).
% finU
thf(fact_837_IntI,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_838_IntI,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_839_IntI,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A2 )
=> ( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_840_IntI,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_841_Int__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_842_Int__iff,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) )
= ( ( member_set_nat @ C @ A2 )
& ( member_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_843_Int__iff,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
= ( ( member_set_set_nat @ C @ A2 )
& ( member_set_set_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_844_Int__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) )
= ( ( member_nat_nat @ C @ A2 )
& ( member_nat_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_845_UnCI,axiom,
! [C: set_set_nat,B: set_set_set_nat,A2: set_set_set_nat] :
( ( ~ ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ A2 ) )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_846_UnCI,axiom,
! [C: set_nat,B: set_set_nat,A2: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ A2 ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_847_UnCI,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_848_UnCI,axiom,
! [C: nat > nat,B: set_nat_nat,A2: set_nat_nat] :
( ( ~ ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ A2 ) )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_849_Un__iff,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
= ( ( member_set_set_nat @ C @ A2 )
| ( member_set_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_850_Un__iff,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( ( member_set_nat @ C @ A2 )
| ( member_set_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_851_Un__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_852_Un__iff,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) )
= ( ( member_nat_nat @ C @ A2 )
| ( member_nat_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_853_vplus__dsU,axiom,
( ( clique8462013130872731469t_v_gs @ u )
= s ) ).
% vplus_dsU
thf(fact_854_r__def,axiom,
( r
= ( finite1149291290879098388et_nat @ u ) ) ).
% r_def
thf(fact_855__092_060open_062card_A_Iv__gs_AU_J_A_061_Acard_AS_092_060close_062,axiom,
( ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ u ) )
= ( finite_card_set_nat @ s ) ) ).
% \<open>card (v_gs U) = card S\<close>
thf(fact_856_G_I3_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( member_set_set_nat @ ( g @ I2 ) @ u ) ) ).
% G(3)
thf(fact_857__092_060open_062f_A_092_060notin_062_AACC__cf_AU_092_060close_062,axiom,
~ ( member_nat_nat @ f @ ( clique951075384711337423ACC_cf @ k @ u ) ) ).
% \<open>f \<notin> ACC_cf U\<close>
thf(fact_858_UCf,axiom,
! [D2: set_set_nat] :
( ( member_set_set_nat @ D2 @ u )
=> ~ ( ord_le6893508408891458716et_nat @ D2 @ ( clique5033774636164728462irst_C @ k @ f ) ) ) ).
% UCf
thf(fact_859__092_060open_062_092_060not_062_A_I_092_060exists_062D_092_060in_062U_O_AD_A_092_060subseteq_062_AC_Af_J_092_060close_062,axiom,
~ ? [X4: set_set_nat] :
( ( member_set_set_nat @ X4 @ u )
& ( ord_le6893508408891458716et_nat @ X4 @ ( clique5033774636164728462irst_C @ k @ f ) ) ) ).
% \<open>\<not> (\<exists>D\<in>U. D \<subseteq> C f)\<close>
thf(fact_860__092_060open_062_092_060not_062_AU_A_092_060tturnstile_062_AC_Af_092_060close_062,axiom,
~ ( clique3686358387679108662ccepts @ u @ ( clique5033774636164728462irst_C @ k @ f ) ) ).
% \<open>\<not> U \<tturnstile> C f\<close>
thf(fact_861__092_060open_062U_A_092_060subseteq_062_A_092_060G_062_092_060close_062,axiom,
ord_le9131159989063066194et_nat @ u @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% \<open>U \<subseteq> \<G>\<close>
thf(fact_862_Int__subset__iff,axiom,
! [C3: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( ord_less_eq_set_nat @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_863_Int__subset__iff,axiom,
! [C3: set_set_nat,A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ C3 @ ( inf_inf_set_set_nat @ A2 @ B ) )
= ( ( ord_le6893508408891458716et_nat @ C3 @ A2 )
& ( ord_le6893508408891458716et_nat @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_864_Int__subset__iff,axiom,
! [C3: set_set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ C3 @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
= ( ( ord_le9131159989063066194et_nat @ C3 @ A2 )
& ( ord_le9131159989063066194et_nat @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_865_Int__subset__iff,axiom,
! [C3: set_nat_nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ C3 @ ( inf_inf_set_nat_nat @ A2 @ B ) )
= ( ( ord_le9059583361652607317at_nat @ C3 @ A2 )
& ( ord_le9059583361652607317at_nat @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_866_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_867_finite__Int,axiom,
! [F2: set_set_nat,G2: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ F2 )
| ( finite1152437895449049373et_nat @ G2 ) )
=> ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_868_finite__Int,axiom,
! [F2: set_set_set_nat,G2: set_set_set_nat] :
( ( ( finite6739761609112101331et_nat @ F2 )
| ( finite6739761609112101331et_nat @ G2 ) )
=> ( finite6739761609112101331et_nat @ ( inf_in5711780100303410308et_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_869_finite__Int,axiom,
! [F2: set_nat_nat,G2: set_nat_nat] :
( ( ( finite2115694454571419734at_nat @ F2 )
| ( finite2115694454571419734at_nat @ G2 ) )
=> ( finite2115694454571419734at_nat @ ( inf_inf_set_nat_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_870_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_871_Int__insert__right__if1,axiom,
! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B ) )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_872_Int__insert__right__if1,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ A @ A2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ ( insert_set_set_nat @ A @ B ) )
= ( insert_set_set_nat @ A @ ( inf_in5711780100303410308et_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_873_Int__insert__right__if1,axiom,
! [A: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ A @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ ( insert_nat_nat @ A @ B ) )
= ( insert_nat_nat @ A @ ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_874_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_875_Int__insert__right__if0,axiom,
! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ A @ B ) )
= ( inf_inf_set_set_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_876_Int__insert__right__if0,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ A @ A2 )
=> ( ( inf_in5711780100303410308et_nat @ A2 @ ( insert_set_set_nat @ A @ B ) )
= ( inf_in5711780100303410308et_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_877_Int__insert__right__if0,axiom,
! [A: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ A @ A2 )
=> ( ( inf_inf_set_nat_nat @ A2 @ ( insert_nat_nat @ A @ B ) )
= ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_878_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_879_insert__inter__insert,axiom,
! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ ( insert_set_nat @ A @ B ) )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_880_insert__inter__insert,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A @ A2 ) @ ( insert_set_set_nat @ A @ B ) )
= ( insert_set_set_nat @ A @ ( inf_in5711780100303410308et_nat @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_881_insert__inter__insert,axiom,
! [A: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A @ A2 ) @ ( insert_nat_nat @ A @ B ) )
= ( insert_nat_nat @ A @ ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_882_Int__insert__left__if1,axiom,
! [A: nat,C3: set_nat,B: set_nat] :
( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C3 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_883_Int__insert__left__if1,axiom,
! [A: set_nat,C3: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ A @ C3 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B ) @ C3 )
= ( insert_set_nat @ A @ ( inf_inf_set_set_nat @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_884_Int__insert__left__if1,axiom,
! [A: set_set_nat,C3: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ A @ C3 )
=> ( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A @ B ) @ C3 )
= ( insert_set_set_nat @ A @ ( inf_in5711780100303410308et_nat @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_885_Int__insert__left__if1,axiom,
! [A: nat > nat,C3: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ A @ C3 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A @ B ) @ C3 )
