TPTP Problem File: SLH0568^1.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 : Frequency_Moments/0083_Probability_Ext/prob_00265_009695__19853786_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1470 ( 529 unt; 192 typ; 0 def)
% Number of atoms : 3973 (1517 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 12693 ( 512 ~; 59 |; 402 &;9836 @)
% ( 0 <=>;1884 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 21 ( 20 usr)
% Number of type conns : 990 ( 990 >; 0 *; 0 +; 0 <<)
% Number of symbols : 175 ( 172 usr; 25 con; 0-5 aty)
% Number of variables : 3837 ( 386 ^;3325 !; 126 ?;3837 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:14:32.879
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
sigma_measure_real: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
sigma_measure_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
set_a_real: $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__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__b_J,type,
sigma_measure_b: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__a_J,type,
sigma_measure_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
set_set_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Sigma____Algebra__Omeasure_I_Eo_J,type,
sigma_measure_o: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (172)
thf(sy_c_Bochner__Integration_Ointegrable_001tf__a_001t__Real__Oreal,type,
bochne2139062162225249880a_real: sigma_measure_a > ( a > real ) > $o ).
thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
borel_5078946678739801102l_real: sigma_measure_real ).
thf(sy_c_Complete__Measure_Ocompletion_001tf__a,type,
comple3428971583294703880tion_a: sigma_measure_a > sigma_measure_a ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mt__Real__Oreal_J,type,
finite_finite_a_real: set_a_real > $o ).
thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
finite_finite_o: set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
finite_finite_real: set_real > $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_Itf__a_J_J,type,
finite7209287970140883943_set_a: set_set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__b_J,type,
finite_finite_set_b: set_set_b > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
finite_finite_b: set_b > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001_Eo,type,
giry_subprob_space_o: sigma_measure_o > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001t__Nat__Onat,type,
giry_s459323515522551452ce_nat: sigma_measure_nat > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001tf__a,type,
giry_subprob_space_a: sigma_measure_a > $o ).
thf(sy_c_Giry__Monad_Osubprob__space_001tf__b,type,
giry_subprob_space_b: sigma_measure_b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_Itf__a_Mt__Real__Oreal_J_M_Eo_J,type,
minus_minus_a_real_o: ( ( a > real ) > $o ) > ( ( a > real ) > $o ) > ( a > real ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_Eo_M_Eo_J,type,
minus_minus_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
minus_minus_set_a_o: ( set_a > $o ) > ( set_a > $o ) > set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__b_M_Eo_J,type,
minus_minus_b_o: ( b > $o ) > ( b > $o ) > b > $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__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
minus_4124197362600706274a_real: set_a_real > set_a_real > set_a_real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
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_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__b_J,type,
minus_minus_set_b: set_b > set_b > set_b ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Real__Oreal_J,type,
plus_plus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_Itf__a_Mt__Real__Oreal_J_001t__Nat__Onat,type,
groups1701885688937111089al_nat: ( ( a > real ) > nat ) > set_a_real > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_Itf__a_Mt__Real__Oreal_J_001t__Real__Oreal,type,
groups6125259628802515085l_real: ( ( a > real ) > real ) > set_a_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Nat__Onat,type,
groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_Eo_001t__Real__Oreal,type,
groups8691415230153176458o_real: ( $o > real ) > set_o > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Real__Oreal,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
groups2637260376230714770et_nat: ( nat > set_nat ) > set_nat > set_nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Real__Oreal_001t__Real__Oreal,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_Itf__a_J_001t__Nat__Onat,type,
groups6141743369313575924_a_nat: ( set_a > nat ) > set_set_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_Itf__a_J_001t__Real__Oreal,type,
groups9174420418583655632a_real: ( set_a > real ) > set_set_a > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Nat__Onat,type,
groups6334556678337121940_a_nat: ( a > nat ) > set_a > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Real__Oreal,type,
groups2740460157737275248a_real: ( a > real ) > set_a > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__a_001t__Set__Oset_It__Nat__Onat_J,type,
groups7609827518827096650et_nat: ( a > set_nat ) > set_a > set_nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__b_001t__Nat__Onat,type,
groups7570001007293516437_b_nat: ( b > nat ) > set_b > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__b_001t__Real__Oreal,type,
groups8336678772925405937b_real: ( b > real ) > set_b > real ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001tf__b_001t__Set__Oset_It__Nat__Onat_J,type,
groups1208830032083771979et_nat: ( b > set_nat ) > set_b > set_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__set_001tf__a,type,
indepe2041756565122539606_set_a: sigma_measure_a > set_set_a > set_set_a > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_Eo,type,
indepe7780107833195774214ts_a_o: sigma_measure_a > ( $o > set_set_a ) > set_o > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Nat__Onat,type,
indepe6267730027088848354_a_nat: sigma_measure_a > ( nat > set_set_a ) > set_nat > $o ).
thf(sy_c_Independent__Family_Oprob__space_Oindep__var_001tf__a_001t__Real__Oreal,type,
indepe8958435565499147358a_real: sigma_measure_a > sigma_measure_real > ( a > real ) > sigma_measure_real > ( a > real ) > $o ).
thf(sy_c_Independent__Family_Oprob__space_Otail__events_001tf__a_001t__Nat__Onat,type,
indepe7538416700049374166_a_nat: sigma_measure_a > ( nat > set_set_a ) > set_set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Real__Oreal,type,
measur1776380161843274167a_real: set_set_a > ( set_a > real ) > $o ).
thf(sy_c_Measure__Space_Onull__measure_001_Eo,type,
measur6133975857628879380sure_o: sigma_measure_o > sigma_measure_o ).
thf(sy_c_Measure__Space_Onull__measure_001t__Nat__Onat,type,
measur6922722954359385172re_nat: sigma_measure_nat > sigma_measure_nat ).
thf(sy_c_Measure__Space_Onull__measure_001tf__a,type,
measur3836006170472588154sure_a: sigma_measure_a > sigma_measure_a ).
thf(sy_c_Measure__Space_Onull__measure_001tf__b,type,
measur3836006170472588155sure_b: sigma_measure_b > sigma_measure_b ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001_Eo,type,
nonneg5198678888045619090sure_o: set_o > sigma_measure_o ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001t__Nat__Onat,type,
nonneg7031465154080143958re_nat: set_nat > sigma_measure_nat ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001tf__a,type,
nonneg7367794086797660664sure_a: set_a > sigma_measure_a ).
thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001tf__b,type,
nonneg7367794086797660665sure_b: set_b > sigma_measure_b ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_Itf__a_Mt__Real__Oreal_J_M_Eo_J,type,
bot_bot_a_real_o: ( a > real ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
bot_bot_set_a_o: set_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
bot_bot_b_o: b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
bot_bot_set_a_real: set_a_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
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__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo3380559777022489994_set_a: set_set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
bot_bot_set_b: set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_I_Eo_J,type,
bot_bo5758314138661044393sure_o: sigma_measure_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
bot_bo6718502177978453909re_nat: sigma_measure_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_Itf__a_J,type,
bot_bo2108912051383640591sure_a: sigma_measure_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_Itf__b_J,type,
bot_bo2108912055686869392sure_b: sigma_measure_b ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_Mt__Real__Oreal_J,type,
ord_less_eq_a_real: ( a > real ) > ( a > real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
ord_le3334967407727675675a_real: set_a_real > set_a_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $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_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
ord_less_eq_set_b: set_b > set_b > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(sy_c_Probability__Ext_Oprob__space_Ocovariance_001tf__a_001t__Real__Oreal,type,
probab3938396695707481060a_real: sigma_measure_a > ( a > real ) > ( a > real ) > real ).
thf(sy_c_Probability__Measure_Oprob__space_001_Eo,type,
probab1190487603588612059pace_o: sigma_measure_o > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001t__Nat__Onat,type,
probab2904919403188438605ce_nat: sigma_measure_nat > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001tf__a,type,
probab7247484486040049089pace_a: sigma_measure_a > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001tf__b,type,
probab7247484486040049090pace_b: sigma_measure_b > $o ).