= ( insert_nat_nat @ A @ ( inf_inf_set_nat_nat @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_886_Int__insert__left__if0,axiom,
! [A: nat,C3: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B ) @ C3 )
= ( inf_inf_set_nat @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_887_Int__insert__left__if0,axiom,
! [A: set_nat,C3: set_set_nat,B: set_set_nat] :
( ~ ( member_set_nat @ A @ C3 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ B ) @ C3 )
= ( inf_inf_set_set_nat @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_888_Int__insert__left__if0,axiom,
! [A: set_set_nat,C3: set_set_set_nat,B: set_set_set_nat] :
( ~ ( member_set_set_nat @ A @ C3 )
=> ( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A @ B ) @ C3 )
= ( inf_in5711780100303410308et_nat @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_889_Int__insert__left__if0,axiom,
! [A: nat > nat,C3: set_nat_nat,B: set_nat_nat] :
( ~ ( member_nat_nat @ A @ C3 )
=> ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A @ B ) @ C3 )
= ( inf_inf_set_nat_nat @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_890_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( ( ord_less_eq_set_nat @ A2 @ C3 )
& ( ord_less_eq_set_nat @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_891_Un__subset__iff,axiom,
! [A2: set_set_nat,B: set_set_nat,C3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ C3 )
= ( ( ord_le6893508408891458716et_nat @ A2 @ C3 )
& ( ord_le6893508408891458716et_nat @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_892_Un__subset__iff,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C3: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B ) @ C3 )
= ( ( ord_le9131159989063066194et_nat @ A2 @ C3 )
& ( ord_le9131159989063066194et_nat @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_893_Un__subset__iff,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ord_le9059583361652607317at_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ C3 )
= ( ( ord_le9059583361652607317at_nat @ A2 @ C3 )
& ( ord_le9059583361652607317at_nat @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_894_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_895_Un__empty,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat )
= ( ( A2 = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_896_Un__empty,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ A2 @ B )
= bot_bo7198184520161983622et_nat )
= ( ( A2 = bot_bo7198184520161983622et_nat )
& ( B = bot_bo7198184520161983622et_nat ) ) ) ).
% Un_empty
thf(fact_897_Un__empty,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ A2 @ B )
= bot_bot_set_nat_nat )
= ( ( A2 = bot_bot_set_nat_nat )
& ( B = bot_bot_set_nat_nat ) ) ) ).
% Un_empty
thf(fact_898_finite__Un,axiom,
! [F2: set_set_set_nat,G2: set_set_set_nat] :
( ( finite6739761609112101331et_nat @ ( sup_su4213647025997063966et_nat @ F2 @ G2 ) )
= ( ( finite6739761609112101331et_nat @ F2 )
& ( finite6739761609112101331et_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_899_finite__Un,axiom,
! [F2: set_set_nat,G2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F2 @ G2 ) )
= ( ( finite1152437895449049373et_nat @ F2 )
& ( finite1152437895449049373et_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_900_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_901_finite__Un,axiom,
! [F2: set_nat_nat,G2: set_nat_nat] :
( ( finite2115694454571419734at_nat @ ( sup_sup_set_nat_nat @ F2 @ G2 ) )
= ( ( finite2115694454571419734at_nat @ F2 )
& ( finite2115694454571419734at_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_902_Un__insert__right,axiom,
! [A2: set_set_set_nat,A: set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ ( insert_set_set_nat @ A @ B ) )
= ( insert_set_set_nat @ A @ ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_903_Un__insert__right,axiom,
! [A2: set_set_nat,A: set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( insert_set_nat @ A @ B ) )
= ( insert_set_nat @ A @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_904_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_905_Un__insert__right,axiom,
! [A2: set_nat_nat,A: nat > nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ ( insert_nat_nat @ A @ B ) )
= ( insert_nat_nat @ A @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_906_Un__insert__left,axiom,
! [A: set_set_nat,B: set_set_set_nat,C3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( insert_set_set_nat @ A @ B ) @ C3 )
= ( insert_set_set_nat @ A @ ( sup_su4213647025997063966et_nat @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_907_Un__insert__left,axiom,
! [A: set_nat,B: set_set_nat,C3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( insert_set_nat @ A @ B ) @ C3 )
= ( insert_set_nat @ A @ ( sup_sup_set_set_nat @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_908_Un__insert__left,axiom,
! [A: nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B ) @ C3 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_909_Un__insert__left,axiom,
! [A: nat > nat,B: set_nat_nat,C3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( insert_nat_nat @ A @ B ) @ C3 )
= ( insert_nat_nat @ A @ ( sup_sup_set_nat_nat @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_910_Int__Un__eq_I4_J,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ T2 @ ( inf_in5711780100303410308et_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_911_Int__Un__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( sup_sup_set_set_nat @ T2 @ ( inf_inf_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_912_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_913_Int__Un__eq_I4_J,axiom,
! [T2: set_nat_nat,S: set_nat_nat] :
( ( sup_sup_set_nat_nat @ T2 @ ( inf_inf_set_nat_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_914_Int__Un__eq_I3_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ S @ ( inf_in5711780100303410308et_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_915_Int__Un__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ S @ ( inf_inf_set_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_916_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_917_Int__Un__eq_I3_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ S @ ( inf_inf_set_nat_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_918_Int__Un__eq_I2_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_919_Int__Un__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_920_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_921_Int__Un__eq_I2_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_922_Int__Un__eq_I1_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_923_Int__Un__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_924_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_925_Int__Un__eq_I1_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_926_Un__Int__eq_I4_J,axiom,
! [T2: set_set_set_nat,S: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ T2 @ ( sup_su4213647025997063966et_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_927_Un__Int__eq_I4_J,axiom,
! [T2: set_set_nat,S: set_set_nat] :
( ( inf_inf_set_set_nat @ T2 @ ( sup_sup_set_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_928_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_929_Un__Int__eq_I4_J,axiom,
! [T2: set_nat_nat,S: set_nat_nat] :
( ( inf_inf_set_nat_nat @ T2 @ ( sup_sup_set_nat_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_930_Un__Int__eq_I3_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ S @ ( sup_su4213647025997063966et_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_931_Un__Int__eq_I3_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ S @ ( sup_sup_set_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_932_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_933_Un__Int__eq_I3_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ S @ ( sup_sup_set_nat_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_934_Un__Int__eq_I2_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_935_Un__Int__eq_I2_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_936_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_937_Un__Int__eq_I2_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_938_Un__Int__eq_I1_J,axiom,
! [S: set_set_set_nat,T2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( sup_su4213647025997063966et_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_939_Un__Int__eq_I1_J,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( inf_inf_set_set_nat @ ( sup_sup_set_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_940_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_941_Un__Int__eq_I1_J,axiom,
! [S: set_nat_nat,T2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( sup_sup_set_nat_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_942_Un__Diff__cancel2,axiom,
! [B: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_943_Un__Diff__cancel2,axiom,
! [B: set_nat_nat,A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ B @ A2 ) @ A2 )
= ( sup_sup_set_nat_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_944_Un__Diff__cancel2,axiom,
! [B: set_set_set_nat,A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ B @ A2 ) @ A2 )
= ( sup_su4213647025997063966et_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_945_Un__Diff__cancel2,axiom,
! [B: set_set_nat,A2: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ B @ A2 ) @ A2 )
= ( sup_sup_set_set_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_946_Un__Diff__cancel,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_947_Un__Diff__cancel,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ ( minus_8121590178497047118at_nat @ B @ A2 ) )
= ( sup_sup_set_nat_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_948_Un__Diff__cancel,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ ( minus_2447799839930672331et_nat @ B @ A2 ) )
= ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_949_Un__Diff__cancel,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ B @ A2 ) )
= ( sup_sup_set_set_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_950_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_951_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_952_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_953_insert__disjoint_I1_J,axiom,
! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A @ B )
& ( ( inf_inf_set_set_nat @ A2 @ B )
= bot_bot_set_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_954_insert__disjoint_I1_J,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A @ A2 ) @ B )
= bot_bo7198184520161983622et_nat )
= ( ~ ( member_set_set_nat @ A @ B )
& ( ( inf_in5711780100303410308et_nat @ A2 @ B )
= bot_bo7198184520161983622et_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_955_insert__disjoint_I1_J,axiom,
! [A: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A @ A2 ) @ B )
= bot_bot_set_nat_nat )
= ( ~ ( member_nat_nat @ A @ B )
& ( ( inf_inf_set_nat_nat @ A2 @ B )
= bot_bot_set_nat_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_956_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A2 ) @ B ) )
= ( ~ ( member_nat @ A @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_957_insert__disjoint_I2_J,axiom,
! [A: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ ( insert_set_nat @ A @ A2 ) @ B ) )
= ( ~ ( member_set_nat @ A @ B )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_958_insert__disjoint_I2_J,axiom,
! [A: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( inf_in5711780100303410308et_nat @ ( insert_set_set_nat @ A @ A2 ) @ B ) )
= ( ~ ( member_set_set_nat @ A @ B )
& ( bot_bo7198184520161983622et_nat
= ( inf_in5711780100303410308et_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_959_insert__disjoint_I2_J,axiom,
! [A: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ ( insert_nat_nat @ A @ A2 ) @ B ) )
= ( ~ ( member_nat_nat @ A @ B )
& ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_960_disjoint__insert_I1_J,axiom,
! [B: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ B @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_961_disjoint__insert_I1_J,axiom,
! [B: set_set_nat,A: set_nat,A2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ B @ ( insert_set_nat @ A @ A2 ) )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A @ B )
& ( ( inf_inf_set_set_nat @ B @ A2 )
= bot_bot_set_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_962_disjoint__insert_I1_J,axiom,
! [B: set_set_set_nat,A: set_set_nat,A2: set_set_set_nat] :
( ( ( inf_in5711780100303410308et_nat @ B @ ( insert_set_set_nat @ A @ A2 ) )
= bot_bo7198184520161983622et_nat )
= ( ~ ( member_set_set_nat @ A @ B )
& ( ( inf_in5711780100303410308et_nat @ B @ A2 )
= bot_bo7198184520161983622et_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_963_disjoint__insert_I1_J,axiom,
! [B: set_nat_nat,A: nat > nat,A2: set_nat_nat] :
( ( ( inf_inf_set_nat_nat @ B @ ( insert_nat_nat @ A @ A2 ) )
= bot_bot_set_nat_nat )
= ( ~ ( member_nat_nat @ A @ B )
& ( ( inf_inf_set_nat_nat @ B @ A2 )
= bot_bot_set_nat_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_964_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_965_disjoint__insert_I2_J,axiom,
! [A2: set_set_nat,B2: set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) )
= ( ~ ( member_set_nat @ B2 @ A2 )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_966_disjoint__insert_I2_J,axiom,
! [A2: set_set_set_nat,B2: set_set_nat,B: set_set_set_nat] :
( ( bot_bo7198184520161983622et_nat
= ( inf_in5711780100303410308et_nat @ A2 @ ( insert_set_set_nat @ B2 @ B ) ) )
= ( ~ ( member_set_set_nat @ B2 @ A2 )
& ( bot_bo7198184520161983622et_nat
= ( inf_in5711780100303410308et_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_967_disjoint__insert_I2_J,axiom,
! [A2: set_nat_nat,B2: nat > nat,B: set_nat_nat] :
( ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ A2 @ ( insert_nat_nat @ B2 @ B ) ) )
= ( ~ ( member_nat_nat @ B2 @ A2 )
& ( bot_bot_set_nat_nat
= ( inf_inf_set_nat_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_968_Diff__disjoint,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_969_Diff__disjoint,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ ( minus_8121590178497047118at_nat @ B @ A2 ) )
= bot_bot_set_nat_nat ) ).
% Diff_disjoint
thf(fact_970_Diff__disjoint,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ ( minus_2447799839930672331et_nat @ B @ A2 ) )
= bot_bo7198184520161983622et_nat ) ).
% Diff_disjoint
thf(fact_971_Diff__disjoint,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ B @ A2 ) )
= bot_bot_set_set_nat ) ).
% Diff_disjoint
thf(fact_972_Un__Int__assoc__eq,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C3 )
= ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) ) )
= ( ord_less_eq_set_nat @ C3 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_973_Un__Int__assoc__eq,axiom,
! [A2: set_set_nat,B: set_set_nat,C3: set_set_nat] :
( ( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B ) @ C3 )
= ( inf_inf_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C3 ) ) )
= ( ord_le6893508408891458716et_nat @ C3 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_974_Un__Int__assoc__eq,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C3: set_set_set_nat] :
( ( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ C3 )
= ( inf_in5711780100303410308et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B @ C3 ) ) )
= ( ord_le9131159989063066194et_nat @ C3 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_975_Un__Int__assoc__eq,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ C3 )
= ( inf_inf_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B @ C3 ) ) )
= ( ord_le9059583361652607317at_nat @ C3 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_976_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_977_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_978_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_979_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_980_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_981_Un__Diff__Int,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_982_Un__Diff__Int,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) @ ( inf_inf_set_nat_nat @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_983_Un__Diff__Int,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_984_Un__Diff__Int,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) @ ( inf_inf_set_set_nat @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_985_Int__Diff__Un,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_986_Int__Diff__Un,axiom,
! [A2: set_nat_nat,B: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ ( minus_8121590178497047118at_nat @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_987_Int__Diff__Un,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ ( minus_2447799839930672331et_nat @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_988_Int__Diff__Un,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( sup_sup_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B ) @ ( minus_2163939370556025621et_nat @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_989_Diff__Int,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C3 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ A2 @ C3 ) ) ) ).