thf(sy_c_Product__Type_Obool_Ocase__bool_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
produc6113963288868236716_set_a: set_set_a > set_set_a > $o > set_set_a ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
real_V1485227260804924795R_real: real > real > real ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__Real__Oreal_J,type,
collect_a_real: ( ( a > real ) > $o ) > set_a_real ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
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_Itf__a_J_J,type,
collect_set_set_a: ( set_set_a > $o ) > set_set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__b_J,type,
collect_set_b: ( set_b > $o ) > set_set_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oinsert_001_062_Itf__a_Mt__Real__Oreal_J,type,
insert_a_real: ( a > real ) > set_a_real > set_a_real ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Set_Ois__empty_001_Eo,type,
is_empty_o: set_o > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__b,type,
is_empty_b: set_b > $o ).
thf(sy_c_Set_Ois__singleton_001_062_Itf__a_Mt__Real__Oreal_J,type,
is_singleton_a_real: set_a_real > $o ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
is_singleton_set_a: set_set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__b,type,
is_singleton_b: set_b > $o ).
thf(sy_c_Set_Oremove_001_062_Itf__a_Mt__Real__Oreal_J,type,
remove_a_real: ( a > real ) > set_a_real > set_a_real ).
thf(sy_c_Set_Oremove_001_Eo,type,
remove_o: $o > set_o > set_o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001t__Set__Oset_Itf__a_J,type,
remove_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Oremove_001tf__b,type,
remove_b: b > set_b > set_b ).
thf(sy_c_Set_Othe__elem_001_062_Itf__a_Mt__Real__Oreal_J,type,
the_elem_a_real: set_a_real > a > real ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_Itf__a_J,type,
the_elem_set_a: set_set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set_Othe__elem_001tf__b,type,
the_elem_b: set_b > b ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001t__Real__Oreal,type,
sigma_9116425665531756122a_real: sigma_measure_a > sigma_measure_real > set_a_real ).
thf(sy_c_Sigma__Algebra_Omeasure_001tf__a,type,
sigma_measure_a2: sigma_measure_a > set_a > real ).
thf(sy_c_Sigma__Algebra_Osets_001tf__a,type,
sigma_sets_a: sigma_measure_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001tf__a,type,
sigma_4968961713055010667ebra_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Ospace_001_Eo,type,
sigma_space_o: sigma_measure_o > set_o ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Nat__Onat,type,
sigma_space_nat: sigma_measure_nat > set_nat ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__b,type,
sigma_space_b: sigma_measure_b > set_b ).
thf(sy_c_member_001_062_Itf__a_Mt__Real__Oreal_J,type,
member_a_real: ( a > real ) > set_a_real > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_I,type,
i: set_b ).
thf(sy_v_M,type,
m: sigma_measure_a ).
thf(sy_v_f,type,
f: b > a > real ).
% Relevant facts (1272)
thf(fact_0_assms_I1_J,axiom,
finite_finite_b @ i ).
% assms(1)
thf(fact_1_prob__space__axioms,axiom,
probab7247484486040049089pace_a @ m ).
% prob_space_axioms
thf(fact_2_prob__space_Ocovariance_Ocong,axiom,
probab3938396695707481060a_real = probab3938396695707481060a_real ).
% prob_space.covariance.cong
thf(fact_3_insert__Diff__single,axiom,
! [A: a > real,A2: set_a_real] :
( ( insert_a_real @ A @ ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) )
= ( insert_a_real @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_4_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_5_insert__Diff__single,axiom,
! [A: set_a,A2: set_set_a] :
( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= ( insert_set_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_6_insert__Diff__single,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( insert_o @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_7_insert__Diff__single,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( insert_b @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_8_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_9_singleton__conv,axiom,
! [A: a > real] :
( ( collect_a_real
@ ^ [X: a > real] : ( X = A ) )
= ( insert_a_real @ A @ bot_bot_set_a_real ) ) ).
% singleton_conv
thf(fact_10_singleton__conv,axiom,
! [A: set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( X = A ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singleton_conv
thf(fact_11_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_12_singleton__conv,axiom,
! [A: b] :
( ( collect_b
@ ^ [X: b] : ( X = A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv
thf(fact_13_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X: a] : ( X = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_14_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X: $o] : ( X = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_15_singleton__conv2,axiom,
! [A: a > real] :
( ( collect_a_real
@ ( ^ [Y: a > real,Z: a > real] : ( Y = Z )
@ A ) )
= ( insert_a_real @ A @ bot_bot_set_a_real ) ) ).
% singleton_conv2
thf(fact_16_singleton__conv2,axiom,
! [A: set_a] :
( ( collect_set_a
@ ( ^ [Y: set_a,Z: set_a] : ( Y = Z )
@ A ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singleton_conv2
thf(fact_17_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y: nat,Z: nat] : ( Y = Z )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_18_singleton__conv2,axiom,
! [A: b] :
( ( collect_b
@ ( ^ [Y: b,Z: b] : ( Y = Z )
@ A ) )
= ( insert_b @ A @ bot_bot_set_b ) ) ).
% singleton_conv2
thf(fact_19_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y: a,Z: a] : ( Y = Z )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_20_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y: $o,Z: $o] : ( Y = Z )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_21_Diff__insert0,axiom,
! [X2: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X2 @ A2 )
=> ( ( minus_minus_set_o @ A2 @ ( insert_o @ X2 @ B ) )
= ( minus_minus_set_o @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_22_Diff__insert0,axiom,
! [X2: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X2 @ B ) )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_23_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_24_Diff__insert0,axiom,
! [X2: a > real,A2: set_a_real,B: set_a_real] :
( ~ ( member_a_real @ X2 @ A2 )
=> ( ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ X2 @ B ) )
= ( minus_4124197362600706274a_real @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_25_Diff__insert0,axiom,
! [X2: b,A2: set_b,B: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ B ) )
= ( minus_minus_set_b @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_26_Diff__insert0,axiom,
! [X2: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_27_insert__Diff1,axiom,
! [X2: $o,B: set_o,A2: set_o] :
( ( member_o @ X2 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A2 ) @ B )
= ( minus_minus_set_o @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_28_insert__Diff1,axiom,
! [X2: set_a,B: set_set_a,A2: set_set_a] :
( ( member_set_a @ X2 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A2 ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_29_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_30_insert__Diff1,axiom,
! [X2: a > real,B: set_a_real,A2: set_a_real] :
( ( member_a_real @ X2 @ B )
=> ( ( minus_4124197362600706274a_real @ ( insert_a_real @ X2 @ A2 ) @ B )
= ( minus_4124197362600706274a_real @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_31_insert__Diff1,axiom,
! [X2: b,B: set_b,A2: set_b] :
( ( member_b @ X2 @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B )
= ( minus_minus_set_b @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_32_insert__Diff1,axiom,
! [X2: a,B: set_a,A2: set_a] :
( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_33_Diff__empty,axiom,
! [A2: set_a_real] :
( ( minus_4124197362600706274a_real @ A2 @ bot_bot_set_a_real )
= A2 ) ).
% Diff_empty
thf(fact_34_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_35_Diff__empty,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ bot_bot_set_set_a )
= A2 ) ).
% Diff_empty
thf(fact_36_Diff__empty,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Diff_empty
thf(fact_37_Diff__empty,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ bot_bot_set_b )
= A2 ) ).
% Diff_empty
thf(fact_38_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_39_empty__Diff,axiom,
! [A2: set_a_real] :
( ( minus_4124197362600706274a_real @ bot_bot_set_a_real @ A2 )
= bot_bot_set_a_real ) ).
% empty_Diff
thf(fact_40_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_41_empty__Diff,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ bot_bot_set_set_a @ A2 )
= bot_bot_set_set_a ) ).