% Diff_Int
thf(fact_990_Diff__Int,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C3 ) )
= ( sup_sup_set_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) @ ( minus_8121590178497047118at_nat @ A2 @ C3 ) ) ) ).
% Diff_Int
thf(fact_991_Diff__Int,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C3: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B @ C3 ) )
= ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) @ ( minus_2447799839930672331et_nat @ A2 @ C3 ) ) ) ).
% Diff_Int
thf(fact_992_Diff__Int,axiom,
! [A2: set_set_nat,B: set_set_nat,C3: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( inf_inf_set_set_nat @ B @ C3 ) )
= ( sup_sup_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) @ ( minus_2163939370556025621et_nat @ A2 @ C3 ) ) ) ).
% Diff_Int
thf(fact_993_Diff__Un,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ A2 @ C3 ) ) ) ).
% Diff_Un
thf(fact_994_Diff__Un,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( minus_8121590178497047118at_nat @ A2 @ ( sup_sup_set_nat_nat @ B @ C3 ) )
= ( inf_inf_set_nat_nat @ ( minus_8121590178497047118at_nat @ A2 @ B ) @ ( minus_8121590178497047118at_nat @ A2 @ C3 ) ) ) ).
% Diff_Un
thf(fact_995_Diff__Un,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C3: set_set_set_nat] :
( ( minus_2447799839930672331et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B @ C3 ) )
= ( inf_in5711780100303410308et_nat @ ( minus_2447799839930672331et_nat @ A2 @ B ) @ ( minus_2447799839930672331et_nat @ A2 @ C3 ) ) ) ).
% Diff_Un
thf(fact_996_Diff__Un,axiom,
! [A2: set_set_nat,B: set_set_nat,C3: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C3 ) )
= ( inf_inf_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ B ) @ ( minus_2163939370556025621et_nat @ A2 @ C3 ) ) ) ).
% Diff_Un
thf(fact_997_UnE,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
=> ( ~ ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_998_UnE,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) )
=> ( ~ ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ B ) ) ) ).
% UnE
thf(fact_999_UnE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_1000_UnE,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) )
=> ( ~ ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ B ) ) ) ).
% UnE
thf(fact_1001_IntE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_1002_IntE,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) )
=> ~ ( ( member_set_nat @ C @ A2 )
=> ~ ( member_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_1003_IntE,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
=> ~ ( ( member_set_set_nat @ C @ A2 )
=> ~ ( member_set_set_nat @ C @ B ) ) ) ).
% IntE
thf(fact_1004_IntE,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) )
=> ~ ( ( member_nat_nat @ C @ A2 )
=> ~ ( member_nat_nat @ C @ B ) ) ) ).
% IntE
thf(fact_1005_UnI1,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ A2 )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1006_UnI1,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1007_UnI1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1008_UnI1,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ A2 )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_1009_UnI2,axiom,
! [C: set_set_nat,B: set_set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ C @ B )
=> ( member_set_set_nat @ C @ ( sup_su4213647025997063966et_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1010_UnI2,axiom,
! [C: set_nat,B: set_set_nat,A2: set_set_nat] :
( ( member_set_nat @ C @ B )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1011_UnI2,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1012_UnI2,axiom,
! [C: nat > nat,B: set_nat_nat,A2: set_nat_nat] :
( ( member_nat_nat @ C @ B )
=> ( member_nat_nat @ C @ ( sup_sup_set_nat_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_1013_IntD1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1014_IntD1,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) )
=> ( member_set_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1015_IntD1,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
=> ( member_set_set_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1016_IntD1,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) )
=> ( member_nat_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1017_IntD2,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_1018_IntD2,axiom,
! [C: set_nat,A2: set_set_nat,B: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A2 @ B ) )
=> ( member_set_nat @ C @ B ) ) ).
% IntD2
thf(fact_1019_IntD2,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( inf_in5711780100303410308et_nat @ A2 @ B ) )
=> ( member_set_set_nat @ C @ B ) ) ).
% IntD2
thf(fact_1020_IntD2,axiom,
! [C: nat > nat,A2: set_nat_nat,B: set_nat_nat] :
( ( member_nat_nat @ C @ ( inf_inf_set_nat_nat @ A2 @ B ) )
=> ( member_nat_nat @ C @ B ) ) ).
% IntD2
thf(fact_1021_bex__Un,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1022_bex__Un,axiom,
! [A2: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1023_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1024_bex__Un,axiom,
! [A2: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_1025_ball__Un,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,P: set_set_nat > $o] :
( ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( sup_su4213647025997063966et_nat @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1026_ball__Un,axiom,
! [A2: set_set_nat,B: set_set_nat,P: set_nat > $o] :
( ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( sup_sup_set_set_nat @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: set_nat] :
( ( member_set_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1027_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1028_ball__Un,axiom,
! [A2: set_nat_nat,B: set_nat_nat,P: ( nat > nat ) > $o] :
( ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ ( sup_sup_set_nat_nat @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: nat > nat] :
( ( member_nat_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_1029_Un__assoc,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C3: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ ( sup_su4213647025997063966et_nat @ A2 @ B ) @ C3 )
= ( sup_su4213647025997063966et_nat @ A2 @ ( sup_su4213647025997063966et_nat @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_1030_Un__assoc,axiom,
! [A2: set_set_nat,B: set_set_nat,C3: set_set_nat] :
( ( sup_sup_set_set_nat @ ( sup_sup_set_set_nat @ A2 @ B ) @ C3 )
= ( sup_sup_set_set_nat @ A2 @ ( sup_sup_set_set_nat @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_1031_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_1032_Un__assoc,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( sup_sup_set_nat_nat @ ( sup_sup_set_nat_nat @ A2 @ B ) @ C3 )
= ( sup_sup_set_nat_nat @ A2 @ ( sup_sup_set_nat_nat @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_1033_Int__assoc,axiom,
! [A2: set_set_nat,B: set_set_nat,C3: set_set_nat] :
( ( inf_inf_set_set_nat @ ( inf_inf_set_set_nat @ A2 @ B ) @ C3 )
= ( inf_inf_set_set_nat @ A2 @ ( inf_inf_set_set_nat @ B @ C3 ) ) ) ).
% Int_assoc
thf(fact_1034_Int__assoc,axiom,
! [A2: set_set_set_nat,B: set_set_set_nat,C3: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ ( inf_in5711780100303410308et_nat @ A2 @ B ) @ C3 )
= ( inf_in5711780100303410308et_nat @ A2 @ ( inf_in5711780100303410308et_nat @ B @ C3 ) ) ) ).
% Int_assoc
thf(fact_1035_Int__assoc,axiom,
! [A2: set_nat_nat,B: set_nat_nat,C3: set_nat_nat] :
( ( inf_inf_set_nat_nat @ ( inf_inf_set_nat_nat @ A2 @ B ) @ C3 )
= ( inf_inf_set_nat_nat @ A2 @ ( inf_inf_set_nat_nat @ B @ C3 ) ) ) ).
% Int_assoc
thf(fact_1036_Un__absorb,axiom,
! [A2: set_set_set_nat] :
( ( sup_su4213647025997063966et_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1037_Un__absorb,axiom,
! [A2: set_set_nat] :
( ( sup_sup_set_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1038_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1039_Un__absorb,axiom,
! [A2: set_nat_nat] :
( ( sup_sup_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_1040_Int__absorb,axiom,
! [A2: set_set_nat] :
( ( inf_inf_set_set_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1041_Int__absorb,axiom,
! [A2: set_set_set_nat] :
( ( inf_in5711780100303410308et_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1042_Int__absorb,axiom,
! [A2: set_nat_nat] :
( ( inf_inf_set_nat_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1043_Un__commute,axiom,
( sup_sup_set_set_nat
= ( ^ [A3: set_set_nat,B4: set_set_nat] : ( sup_sup_set_set_nat @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_1044_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_1045_Un__commute,axiom,
( sup_sup_set_nat_nat
= ( ^ [A3: set_nat_nat,B4: set_nat_nat] : ( sup_sup_set_nat_nat @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_1046_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1047_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1048_diff__mult__distrib,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1049_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1050_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B2 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1051_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_1052_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_1053_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_1054_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1055_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_1056_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_1057_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1058_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1059_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1060_less__not__refl3,axiom,
! [S2: nat,T4: nat] :
( ( ord_less_nat @ S2 @ T4 )
=> ( S2 != T4 ) ) ).
% less_not_refl3
thf(fact_1061_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_1062_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1063_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1064_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1065_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1066_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1067_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_1068_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1069_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1070_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1071_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1072_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1073_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1074_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1075_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_1076_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1077_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1078_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1079_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1080_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1081_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1082_ACC__odot,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ k @ X5 ) @ ( clique3210737319928189260st_ACC @ k @ Y4 ) ) ) ).
% ACC_odot
thf(fact_1083_ACC__union,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( clique3210737319928189260st_ACC @ k @ ( sup_su4213647025997063966et_nat @ X5 @ Y4 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ k @ X5 ) @ ( clique3210737319928189260st_ACC @ k @ Y4 ) ) ) ).
% ACC_union
thf(fact_1084_v__union,axiom,
! [G2: set_set_nat,H: set_set_nat] :
( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G2 @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ).
% v_union
thf(fact_1085_v__gs__union,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X5 @ Y4 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y4 ) ) ) ).
% v_gs_union
thf(fact_1086_ACC__cf__union,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( sup_su4213647025997063966et_nat @ X5 @ Y4 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique951075384711337423ACC_cf @ k @ Y4 ) ) ) ).
% ACC_cf_union
thf(fact_1087_ACC__cf__odot,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( clique951075384711337423ACC_cf @ k @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ k @ X5 ) @ ( clique951075384711337423ACC_cf @ k @ Y4 ) ) ) ).
% ACC_cf_odot
thf(fact_1088_POS__CLIQUE,axiom,
ord_le152980574450754630et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique363107459185959606CLIQUE @ k ) ).
% POS_CLIQUE
thf(fact_1089_ACC__empty,axiom,
( ( clique3210737319928189260st_ACC @ k @ bot_bo7198184520161983622et_nat )
= bot_bo7198184520161983622et_nat ) ).
% ACC_empty
thf(fact_1090_union___092_060G_062,axiom,
! [G2: set_set_nat,H: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G2 @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ) ) ) ).
% union_\<G>
thf(fact_1091_ACC__I,axiom,
! [G2: set_set_nat,X5: set_set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X5 @ G2 )
=> ( member_set_set_nat @ G2 @ ( clique3210737319928189260st_ACC @ k @ X5 ) ) ) ) ).
% ACC_I
thf(fact_1092_first__assumptions_OACC_Ocong,axiom,
clique3210737319928189260st_ACC = clique3210737319928189260st_ACC ).
% first_assumptions.ACC.cong
thf(fact_1093__092_060open_062v__gs_A_IX_A_N_AU_A_092_060union_062_A_123Gs_125_J_A_061_Av__gs_A_IX_A_N_AU_J_A_092_060union_062_Av__gs_A_123Gs_125_092_060close_062,axiom,
( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) @ ( clique8462013130872731469t_v_gs @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ) ).
% \<open>v_gs (X - U \<union> {Gs}) = v_gs (X - U) \<union> v_gs {Gs}\<close>
thf(fact_1094_finX,axiom,
finite6739761609112101331et_nat @ x ).
% finX
thf(fact_1095_fin1,axiom,
finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ x ) ).
% fin1
thf(fact_1096__092_060open_062_092_060And_062A_O_AA_A_092_060subseteq_062_AX_A_092_060Longrightarrow_062_Afinite_AA_092_060close_062,axiom,
! [A2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ A2 @ x )
=> ( finite6739761609112101331et_nat @ A2 ) ) ).
% \<open>\<And>A. A \<subseteq> X \<Longrightarrow> finite A\<close>
thf(fact_1097_UX,axiom,
ord_le9131159989063066194et_nat @ u @ x ).
% UX
thf(fact_1098_S_I1_J,axiom,
ord_le6893508408891458716et_nat @ s @ ( clique8462013130872731469t_v_gs @ x ) ).
% S(1)
thf(fact_1099_vplus__dsXU,axiom,
( ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) )
= ( minus_2163939370556025621et_nat @ ( clique8462013130872731469t_v_gs @ x ) @ ( clique8462013130872731469t_v_gs @ u ) ) ) ).
% vplus_dsXU
thf(fact_1100_G_I1_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( member_set_set_nat @ ( g @ I2 ) @ x ) ) ).
% G(1)
thf(fact_1101__092_060open_062card_A_Iv__gs_AX_A_N_Av__gs_AU_J_A_061_Acard_A_Iv__gs_AX_J_A_N_Acard_A_Iv__gs_AU_J_092_060close_062,axiom,
( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( clique8462013130872731469t_v_gs @ x ) @ ( clique8462013130872731469t_v_gs @ u ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ u ) ) ) ) ).
% \<open>card (v_gs X - v_gs U) = card (v_gs X) - card (v_gs U)\<close>
thf(fact_1102_XD,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ k ) ) ) ).
% XD
thf(fact_1103_assms_I3_J,axiom,
( y
= ( clique4095374090462327202g_step @ p @ x ) ) ).
% assms(3)
thf(fact_1104__092_060open_062f_A_092_060in_062_AACC__cf_AY_A_N_AACC__cf_AX_092_060close_062,axiom,
member_nat_nat @ f @ ( minus_8121590178497047118at_nat @ ( clique951075384711337423ACC_cf @ k @ y ) @ ( clique951075384711337423ACC_cf @ k @ x ) ) ).