% empty_Diff
thf(fact_42_empty__Diff,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_43_empty__Diff,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ bot_bot_set_b @ A2 )
= bot_bot_set_b ) ).
% empty_Diff
thf(fact_44_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_45_Diff__cancel,axiom,
! [A2: set_b] :
( ( minus_minus_set_b @ A2 @ A2 )
= bot_bot_set_b ) ).
% Diff_cancel
thf(fact_46_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_47_Diff__cancel,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ A2 )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_48_Diff__cancel,axiom,
! [A2: set_a_real] :
( ( minus_4124197362600706274a_real @ A2 @ A2 )
= bot_bot_set_a_real ) ).
% Diff_cancel
thf(fact_49_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_50_Diff__cancel,axiom,
! [A2: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ A2 )
= bot_bot_set_set_a ) ).
% Diff_cancel
thf(fact_51_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_52_singletonI,axiom,
! [A: a > real] : ( member_a_real @ A @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) ).
% singletonI
thf(fact_53_singletonI,axiom,
! [A: b] : ( member_b @ A @ ( insert_b @ A @ bot_bot_set_b ) ) ).
% singletonI
thf(fact_54_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_55_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_56_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_57_add__diff__cancel,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel
thf(fact_58_diff__add__cancel,axiom,
! [A: real,B2: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ B2 )
= A ) ).
% diff_add_cancel
thf(fact_59_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_60_add__diff__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
= ( minus_minus_real @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_61_add__right__cancel,axiom,
! [B2: real,A: real,C: real] :
( ( ( plus_plus_real @ B2 @ A )
= ( plus_plus_real @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_62_add__right__cancel,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_63_add__left__cancel,axiom,
! [A: real,B2: real,C: real] :
( ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_64_add__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_65_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_66_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_67_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_68_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_69_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_70_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_71_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_72_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_73_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_74_all__not__in__conv,axiom,
! [A2: set_a_real] :
( ( ! [X: a > real] :
~ ( member_a_real @ X @ A2 ) )
= ( A2 = bot_bot_set_a_real ) ) ).
% all_not_in_conv
thf(fact_75_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X: b] :
~ ( member_b @ X @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_76_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_77_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X: $o] :
~ ( member_o @ X @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_78_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_79_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_80_empty__iff,axiom,
! [C: a > real] :
~ ( member_a_real @ C @ bot_bot_set_a_real ) ).
% empty_iff
thf(fact_81_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_82_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_83_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_84_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_85_insert__absorb2,axiom,
! [X2: b,A2: set_b] :
( ( insert_b @ X2 @ ( insert_b @ X2 @ A2 ) )
= ( insert_b @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_86_insert__absorb2,axiom,
! [X2: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A2 ) )
= ( insert_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_87_insert__absorb2,axiom,
! [X2: $o,A2: set_o] :
( ( insert_o @ X2 @ ( insert_o @ X2 @ A2 ) )
= ( insert_o @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_88_insert__absorb2,axiom,
! [X2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A2 ) )
= ( insert_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_89_insert__absorb2,axiom,
! [X2: a > real,A2: set_a_real] :
( ( insert_a_real @ X2 @ ( insert_a_real @ X2 @ A2 ) )
= ( insert_a_real @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_90_insert__absorb2,axiom,
! [X2: set_a,A2: set_set_a] :
( ( insert_set_a @ X2 @ ( insert_set_a @ X2 @ A2 ) )
= ( insert_set_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_91_insert__iff,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_92_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_93_insert__iff,axiom,
! [A: a > real,B2: a > real,A2: set_a_real] :
( ( member_a_real @ A @ ( insert_a_real @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_a_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_94_insert__iff,axiom,
! [A: b,B2: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_95_insert__iff,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_96_insert__iff,axiom,
! [A: $o,B2: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_97_insertCI,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( ~ ( member_set_a @ A @ B )
=> ( A = B2 ) )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_98_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_99_insertCI,axiom,
! [A: a > real,B: set_a_real,B2: a > real] :
( ( ~ ( member_a_real @ A @ B )
=> ( A = B2 ) )
=> ( member_a_real @ A @ ( insert_a_real @ B2 @ B ) ) ) ).
% insertCI
thf(fact_100_insertCI,axiom,
! [A: b,B: set_b,B2: b] :
( ( ~ ( member_b @ A @ B )
=> ( A = B2 ) )
=> ( member_b @ A @ ( insert_b @ B2 @ B ) ) ) ).
% insertCI
thf(fact_101_insertCI,axiom,
! [A: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A @ B )
=> ( A = B2 ) )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_102_insertCI,axiom,
! [A: $o,B: set_o,B2: $o] :
( ( ~ ( member_o @ A @ B )
=> ( A = B2 ) )
=> ( member_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% insertCI
thf(fact_103_Diff__idemp,axiom,
! [A2: set_b,B: set_b] :
( ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B ) @ B )
= ( minus_minus_set_b @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_104_Diff__idemp,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_105_Diff__idemp,axiom,
! [A2: set_o,B: set_o] :
( ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B ) @ B )
= ( minus_minus_set_o @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_106_Diff__idemp,axiom,
! [A2: set_a_real,B: set_a_real] :
( ( minus_4124197362600706274a_real @ ( minus_4124197362600706274a_real @ A2 @ B ) @ B )
= ( minus_4124197362600706274a_real @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_107_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_108_Diff__idemp,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_109_Diff__iff,axiom,
! [C: b,A2: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) )
= ( ( member_b @ C @ A2 )
& ~ ( member_b @ C @ B ) ) ) ).
% Diff_iff
thf(fact_110_Diff__iff,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_111_Diff__iff,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) )
= ( ( member_o @ C @ A2 )
& ~ ( member_o @ C @ B ) ) ) ).
% Diff_iff
thf(fact_112_Diff__iff,axiom,
! [C: a > real,A2: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B ) )
= ( ( member_a_real @ C @ A2 )
& ~ ( member_a_real @ C @ B ) ) ) ).
% Diff_iff
thf(fact_113_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_114_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_115_DiffI,axiom,
! [C: b,A2: set_b,B: set_b] :
( ( member_b @ C @ A2 )
=> ( ~ ( member_b @ C @ B )
=> ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_116_DiffI,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_117_DiffI,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ A2 )
=> ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_118_DiffI,axiom,
! [C: a > real,A2: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ A2 )
=> ( ~ ( member_a_real @ C @ B )
=> ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_119_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_120_DiffI,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_121_add__diff__cancel__right_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_122_add__diff__cancel__right_H,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_123_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_124_add__diff__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( minus_minus_real @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_125_add__diff__cancel__left_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_126_add__diff__cancel__left_H,axiom,
! [A: real,B2: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_127_minus__set__def,axiom,
( minus_minus_set_b
= ( ^ [A3: set_b,B3: set_b] :
( collect_b
@ ( minus_minus_b_o
@ ^ [X: b] : ( member_b @ X @ A3 )
@ ^ [X: b] : ( member_b @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_128_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X: a] : ( member_a @ X @ A3 )
@ ^ [X: a] : ( member_a @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_129_minus__set__def,axiom,
( minus_minus_set_o
= ( ^ [A3: set_o,B3: set_o] :
( collect_o
@ ( minus_minus_o_o
@ ^ [X: $o] : ( member_o @ X @ A3 )
@ ^ [X: $o] : ( member_o @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_130_minus__set__def,axiom,
( minus_4124197362600706274a_real
= ( ^ [A3: set_a_real,B3: set_a_real] :
( collect_a_real
@ ( minus_minus_a_real_o
@ ^ [X: a > real] : ( member_a_real @ X @ A3 )
@ ^ [X: a > real] : ( member_a_real @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_131_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_132_minus__set__def,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ( minus_minus_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A3 )
@ ^ [X: set_a] : ( member_set_a @ X @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_133_add__right__imp__eq,axiom,
! [B2: real,A: real,C: real] :
( ( ( plus_plus_real @ B2 @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_134_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_135_add__left__imp__eq,axiom,
! [A: real,B2: real,C: real] :
( ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_136_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_137_add_Oleft__commute,axiom,
! [B2: real,A: real,C: real] :
( ( plus_plus_real @ B2 @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_138_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_139_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_140_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_141_add_Oright__cancel,axiom,
! [B2: real,A: real,C: real] :
( ( ( plus_plus_real @ B2 @ A )
= ( plus_plus_real @ C @ A ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_142_add_Oleft__cancel,axiom,
! [A: real,B2: real,C: real] :
( ( ( plus_plus_real @ A @ B2 )
= ( plus_plus_real @ A @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_143_add_Oassoc,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_144_add_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_145_group__cancel_Oadd2,axiom,
! [B: real,K: real,B2: real,A: real] :
( ( B
= ( plus_plus_real @ K @ B2 ) )
=> ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_146_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_147_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B2: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B2 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_148_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_149_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_150_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_151_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B2 ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_152_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_153_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_154_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 )
= ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_155_diff__eq__diff__eq,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D ) )
=> ( ( A = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_156_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_157_ex__in__conv,axiom,
! [A2: set_a_real] :
( ( ? [X: a > real] : ( member_a_real @ X @ A2 ) )
= ( A2 != bot_bot_set_a_real ) ) ).
% ex_in_conv
thf(fact_158_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X: b] : ( member_b @ X @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_159_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X: a] : ( member_a @ X @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_160_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X: $o] : ( member_o @ X @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_161_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_162_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_163_mem__Collect__eq,axiom,
! [A: a > real,P: ( a > real ) > $o] :
( ( member_a_real @ A @ ( collect_a_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_164_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_165_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_166_mem__Collect__eq,axiom,
! [A: $o,P: $o > $o] :
( ( member_o @ A @ ( collect_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_167_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_168_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
! [A2: set_a_real] :
( ( collect_a_real
@ ^ [X: a > real] : ( member_a_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_171_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_172_Collect__mem__eq,axiom,
! [A2: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_173_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X: b] : ( member_b @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_174_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_175_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_176_Collect__cong,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X3: $o] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_o @ P )
= ( collect_o @ Q ) ) ) ).
% Collect_cong
thf(fact_177_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_178_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_179_equals0I,axiom,
! [A2: set_a_real] :
( ! [Y2: a > real] :
~ ( member_a_real @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a_real ) ) ).
% equals0I
thf(fact_180_equals0I,axiom,
! [A2: set_b] :
( ! [Y2: b] :
~ ( member_b @ Y2 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_181_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_182_equals0I,axiom,
! [A2: set_o] :
( ! [Y2: $o] :
~ ( member_o @ Y2 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_183_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_184_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_185_equals0D,axiom,
! [A2: set_a_real,A: a > real] :
( ( A2 = bot_bot_set_a_real )
=> ~ ( member_a_real @ A @ A2 ) ) ).
% equals0D
thf(fact_186_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_187_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_188_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_189_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_190_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_191_emptyE,axiom,
! [A: a > real] :
~ ( member_a_real @ A @ bot_bot_set_a_real ) ).
% emptyE
thf(fact_192_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_193_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_194_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_195_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_196_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B5: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B5 ) )
& ~ ( member_set_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_197_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B5: set_nat] :
( ( A2
= ( insert_nat @ A @ B5 ) )
& ~ ( member_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_198_mk__disjoint__insert,axiom,
! [A: a > real,A2: set_a_real] :
( ( member_a_real @ A @ A2 )
=> ? [B5: set_a_real] :
( ( A2
= ( insert_a_real @ A @ B5 ) )
& ~ ( member_a_real @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_199_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B5: set_b] :
( ( A2
= ( insert_b @ A @ B5 ) )
& ~ ( member_b @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_200_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B5: set_a] :
( ( A2
= ( insert_a @ A @ B5 ) )
& ~ ( member_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_201_mk__disjoint__insert,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ? [B5: set_o] :
( ( A2
= ( insert_o @ A @ B5 ) )
& ~ ( member_o @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_202_insert__commute,axiom,
! [X2: b,Y3: b,A2: set_b] :
( ( insert_b @ X2 @ ( insert_b @ Y3 @ A2 ) )
= ( insert_b @ Y3 @ ( insert_b @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_203_insert__commute,axiom,
! [X2: a,Y3: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y3 @ A2 ) )
= ( insert_a @ Y3 @ ( insert_a @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_204_insert__commute,axiom,
! [X2: $o,Y3: $o,A2: set_o] :
( ( insert_o @ X2 @ ( insert_o @ Y3 @ A2 ) )
= ( insert_o @ Y3 @ ( insert_o @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_205_insert__commute,axiom,
! [X2: nat,Y3: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y3 @ A2 ) )
= ( insert_nat @ Y3 @ ( insert_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_206_insert__commute,axiom,
! [X2: a > real,Y3: a > real,A2: set_a_real] :
( ( insert_a_real @ X2 @ ( insert_a_real @ Y3 @ A2 ) )
= ( insert_a_real @ Y3 @ ( insert_a_real @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_207_insert__commute,axiom,
! [X2: set_a,Y3: set_a,A2: set_set_a] :
( ( insert_set_a @ X2 @ ( insert_set_a @ Y3 @ A2 ) )
= ( insert_set_a @ Y3 @ ( insert_set_a @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_208_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B2: set_a,B: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B2 @ B )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_set_a] :
( ( A2
= ( insert_set_a @ B2 @ C2 ) )
& ~ ( member_set_a @ B2 @ C2 )
& ( B
= ( insert_set_a @ A @ C2 ) )
& ~ ( member_set_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_209_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_210_insert__eq__iff,axiom,
! [A: a > real,A2: set_a_real,B2: a > real,B: set_a_real] :
( ~ ( member_a_real @ A @ A2 )
=> ( ~ ( member_a_real @ B2 @ B )
=> ( ( ( insert_a_real @ A @ A2 )
= ( insert_a_real @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_a_real] :
( ( A2
= ( insert_a_real @ B2 @ C2 ) )
& ~ ( member_a_real @ B2 @ C2 )
& ( B
= ( insert_a_real @ A @ C2 ) )
& ~ ( member_a_real @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_211_insert__eq__iff,axiom,
! [A: b,A2: set_b,B2: b,B: set_b] :
( ~ ( member_b @ A @ A2 )
=> ( ~ ( member_b @ B2 @ B )
=> ( ( ( insert_b @ A @ A2 )
= ( insert_b @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_b] :
( ( A2
= ( insert_b @ B2 @ C2 ) )
& ~ ( member_b @ B2 @ C2 )
& ( B
= ( insert_b @ A @ C2 ) )
& ~ ( member_b @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_212_insert__eq__iff,axiom,
! [A: a,A2: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_a] :
( ( A2
= ( insert_a @ B2 @ C2 ) )
& ~ ( member_a @ B2 @ C2 )
& ( B
= ( insert_a @ A @ C2 ) )
& ~ ( member_a @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_213_insert__eq__iff,axiom,
! [A: $o,A2: set_o,B2: $o,B: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ~ ( member_o @ B2 @ B )
=> ( ( ( insert_o @ A @ A2 )
= ( insert_o @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A = ~ B2 )
=> ? [C2: set_o] :
( ( A2
= ( insert_o @ B2 @ C2 ) )
& ~ ( member_o @ B2 @ C2 )
& ( B
= ( insert_o @ A @ C2 ) )