% \<open>f \<in> ACC_cf Y - ACC_cf X\<close>
thf(fact_1105__092_060open_062card_A_Iv__gs_AY_J_A_061_Acard_A_Iv__gs_A_IX_A_N_AU_A_092_060union_062_A_123Gs_125_J_J_092_060close_062,axiom,
( ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) )
= ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ) ) ).
% \<open>card (v_gs Y) = card (v_gs (X - U \<union> {Gs}))\<close>
thf(fact_1106_Y,axiom,
( y
= ( sup_su4213647025997063966et_nat @ ( minus_2447799839930672331et_nat @ x @ u ) @ ( insert_set_set_nat @ gs @ bot_bo7198184520161983622et_nat ) ) ) ).
% Y
thf(fact_1107__092_060open_062POS_A_092_060inter_062_AACC_AX_A_092_060subseteq_062_AACC_AY_092_060close_062,axiom,
ord_le9131159989063066194et_nat @ ( inf_in5711780100303410308et_nat @ ( clique3326749438856946062irst_K @ k ) @ ( clique3210737319928189260st_ACC @ k @ x ) ) @ ( clique3210737319928189260st_ACC @ k @ y ) ).
% \<open>POS \<inter> ACC X \<subseteq> ACC Y\<close>
thf(fact_1108__092_060open_062card_A_Iv__gs_A_IX_A_N_AU_J_A_092_060union_062_A_123v_AGs_125_J_A_092_060le_062_Acard_A_Iv__gs_A_IX_A_N_AU_J_J_A_L_Acard_A_123v_AGs_125_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) @ ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) ) @ ( finite_card_set_nat @ ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ) ).
% \<open>card (v_gs (X - U) \<union> {v Gs}) \<le> card (v_gs (X - U)) + card {v Gs}\<close>
thf(fact_1109__092_060open_062Y_A_092_060noteq_062_A_123_125_092_060close_062,axiom,
y != bot_bo7198184520161983622et_nat ).
% \<open>Y \<noteq> {}\<close>
thf(fact_1110__092_060open_062_092_060And_062G_O_A_092_060lbrakk_062G_A_092_060in_062_AACC_AX_059_AG_A_092_060in_062_APOS_092_060rbrakk_062_A_092_060Longrightarrow_062_AG_A_092_060in_062_AACC_AY_092_060close_062,axiom,
! [G2: set_set_nat] :
( ( member_set_set_nat @ G2 @ ( clique3210737319928189260st_ACC @ k @ x ) )
=> ( ( member_set_set_nat @ G2 @ ( clique3326749438856946062irst_K @ k ) )
=> ( member_set_set_nat @ G2 @ ( clique3210737319928189260st_ACC @ k @ y ) ) ) ) ).
% \<open>\<And>G. \<lbrakk>G \<in> ACC X; G \<in> POS\<rbrakk> \<Longrightarrow> G \<in> ACC Y\<close>
thf(fact_1111_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1112_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1113_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1114_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1115_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1116_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1117__092_060open_062card_A_Iv__gs_AY_J_A_092_060le_062_Acard_A_Iv__gs_AX_J_A_N_Ap_A_L_A1_092_060close_062,axiom,
ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ y ) ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) @ p ) @ one_one_nat ) ).
% \<open>card (v_gs Y) \<le> card (v_gs X) - p + 1\<close>
thf(fact_1118__092_060open_062card_A_Iv__gs_A_IX_A_N_AU_J_J_A_L_Acard_A_123v_AGs_125_A_092_060le_062_Acard_A_Iv__gs_A_IX_A_N_AU_J_J_A_L_A1_092_060close_062,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) ) @ ( finite_card_set_nat @ ( insert_set_nat @ ( clique5033774636164728513irst_v @ gs ) @ bot_bot_set_set_nat ) ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ ( minus_2447799839930672331et_nat @ x @ u ) ) ) @ one_one_nat ) ).
% \<open>card (v_gs (X - U)) + card {v Gs} \<le> card (v_gs (X - U)) + 1\<close>
thf(fact_1119_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1120_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1121_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1122_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1123_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_1124_trans__le__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1125_trans__le__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1126_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1127_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1128_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1129_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1130_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1131_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1132_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1133_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1134_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_1135_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1136_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_1137_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_1138_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1139_trans__less__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1140_trans__less__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1141_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1142_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1143_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1144_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1145_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1146_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1147_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1148_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1149_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1150_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1151__092_060open_062S_A_092_060subseteq_062_Av__gs_AX_A_092_060and_062_Asunflower_AS_A_092_060and_062_Acard_AS_A_061_Ap_092_060close_062,axiom,
( ( ord_le6893508408891458716et_nat @ s @ ( clique8462013130872731469t_v_gs @ x ) )
& ( sunflower_nat @ s )
& ( ( finite_card_set_nat @ s )
= p ) ) ).
% \<open>S \<subseteq> v_gs X \<and> sunflower S \<and> card S = p\<close>
thf(fact_1152_sunflower,axiom,
? [S3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S3 @ ( clique8462013130872731469t_v_gs @ x ) )
& ( sunflower_nat @ S3 )
& ( ( finite_card_set_nat @ S3 )
= p ) ) ).
% sunflower
thf(fact_1153_S_I2_J,axiom,
sunflower_nat @ s ).
% S(2)
thf(fact_1154_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1155_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_less_add_iff1
thf(fact_1156_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1157_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1158_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1159_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M2 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1160_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1161_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M2 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1162_card__POS,axiom,
( ( finite1149291290879098388et_nat @ ( clique3326749438856946062irst_K @ k ) )
= ( binomial @ ( assump1710595444109740334irst_m @ k ) @ k ) ) ).
% card_POS
thf(fact_1163_binomial__n__n,axiom,
! [N2: nat] :
( ( binomial @ N2 @ N2 )
= one_one_nat ) ).
% binomial_n_n
thf(fact_1164_binomial__absorb__comp,axiom,
! [N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
= ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_1165_choose__one,axiom,
! [N2: nat] :
( ( binomial @ N2 @ one_one_nat )
= N2 ) ).
% choose_one
thf(fact_1166_binomial__symmetric,axiom,
! [K: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ N2 )
=> ( ( binomial @ N2 @ K )
= ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_1167_choose__mult,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( times_times_nat @ ( binomial @ N2 @ M2 ) @ ( binomial @ M2 @ K ) )
= ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M2 @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_1168_sf__precond,axiom,
! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( clique8462013130872731469t_v_gs @ x ) )
=> ( ( finite_finite_nat @ X4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ X4 ) @ l ) ) ) ).
% sf_precond
thf(fact_1169_k,axiom,
ord_less_nat @ l @ k ).
% k
thf(fact_1170_pl,axiom,
ord_less_nat @ l @ p ).
% pl
thf(fact_1171_card__Vs,axiom,
ord_less_eq_nat @ ( finite_card_nat @ vs ) @ l ).