& ~ ( member_o @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_214_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_215_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_216_insert__absorb,axiom,
! [A: a > real,A2: set_a_real] :
( ( member_a_real @ A @ A2 )
=> ( ( insert_a_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_217_insert__absorb,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_218_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_219_insert__absorb,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_220_insert__ident,axiom,
! [X2: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A2 )
=> ( ~ ( member_set_a @ X2 @ B )
=> ( ( ( insert_set_a @ X2 @ A2 )
= ( insert_set_a @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_221_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_222_insert__ident,axiom,
! [X2: a > real,A2: set_a_real,B: set_a_real] :
( ~ ( member_a_real @ X2 @ A2 )
=> ( ~ ( member_a_real @ X2 @ B )
=> ( ( ( insert_a_real @ X2 @ A2 )
= ( insert_a_real @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_223_insert__ident,axiom,
! [X2: b,A2: set_b,B: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ~ ( member_b @ X2 @ B )
=> ( ( ( insert_b @ X2 @ A2 )
= ( insert_b @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_224_insert__ident,axiom,
! [X2: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ~ ( member_a @ X2 @ B )
=> ( ( ( insert_a @ X2 @ A2 )
= ( insert_a @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_225_insert__ident,axiom,
! [X2: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X2 @ A2 )
=> ( ~ ( member_o @ X2 @ B )
=> ( ( ( insert_o @ X2 @ A2 )
= ( insert_o @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_226_Set_Oset__insert,axiom,
! [X2: set_a,A2: set_set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ! [B5: set_set_a] :
( ( A2
= ( insert_set_a @ X2 @ B5 ) )
=> ( member_set_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_227_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ! [B5: set_nat] :
( ( A2
= ( insert_nat @ X2 @ B5 ) )
=> ( member_nat @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_228_Set_Oset__insert,axiom,
! [X2: a > real,A2: set_a_real] :
( ( member_a_real @ X2 @ A2 )
=> ~ ! [B5: set_a_real] :
( ( A2
= ( insert_a_real @ X2 @ B5 ) )
=> ( member_a_real @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_229_Set_Oset__insert,axiom,
! [X2: b,A2: set_b] :
( ( member_b @ X2 @ A2 )
=> ~ ! [B5: set_b] :
( ( A2
= ( insert_b @ X2 @ B5 ) )
=> ( member_b @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_230_Set_Oset__insert,axiom,
! [X2: a,A2: set_a] :
( ( member_a @ X2 @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a @ X2 @ B5 ) )
=> ( member_a @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_231_Set_Oset__insert,axiom,
! [X2: $o,A2: set_o] :
( ( member_o @ X2 @ A2 )
=> ~ ! [B5: set_o] :
( ( A2
= ( insert_o @ X2 @ B5 ) )
=> ( member_o @ X2 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_232_insertI2,axiom,
! [A: set_a,B: set_set_a,B2: set_a] :
( ( member_set_a @ A @ B )
=> ( member_set_a @ A @ ( insert_set_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_233_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_234_insertI2,axiom,
! [A: a > real,B: set_a_real,B2: a > real] :
( ( member_a_real @ A @ B )
=> ( member_a_real @ A @ ( insert_a_real @ B2 @ B ) ) ) ).
% insertI2
thf(fact_235_insertI2,axiom,
! [A: b,B: set_b,B2: b] :
( ( member_b @ A @ B )
=> ( member_b @ A @ ( insert_b @ B2 @ B ) ) ) ).
% insertI2
thf(fact_236_insertI2,axiom,
! [A: a,B: set_a,B2: a] :
( ( member_a @ A @ B )
=> ( member_a @ A @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_237_insertI2,axiom,
! [A: $o,B: set_o,B2: $o] :
( ( member_o @ A @ B )
=> ( member_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% insertI2
thf(fact_238_insertI1,axiom,
! [A: set_a,B: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B ) ) ).
% insertI1
thf(fact_239_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_240_insertI1,axiom,
! [A: a > real,B: set_a_real] : ( member_a_real @ A @ ( insert_a_real @ A @ B ) ) ).
% insertI1
thf(fact_241_insertI1,axiom,
! [A: b,B: set_b] : ( member_b @ A @ ( insert_b @ A @ B ) ) ).
% insertI1
thf(fact_242_insertI1,axiom,
! [A: a,B: set_a] : ( member_a @ A @ ( insert_a @ A @ B ) ) ).
% insertI1
thf(fact_243_insertI1,axiom,
! [A: $o,B: set_o] : ( member_o @ A @ ( insert_o @ A @ B ) ) ).
% insertI1
thf(fact_244_insertE,axiom,
! [A: set_a,B2: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
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: a > real,B2: a > real,A2: set_a_real] :
( ( member_a_real @ A @ ( insert_a_real @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_a_real @ A @ A2 ) ) ) ).
% insertE
thf(fact_247_insertE,axiom,
! [A: b,B2: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_b @ A @ A2 ) ) ) ).
% insertE
thf(fact_248_insertE,axiom,
! [A: a,B2: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_249_insertE,axiom,
! [A: $o,B2: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B2 @ A2 ) )
=> ( ( A = ~ B2 )
=> ( member_o @ A @ A2 ) ) ) ).
% insertE
thf(fact_250_DiffD2,axiom,
! [C: b,A2: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) )
=> ~ ( member_b @ C @ B ) ) ).
% DiffD2
thf(fact_251_DiffD2,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_252_DiffD2,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) )
=> ~ ( member_o @ C @ B ) ) ).
% DiffD2
thf(fact_253_DiffD2,axiom,
! [C: a > real,A2: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B ) )
=> ~ ( member_a_real @ C @ B ) ) ).
% DiffD2
thf(fact_254_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_255_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ~ ( member_set_a @ C @ B ) ) ).
% DiffD2
thf(fact_256_DiffD1,axiom,
! [C: b,A2: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) )
=> ( member_b @ C @ A2 ) ) ).
% DiffD1
thf(fact_257_DiffD1,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_258_DiffD1,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) )
=> ( member_o @ C @ A2 ) ) ).
% DiffD1
thf(fact_259_DiffD1,axiom,
! [C: a > real,A2: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B ) )
=> ( member_a_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_260_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_261_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_262_DiffE,axiom,
! [C: b,A2: set_b,B: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B ) )
=> ~ ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B ) ) ) ).
% DiffE
thf(fact_263_DiffE,axiom,
! [C: a,A2: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_264_DiffE,axiom,
! [C: $o,A2: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A2 @ B ) )
=> ~ ( ( member_o @ C @ A2 )
=> ( member_o @ C @ B ) ) ) ).
% DiffE
thf(fact_265_DiffE,axiom,
! [C: a > real,A2: set_a_real,B: set_a_real] :
( ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B ) )
=> ~ ( ( member_a_real @ C @ A2 )
=> ( member_a_real @ C @ B ) ) ) ).
% DiffE
thf(fact_266_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_267_DiffE,axiom,
! [C: set_a,A2: set_set_a,B: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% DiffE
thf(fact_268_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X: b] : $false ) ) ).
% empty_def
thf(fact_269_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% empty_def
thf(fact_270_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X: $o] : $false ) ) ).
% empty_def
thf(fact_271_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_272_insert__Collect,axiom,
! [A: a > real,P: ( a > real ) > $o] :
( ( insert_a_real @ A @ ( collect_a_real @ P ) )
= ( collect_a_real
@ ^ [U: a > real] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_273_insert__Collect,axiom,
! [A: set_a,P: set_a > $o] :
( ( insert_set_a @ A @ ( collect_set_a @ P ) )
= ( collect_set_a
@ ^ [U: set_a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_274_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_275_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U: a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_276_insert__Collect,axiom,
! [A: $o,P: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U: $o] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_277_insert__Collect,axiom,
! [A: b,P: b > $o] :
( ( insert_b @ A @ ( collect_b @ P ) )
= ( collect_b
@ ^ [U: b] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_278_insert__compr,axiom,
( insert_set_a
= ( ^ [A4: set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X: set_a] :
( ( X = A4 )
| ( member_set_a @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_279_insert__compr,axiom,
( insert_a_real
= ( ^ [A4: a > real,B3: set_a_real] :
( collect_a_real
@ ^ [X: a > real] :
( ( X = A4 )
| ( member_a_real @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_280_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A4 )
| ( member_nat @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_281_insert__compr,axiom,
( insert_a
= ( ^ [A4: a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( X = A4 )
| ( member_a @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_282_insert__compr,axiom,
( insert_o
= ( ^ [A4: $o,B3: set_o] :
( collect_o
@ ^ [X: $o] :
( ( X = A4 )
| ( member_o @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_283_insert__compr,axiom,
( insert_b
= ( ^ [A4: b,B3: set_b] :
( collect_b
@ ^ [X: b] :
( ( X = A4 )
| ( member_b @ X @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_284_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A3: set_b,B3: set_b] :
( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A3 )
& ~ ( member_b @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_285_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A3 )
& ~ ( member_a @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_286_set__diff__eq,axiom,
( minus_minus_set_o
= ( ^ [A3: set_o,B3: set_o] :
( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A3 )
& ~ ( member_o @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_287_set__diff__eq,axiom,
( minus_4124197362600706274a_real
= ( ^ [A3: set_a_real,B3: set_a_real] :
( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ A3 )
& ~ ( member_a_real @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_288_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ~ ( member_nat @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_289_set__diff__eq,axiom,
( minus_5736297505244876581_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A3 )
& ~ ( member_set_a @ X @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_290_diff__diff__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_291_diff__diff__eq,axiom,
! [A: real,B2: real,C: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_292_add__implies__diff,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A )
=> ( C
= ( minus_minus_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_293_add__implies__diff,axiom,
! [C: real,B2: real,A: real] :
( ( ( plus_plus_real @ C @ B2 )
= A )
=> ( C
= ( minus_minus_real @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_294_diff__add__eq__diff__diff__swap,axiom,
! [A: real,B2: real,C: real] :
( ( minus_minus_real @ A @ ( plus_plus_real @ B2 @ C ) )
= ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_295_diff__add__eq,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_296_diff__diff__eq2,axiom,
! [A: real,B2: real,C: real] :
( ( minus_minus_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_297_add__diff__eq,axiom,
! [A: real,B2: real,C: real] :
( ( plus_plus_real @ A @ ( minus_minus_real @ B2 @ C ) )
= ( minus_minus_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_298_eq__diff__eq,axiom,
! [A: real,C: real,B2: real] :
( ( A
= ( minus_minus_real @ C @ B2 ) )
= ( ( plus_plus_real @ A @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_299_diff__eq__eq,axiom,
! [A: real,B2: real,C: real] :
( ( ( minus_minus_real @ A @ B2 )
= C )
= ( A
= ( plus_plus_real @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_300_group__cancel_Osub1,axiom,
! [A2: real,K: real,A: real,B2: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( minus_minus_real @ A2 @ B2 )
= ( plus_plus_real @ K @ ( minus_minus_real @ A @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_301_singleton__inject,axiom,
! [A: a > real,B2: a > real] :
( ( ( insert_a_real @ A @ bot_bot_set_a_real )
= ( insert_a_real @ B2 @ bot_bot_set_a_real ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_302_singleton__inject,axiom,
! [A: set_a,B2: set_a] :
( ( ( insert_set_a @ A @ bot_bot_set_set_a )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_303_singleton__inject,axiom,
! [A: b,B2: b] :
( ( ( insert_b @ A @ bot_bot_set_b )
= ( insert_b @ B2 @ bot_bot_set_b ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_304_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_305_singleton__inject,axiom,
! [A: $o,B2: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B2 @ bot_bot_set_o ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_306_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_307_insert__not__empty,axiom,
! [A: a > real,A2: set_a_real] :
( ( insert_a_real @ A @ A2 )
!= bot_bot_set_a_real ) ).
% insert_not_empty
thf(fact_308_insert__not__empty,axiom,
! [A: set_a,A2: set_set_a] :
( ( insert_set_a @ A @ A2 )
!= bot_bot_set_set_a ) ).
% insert_not_empty
thf(fact_309_insert__not__empty,axiom,
! [A: b,A2: set_b] :
( ( insert_b @ A @ A2 )
!= bot_bot_set_b ) ).
% insert_not_empty
thf(fact_310_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_311_insert__not__empty,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ A2 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_312_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_313_doubleton__eq__iff,axiom,
! [A: a > real,B2: a > real,C: a > real,D: a > real] :
( ( ( insert_a_real @ A @ ( insert_a_real @ B2 @ bot_bot_set_a_real ) )
= ( insert_a_real @ C @ ( insert_a_real @ D @ bot_bot_set_a_real ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_314_doubleton__eq__iff,axiom,
! [A: set_a,B2: set_a,C: set_a,D: set_a] :
( ( ( insert_set_a @ A @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( insert_set_a @ C @ ( insert_set_a @ D @ bot_bot_set_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_315_doubleton__eq__iff,axiom,
! [A: b,B2: b,C: b,D: b] :
( ( ( insert_b @ A @ ( insert_b @ B2 @ bot_bot_set_b ) )
= ( insert_b @ C @ ( insert_b @ D @ bot_bot_set_b ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_316_doubleton__eq__iff,axiom,
! [A: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_317_doubleton__eq__iff,axiom,
! [A: $o,B2: $o,C: $o,D: $o] :
( ( ( insert_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_318_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_319_singleton__iff,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_320_singleton__iff,axiom,
! [B2: a > real,A: a > real] :
( ( member_a_real @ B2 @ ( insert_a_real @ A @ bot_bot_set_a_real ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_321_singleton__iff,axiom,
! [B2: b,A: b] :
( ( member_b @ B2 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_322_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_323_singleton__iff,axiom,
! [B2: $o,A: $o] :
( ( member_o @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_324_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_325_singletonD,axiom,
! [B2: set_a,A: set_a] :
( ( member_set_a @ B2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_326_singletonD,axiom,
! [B2: a > real,A: a > real] :
( ( member_a_real @ B2 @ ( insert_a_real @ A @ bot_bot_set_a_real ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_327_singletonD,axiom,
! [B2: b,A: b] :
( ( member_b @ B2 @ ( insert_b @ A @ bot_bot_set_b ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_328_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_329_singletonD,axiom,
! [B2: $o,A: $o] :
( ( member_o @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_330_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_331_insert__Diff__if,axiom,
! [X2: b,B: set_b,A2: set_b] :
( ( ( member_b @ X2 @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B )
= ( minus_minus_set_b @ A2 @ B ) ) )
& ( ~ ( member_b @ X2 @ B )
=> ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B )
= ( insert_b @ X2 @ ( minus_minus_set_b @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_332_insert__Diff__if,axiom,
! [X2: a,B: set_a,A2: set_a] :
( ( ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X2 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_333_insert__Diff__if,axiom,
! [X2: $o,B: set_o,A2: set_o] :
( ( ( member_o @ X2 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A2 ) @ B )
= ( minus_minus_set_o @ A2 @ B ) ) )
& ( ~ ( member_o @ X2 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A2 ) @ B )
= ( insert_o @ X2 @ ( minus_minus_set_o @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_334_insert__Diff__if,axiom,
! [X2: a > real,B: set_a_real,A2: set_a_real] :
( ( ( member_a_real @ X2 @ B )
=> ( ( minus_4124197362600706274a_real @ ( insert_a_real @ X2 @ A2 ) @ B )
= ( minus_4124197362600706274a_real @ A2 @ B ) ) )
& ( ~ ( member_a_real @ X2 @ B )
=> ( ( minus_4124197362600706274a_real @ ( insert_a_real @ X2 @ A2 ) @ B )
= ( insert_a_real @ X2 @ ( minus_4124197362600706274a_real @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_335_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_336_insert__Diff__if,axiom,
! [X2: set_a,B: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X2 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A2 ) @ B )
= ( minus_5736297505244876581_set_a @ A2 @ B ) ) )
& ( ~ ( member_set_a @ X2 @ B )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A2 ) @ B )
= ( insert_set_a @ X2 @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_337_Collect__conv__if2,axiom,
! [P: ( a > real ) > $o,A: a > real] :
( ( ( P @ A )
=> ( ( collect_a_real
@ ^ [X: a > real] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_a_real @ A @ bot_bot_set_a_real ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a_real
@ ^ [X: a > real] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_a_real ) ) ) ).