% card_Vs
thf(fact_1172_second__assumptions__axioms,axiom,
assump2881078719466019805ptions @ l @ p @ k ).
% second_assumptions_axioms
thf(fact_1173_kml,axiom,
ord_less_eq_nat @ k @ ( minus_minus_nat @ ( assump1710595444109740334irst_m @ k ) @ l ) ).
% kml
thf(fact_1174_lm,axiom,
ord_less_nat @ ( plus_plus_nat @ l @ one_one_nat ) @ ( assump1710595444109740334irst_m @ k ) ).
% lm
thf(fact_1175_X,axiom,
ord_le9131159989063066194et_nat @ x @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% X
thf(fact_1176_i__props_I5_J,axiom,
! [I2: nat] :
( ( ord_less_nat @ I2 @ p )
=> ( ord_less_eq_nat @ ( si2 @ I2 ) @ l ) ) ).
% i_props(5)
thf(fact_1177_L,axiom,
ord_less_nat @ ( assump1710595444109740301irst_L @ l @ p ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ x ) ) ).
% L
thf(fact_1178_joinl__join,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] : ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ l @ k @ X5 @ Y4 ) @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) ) ).
% joinl_join
thf(fact_1179_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1180_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1181_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N2: nat] :
( ! [X: nat] :
( ( member_nat @ X @ N )
=> ( ord_less_nat @ X @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_1182_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M4: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ M4 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1183_finite__v__gs__Gl,axiom,
! [X5: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ l @ k ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ).
% finite_v_gs_Gl
thf(fact_1184__092_060open_062Y_A_092_060subseteq_062_A_092_060G_062l_092_060close_062,axiom,
ord_le9131159989063066194et_nat @ y @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% \<open>Y \<subseteq> \<G>l\<close>
thf(fact_1185_Lp,axiom,
ord_less_nat @ p @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lp
thf(fact_1186_GsGl,axiom,
member_set_set_nat @ gs @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% GsGl
thf(fact_1187_U,axiom,
ord_le9131159989063066194et_nat @ u @ ( clique7840962075309931874st_G_l @ l @ k ) ).
% U
thf(fact_1188_Lm,axiom,
ord_less_eq_nat @ ( assump1710595444109740334irst_m @ k ) @ ( assump1710595444109740301irst_L @ l @ p ) ).
% Lm
thf(fact_1189_odotl__def,axiom,
! [X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( clique7966186356931407165_odotl @ l @ k @ X5 @ Y4 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) @ ( clique7840962075309931874st_G_l @ l @ k ) ) ) ).
% odotl_def
thf(fact_1190_first__assumptions_O_092_060G_062l_Ocong,axiom,
clique7840962075309931874st_G_l = clique7840962075309931874st_G_l ).
% first_assumptions.\<G>l.cong
thf(fact_1191_first__assumptions_Oodotl_Ocong,axiom,
clique7966186356931407165_odotl = clique7966186356931407165_odotl ).
% first_assumptions.odotl.cong
thf(fact_1192_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_1193_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_1194_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_1195_first__assumptions_Om_Ocong,axiom,
assump1710595444109740334irst_m = assump1710595444109740334irst_m ).
% first_assumptions.m.cong
thf(fact_1196_first__assumptions__axioms,axiom,
assump5453534214990993103ptions @ l @ p @ k ).
% first_assumptions_axioms
thf(fact_1197_Vs__def,axiom,
( vs
= ( comple7806235888213564991et_nat @ s ) ) ).
% Vs_def
thf(fact_1198_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_1199_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_1200_first__assumptions_Ok,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ K ) ) ).
% first_assumptions.k
thf(fact_1201_first__assumptions_Okp,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ P2 @ K ) ) ).
% first_assumptions.kp
thf(fact_1202_first__assumptions_Opl,axiom,
! [L: nat,P2: nat,K: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_less_nat @ L @ P2 ) ) ).
% first_assumptions.pl
thf(fact_1203_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_1204_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_1205_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_1206_first__assumptions_Oaccepts__def,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3686358387679108662ccepts @ X5 @ G2 )
= ( ? [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ X5 )
& ( ord_le6893508408891458716et_nat @ X3 @ G2 ) ) ) ) ) ).
% first_assumptions.accepts_def
thf(fact_1207_first__assumptions_OacceptsI,axiom,
! [L: nat,P2: nat,K: nat,D2: set_set_nat,G2: set_set_nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le6893508408891458716et_nat @ D2 @ G2 )
=> ( ( member_set_set_nat @ D2 @ X5 )
=> ( clique3686358387679108662ccepts @ X5 @ G2 ) ) ) ) ).
% first_assumptions.acceptsI
thf(fact_1208_first__assumptions_OACC__union,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( sup_su4213647025997063966et_nat @ X5 @ Y4 ) )
= ( sup_su4213647025997063966et_nat @ ( clique3210737319928189260st_ACC @ K @ X5 ) @ ( clique3210737319928189260st_ACC @ K @ Y4 ) ) ) ) ).
% first_assumptions.ACC_union
thf(fact_1209_first__assumptions_Ofinite__numbers,axiom,
! [L: nat,P2: nat,K: nat,N2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite_finite_nat @ ( clique3652268606331196573umbers @ N2 ) ) ) ).
% first_assumptions.finite_numbers
thf(fact_1210_first__assumptions_Ofinite__ACC,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite2115694454571419734at_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) ) ) ).
% first_assumptions.finite_ACC
thf(fact_1211_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_1212_first__assumptions_Ov__gs__mono,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Y4 )
=> ( ord_le6893508408891458716et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y4 ) ) ) ) ).
% first_assumptions.v_gs_mono
thf(fact_1213_first__assumptions_Ov__mono,axiom,
! [L: nat,P2: nat,K: nat,G2: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le6893508408891458716et_nat @ G2 @ H )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_mono
thf(fact_1214_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_1215_first__assumptions_Ofinite__numbers2,axiom,
! [L: nat,P2: nat,K: nat,N2: nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( finite1152437895449049373et_nat @ ( clique6722202388162463298od_nat @ ( clique3652268606331196573umbers @ N2 ) @ ( clique3652268606331196573umbers @ N2 ) ) ) ) ).
% first_assumptions.finite_numbers2
thf(fact_1216_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_1217_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_1218_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_1219_first__assumptions_Ov__gs__union,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ ( sup_su4213647025997063966et_nat @ X5 @ Y4 ) )
= ( sup_sup_set_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) @ ( clique8462013130872731469t_v_gs @ Y4 ) ) ) ) ).
% first_assumptions.v_gs_union
thf(fact_1220_first__assumptions_OACC__cf__mono,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ Y4 )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique951075384711337423ACC_cf @ K @ Y4 ) ) ) ) ).
% first_assumptions.ACC_cf_mono
thf(fact_1221_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_1222_first__assumptions_Ojoinl__join,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9131159989063066194et_nat @ ( clique7966186356931407165_odotl @ L @ K @ X5 @ Y4 ) @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) ) ) ).