% Collect_conv_if2
thf(fact_338_Collect__conv__if2,axiom,
! [P: set_a > $o,A: set_a] :
( ( ( P @ A )
=> ( ( collect_set_a
@ ^ [X: set_a] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_a
@ ^ [X: set_a] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_339_Collect__conv__if2,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if2
thf(fact_340_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_341_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X: $o] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X: $o] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_342_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_343_Collect__conv__if,axiom,
! [P: ( a > real ) > $o,A: a > real] :
( ( ( P @ A )
=> ( ( collect_a_real
@ ^ [X: a > real] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_a_real @ A @ bot_bot_set_a_real ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a_real
@ ^ [X: a > real] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_a_real ) ) ) ).
% Collect_conv_if
thf(fact_344_Collect__conv__if,axiom,
! [P: set_a > $o,A: set_a] :
( ( ( P @ A )
=> ( ( collect_set_a
@ ^ [X: set_a] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_set_a
@ ^ [X: set_a] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_set_a ) ) ) ).
% Collect_conv_if
thf(fact_345_Collect__conv__if,axiom,
! [P: b > $o,A: b] :
( ( ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_b @ A @ bot_bot_set_b ) ) )
& ( ~ ( P @ A )
=> ( ( collect_b
@ ^ [X: b] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_b ) ) ) ).
% Collect_conv_if
thf(fact_346_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X: a] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_347_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X: $o] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X: $o] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_348_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_349_Diff__insert__absorb,axiom,
! [X2: b,A2: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ ( insert_b @ X2 @ bot_bot_set_b ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_350_Diff__insert__absorb,axiom,
! [X2: a,A2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_351_Diff__insert__absorb,axiom,
! [X2: $o,A2: set_o] :
( ~ ( member_o @ X2 @ A2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X2 @ A2 ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_352_Diff__insert__absorb,axiom,
! [X2: a > real,A2: set_a_real] :
( ~ ( member_a_real @ X2 @ A2 )
=> ( ( minus_4124197362600706274a_real @ ( insert_a_real @ X2 @ A2 ) @ ( insert_a_real @ X2 @ bot_bot_set_a_real ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_353_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_354_Diff__insert__absorb,axiom,
! [X2: set_a,A2: set_set_a] :
( ~ ( member_set_a @ X2 @ A2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X2 @ A2 ) @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_355_Diff__insert2,axiom,
! [A2: set_b,A: b,B: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) @ B ) ) ).
% Diff_insert2
thf(fact_356_Diff__insert2,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_357_Diff__insert2,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B ) ) ).
% Diff_insert2
thf(fact_358_Diff__insert2,axiom,
! [A2: set_a_real,A: a > real,B: set_a_real] :
( ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ B ) )
= ( minus_4124197362600706274a_real @ ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) @ B ) ) ).
% Diff_insert2
thf(fact_359_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_360_Diff__insert2,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_361_insert__Diff,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ( ( insert_b @ A @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_362_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_363_insert__Diff,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_364_insert__Diff,axiom,
! [A: a > real,A2: set_a_real] :
( ( member_a_real @ A @ A2 )
=> ( ( insert_a_real @ A @ ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_365_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_366_insert__Diff,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_367_Diff__insert,axiom,
! [A2: set_b,A: b,B: set_b] :
( ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B ) )
= ( minus_minus_set_b @ ( minus_minus_set_b @ A2 @ B ) @ ( insert_b @ A @ bot_bot_set_b ) ) ) ).
% Diff_insert
thf(fact_368_Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_369_Diff__insert,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_370_Diff__insert,axiom,
! [A2: set_a_real,A: a > real,B: set_a_real] :
( ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ B ) )
= ( minus_4124197362600706274a_real @ ( minus_4124197362600706274a_real @ A2 @ B ) @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) ) ).
% Diff_insert
thf(fact_371_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_372_Diff__insert,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B ) )
= ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ).
% Diff_insert
thf(fact_373_sum_Oinsert,axiom,
! [A2: set_o,X2: $o,G: $o > real] :
( ( finite_finite_o @ A2 )
=> ( ~ ( member_o @ X2 @ A2 )
=> ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups8691415230153176458o_real @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_374_sum_Oinsert,axiom,
! [A2: set_nat,X2: nat,G: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X2 @ A2 )
=> ( ( groups6591440286371151544t_real @ G @ ( insert_nat @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_375_sum_Oinsert,axiom,
! [A2: set_o,X2: $o,G: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ~ ( member_o @ X2 @ A2 )
=> ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_376_sum_Oinsert,axiom,
! [A2: set_b,X2: b,G: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups7570001007293516437_b_nat @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups7570001007293516437_b_nat @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_377_sum_Oinsert,axiom,
! [A2: set_nat,X2: nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X2 @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( insert_nat @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_378_sum_Oinsert,axiom,
! [A2: set_a,X2: a,G: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups6334556678337121940_a_nat @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_379_sum_Oinsert,axiom,
! [A2: set_b,X2: b,G: b > real] :
( ( finite_finite_b @ A2 )
=> ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups8336678772925405937b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups8336678772925405937b_real @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_380_sum_Oinsert,axiom,
! [A2: set_a,X2: a,G: a > real] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups2740460157737275248a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups2740460157737275248a_real @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_381_sum_Oinsert,axiom,
! [A2: set_set_a,X2: set_a,G: set_a > real] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X2 @ A2 )
=> ( ( groups9174420418583655632a_real @ G @ ( insert_set_a @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups9174420418583655632a_real @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_382_sum_Oinsert,axiom,
! [A2: set_set_a,X2: set_a,G: set_a > nat] :
( ( finite_finite_set_a @ A2 )
=> ( ~ ( member_set_a @ X2 @ A2 )
=> ( ( groups6141743369313575924_a_nat @ G @ ( insert_set_a @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups6141743369313575924_a_nat @ G @ A2 ) ) ) ) ) ).