% first_assumptions.joinl_join
thf(fact_1223_first__assumptions_OACC__odot,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique3210737319928189260st_ACC @ K @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) )
= ( inf_in5711780100303410308et_nat @ ( clique3210737319928189260st_ACC @ K @ X5 ) @ ( clique3210737319928189260st_ACC @ K @ Y4 ) ) ) ) ).
% first_assumptions.ACC_odot
thf(fact_1224_first__assumptions_OACC__cf___092_060F_062,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ord_le9059583361652607317at_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique2971579238625216137irst_F @ K ) ) ) ).
% first_assumptions.ACC_cf_\<F>
thf(fact_1225_first__assumptions_Ov__gs__def,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique8462013130872731469t_v_gs @ X5 )
= ( image_5842784325960735177et_nat @ clique5033774636164728513irst_v @ X5 ) ) ) ).
% first_assumptions.v_gs_def
thf(fact_1226_first__assumptions_Ov__union,axiom,
! [L: nat,P2: nat,K: nat,G2: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique5033774636164728513irst_v @ ( sup_sup_set_set_nat @ G2 @ H ) )
= ( sup_sup_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ H ) ) ) ) ).
% first_assumptions.v_union
thf(fact_1227_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_1228_first__assumptions_OACC__cf__union,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( sup_su4213647025997063966et_nat @ X5 @ Y4 ) )
= ( sup_sup_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique951075384711337423ACC_cf @ K @ Y4 ) ) ) ) ).
% first_assumptions.ACC_cf_union
thf(fact_1229_first__assumptions_OACC__cf__odot,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique951075384711337423ACC_cf @ K @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) )
= ( inf_inf_set_nat_nat @ ( clique951075384711337423ACC_cf @ K @ X5 ) @ ( clique951075384711337423ACC_cf @ K @ Y4 ) ) ) ) ).
% first_assumptions.ACC_cf_odot
thf(fact_1230_first__assumptions_Ounion___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,G2: set_set_nat,H: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( member_set_set_nat @ H @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ G2 @ H ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.union_\<G>
thf(fact_1231_first__assumptions_Ofinite__members___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite1152437895449049373et_nat @ G2 ) ) ) ).
% first_assumptions.finite_members_\<G>
thf(fact_1232_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_1233_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_1234_first__assumptions_Ofinite__v__gs__Gl,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique7840962075309931874st_G_l @ L @ K ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ) ).
% first_assumptions.finite_v_gs_Gl
thf(fact_1235_first__assumptions_OACC__cf__I,axiom,
! [L: nat,P2: nat,K: nat,F2: nat > nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_nat_nat @ F2 @ ( clique2971579238625216137irst_F @ K ) )
=> ( ( clique3686358387679108662ccepts @ X5 @ ( clique5033774636164728462irst_C @ K @ F2 ) )
=> ( member_nat_nat @ F2 @ ( clique951075384711337423ACC_cf @ K @ X5 ) ) ) ) ) ).
% first_assumptions.ACC_cf_I
thf(fact_1236_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_1237_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_1238_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_1239_first__assumptions_Ofinite__vG,axiom,
! [L: nat,P2: nat,K: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite_finite_nat @ ( clique5033774636164728513irst_v @ G2 ) ) ) ) ).
% first_assumptions.finite_vG
thf(fact_1240_first__assumptions_Ov___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_less_eq_set_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ).
% first_assumptions.v_\<G>
thf(fact_1241_first__assumptions_Oodotl__def,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( clique7966186356931407165_odotl @ L @ K @ X5 @ Y4 )
= ( inf_in5711780100303410308et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) @ ( clique7840962075309931874st_G_l @ L @ K ) ) ) ) ).
% first_assumptions.odotl_def
thf(fact_1242_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_1243_first__assumptions_Oodot___092_060G_062,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le9131159989063066194et_nat @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) ) ) ) ) ).
% first_assumptions.odot_\<G>
thf(fact_1244_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_1245_first__assumptions_OACC__I,axiom,
! [L: nat,P2: nat,K: nat,G2: set_set_nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( clique3686358387679108662ccepts @ X5 @ G2 )
=> ( member_set_set_nat @ G2 @ ( clique3210737319928189260st_ACC @ K @ X5 ) ) ) ) ) ).
% first_assumptions.ACC_I
thf(fact_1246_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_1247_first__assumptions_Ov___092_060G_062__2,axiom,
! [L: nat,P2: nat,K: nat,G2: set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( member_set_set_nat @ G2 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ord_le6893508408891458716et_nat @ G2 @ ( clique6722202388162463298od_nat @ ( clique5033774636164728513irst_v @ G2 ) @ ( clique5033774636164728513irst_v @ G2 ) ) ) ) ) ).
% first_assumptions.v_\<G>_2
thf(fact_1248_first__assumptions_Ofinite__v__gs,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( finite1152437895449049373et_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) ) ) ).
% first_assumptions.finite_v_gs
thf(fact_1249_first__assumptions_Ocard__v__gs__join,axiom,
! [L: nat,P2: nat,K: nat,X5: set_set_set_nat,Y4: set_set_set_nat,Z3: set_set_set_nat] :
( ( assump5453534214990993103ptions @ L @ P2 @ K )
=> ( ( ord_le9131159989063066194et_nat @ X5 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Y4 @ ( clique5786534781347292306Graphs @ ( clique3652268606331196573umbers @ ( assump1710595444109740334irst_m @ K ) ) ) )
=> ( ( ord_le9131159989063066194et_nat @ Z3 @ ( clique5469973757772500719t_odot @ X5 @ Y4 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Z3 ) ) @ ( times_times_nat @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ X5 ) ) @ ( finite_card_set_nat @ ( clique8462013130872731469t_v_gs @ Y4 ) ) ) ) ) ) ) ) ).
% first_assumptions.card_v_gs_join
thf(fact_1250_Inf__nat__def1,axiom,
! [K3: set_nat] :
( ( K3 != bot_bot_set_nat )
=> ( member_nat @ ( complete_Inf_Inf_nat @ K3 ) @ K3 ) ) ).
% Inf_nat_def1
thf(fact_1251_p0,axiom,
p != zero_zero_nat ).
% p0
thf(fact_1252_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1253_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1254_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1255_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1256_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1257_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1258_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1259_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1260_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1261_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1262_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1263_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1264_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1265_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1266_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1267_binomial__eq__0__iff,axiom,
! [N2: nat,K: nat] :
( ( ( binomial @ N2 @ K )
= zero_zero_nat )
= ( ord_less_nat @ N2 @ K ) ) ).
% binomial_eq_0_iff
thf(fact_1268_binomial__n__0,axiom,
! [N2: nat] :
( ( binomial @ N2 @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_1269_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1270_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
% Conjectures (1)
thf(conj_0,conjecture,
~ ( member_nat @ ( w @ i ) @ us ) ).
%------------------------------------------------------------------------------