% sum.insert
thf(fact_383_finite__Diff__insert,axiom,
! [A2: set_b,A: b,B: set_b] :
( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B ) ) )
= ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_384_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_385_finite__Diff__insert,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( finite_finite_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B ) ) )
= ( finite_finite_o @ ( minus_minus_set_o @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_386_finite__Diff__insert,axiom,
! [A2: set_a_real,A: a > real,B: set_a_real] :
( ( finite_finite_a_real @ ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ B ) ) )
= ( finite_finite_a_real @ ( minus_4124197362600706274a_real @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_387_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_388_finite__Diff__insert,axiom,
! [A2: set_set_a,A: set_a,B: set_set_a] :
( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B ) ) )
= ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_389_sum_Odelta__remove,axiom,
! [S: set_b,A: b,B2: b > nat,C: b > nat] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups7570001007293516437_b_nat
@ ^ [K2: b] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_nat @ ( B2 @ A ) @ ( groups7570001007293516437_b_nat @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups7570001007293516437_b_nat
@ ^ [K2: b] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups7570001007293516437_b_nat @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_390_sum_Odelta__remove,axiom,
! [S: set_a,A: a,B2: a > nat,C: a > nat] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_nat @ ( B2 @ A ) @ ( groups6334556678337121940_a_nat @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K2: a] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups6334556678337121940_a_nat @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_391_sum_Odelta__remove,axiom,
! [S: set_o,A: $o,B2: $o > real,C: $o > real] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A @ S )
=> ( ( groups8691415230153176458o_real
@ ^ [K2: $o] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_real @ ( B2 @ A ) @ ( groups8691415230153176458o_real @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ A @ S )
=> ( ( groups8691415230153176458o_real
@ ^ [K2: $o] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups8691415230153176458o_real @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_392_sum_Odelta__remove,axiom,
! [S: set_o,A: $o,B2: $o > nat,C: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ A @ S )
=> ( ( groups8507830703676809646_o_nat
@ ^ [K2: $o] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_nat @ ( B2 @ A ) @ ( groups8507830703676809646_o_nat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
& ( ~ ( member_o @ A @ S )
=> ( ( groups8507830703676809646_o_nat
@ ^ [K2: $o] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups8507830703676809646_o_nat @ C @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_393_sum_Odelta__remove,axiom,
! [S: set_nat,A: nat,B2: nat > real,C: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_real @ ( B2 @ A ) @ ( groups6591440286371151544t_real @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups6591440286371151544t_real
@ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups6591440286371151544t_real @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_394_sum_Odelta__remove,axiom,
! [S: set_nat,A: nat,B2: nat > nat,C: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_nat @ ( B2 @ A ) @ ( groups3542108847815614940at_nat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
& ( ~ ( member_nat @ A @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups3542108847815614940at_nat @ C @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_395_sum_Odelta__remove,axiom,
! [S: set_b,A: b,B2: b > real,C: b > real] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ A @ S )
=> ( ( groups8336678772925405937b_real
@ ^ [K2: b] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_real @ ( B2 @ A ) @ ( groups8336678772925405937b_real @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) )
& ( ~ ( member_b @ A @ S )
=> ( ( groups8336678772925405937b_real
@ ^ [K2: b] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups8336678772925405937b_real @ C @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_396_sum_Odelta__remove,axiom,
! [S: set_a,A: a,B2: a > real,C: a > real] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ A @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [K2: a] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_real @ ( B2 @ A ) @ ( groups2740460157737275248a_real @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) )
& ( ~ ( member_a @ A @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [K2: a] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups2740460157737275248a_real @ C @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_397_sum_Odelta__remove,axiom,
! [S: set_set_a,A: set_a,B2: set_a > real,C: set_a > real] :
( ( finite_finite_set_a @ S )
=> ( ( ( member_set_a @ A @ S )
=> ( ( groups9174420418583655632a_real
@ ^ [K2: set_a] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_real @ ( B2 @ A ) @ ( groups9174420418583655632a_real @ C @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ) )
& ( ~ ( member_set_a @ A @ S )
=> ( ( groups9174420418583655632a_real
@ ^ [K2: set_a] : ( if_real @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups9174420418583655632a_real @ C @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_398_sum_Odelta__remove,axiom,
! [S: set_set_a,A: set_a,B2: set_a > nat,C: set_a > nat] :
( ( finite_finite_set_a @ S )
=> ( ( ( member_set_a @ A @ S )
=> ( ( groups6141743369313575924_a_nat
@ ^ [K2: set_a] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( plus_plus_nat @ ( B2 @ A ) @ ( groups6141743369313575924_a_nat @ C @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ) )
& ( ~ ( member_set_a @ A @ S )
=> ( ( groups6141743369313575924_a_nat
@ ^ [K2: set_a] : ( if_nat @ ( K2 = A ) @ ( B2 @ K2 ) @ ( C @ K2 ) )
@ S )
= ( groups6141743369313575924_a_nat @ C @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_399_sum__diff1,axiom,
! [A2: set_o,A: $o,F: $o > real] :
( ( finite_finite_o @ A2 )
=> ( ( ( member_o @ A @ A2 )
=> ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( minus_minus_real @ ( groups8691415230153176458o_real @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups8691415230153176458o_real @ F @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_400_sum__diff1,axiom,
! [A2: set_a_real,A: a > real,F: ( a > real ) > real] :
( ( finite_finite_a_real @ A2 )
=> ( ( ( member_a_real @ A @ A2 )
=> ( ( groups6125259628802515085l_real @ F @ ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) )
= ( minus_minus_real @ ( groups6125259628802515085l_real @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_a_real @ A @ A2 )
=> ( ( groups6125259628802515085l_real @ F @ ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) )
= ( groups6125259628802515085l_real @ F @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_401_sum__diff1,axiom,
! [A2: set_nat,A: nat,F: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ A @ A2 )
=> ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups6591440286371151544t_real @ F @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_402_sum__diff1,axiom,
! [A2: set_set_a,A: set_a,F: set_a > real] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( member_set_a @ A @ A2 )
=> ( ( groups9174420418583655632a_real @ F @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= ( minus_minus_real @ ( groups9174420418583655632a_real @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( groups9174420418583655632a_real @ F @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= ( groups9174420418583655632a_real @ F @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_403_sum__diff1,axiom,
! [A2: set_b,A: b,F: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ A @ A2 )
=> ( ( groups8336678772925405937b_real @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( minus_minus_real @ ( groups8336678772925405937b_real @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_b @ A @ A2 )
=> ( ( groups8336678772925405937b_real @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( groups8336678772925405937b_real @ F @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_404_sum__diff1,axiom,
! [A2: set_a,A: a,F: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ A @ A2 )
=> ( ( groups2740460157737275248a_real @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( minus_minus_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( groups2740460157737275248a_real @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( groups2740460157737275248a_real @ F @ A2 ) ) ) ) ) ).
% sum_diff1
thf(fact_405_sum_Oinsert__remove,axiom,
! [A2: set_b,G: b > nat,X2: b] :
( ( finite_finite_b @ A2 )
=> ( ( groups7570001007293516437_b_nat @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups7570001007293516437_b_nat @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_406_sum_Oinsert__remove,axiom,
! [A2: set_a,G: a > nat,X2: a] :
( ( finite_finite_a @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups6334556678337121940_a_nat @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_407_sum_Oinsert__remove,axiom,
! [A2: set_o,G: $o > real,X2: $o] :
( ( finite_finite_o @ A2 )
=> ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups8691415230153176458o_real @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_408_sum_Oinsert__remove,axiom,
! [A2: set_o,G: $o > nat,X2: $o] :
( ( finite_finite_o @ A2 )
=> ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups8507830703676809646_o_nat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_409_sum_Oinsert__remove,axiom,
! [A2: set_nat,G: nat > real,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6591440286371151544t_real @ G @ ( insert_nat @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups6591440286371151544t_real @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_410_sum_Oinsert__remove,axiom,
! [A2: set_nat,G: nat > nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( insert_nat @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_411_sum_Oinsert__remove,axiom,
! [A2: set_b,G: b > real,X2: b] :
( ( finite_finite_b @ A2 )
=> ( ( groups8336678772925405937b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups8336678772925405937b_real @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_412_sum_Oinsert__remove,axiom,
! [A2: set_a,G: a > real,X2: a] :
( ( finite_finite_a @ A2 )
=> ( ( groups2740460157737275248a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups2740460157737275248a_real @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_413_sum_Oinsert__remove,axiom,
! [A2: set_set_a,G: set_a > real,X2: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( groups9174420418583655632a_real @ G @ ( insert_set_a @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups9174420418583655632a_real @ G @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_414_sum_Oinsert__remove,axiom,
! [A2: set_set_a,G: set_a > nat,X2: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( groups6141743369313575924_a_nat @ G @ ( insert_set_a @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups6141743369313575924_a_nat @ G @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_415_sum_Oremove,axiom,
! [A2: set_b,X2: b,G: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ( member_b @ X2 @ A2 )
=> ( ( groups7570001007293516437_b_nat @ G @ A2 )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups7570001007293516437_b_nat @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_416_sum_Oremove,axiom,
! [A2: set_a,X2: a,G: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X2 @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G @ A2 )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups6334556678337121940_a_nat @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_417_sum_Oremove,axiom,
! [A2: set_o,X2: $o,G: $o > real] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ X2 @ A2 )
=> ( ( groups8691415230153176458o_real @ G @ A2 )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups8691415230153176458o_real @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_418_sum_Oremove,axiom,
! [A2: set_o,X2: $o,G: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ X2 @ A2 )
=> ( ( groups8507830703676809646_o_nat @ G @ A2 )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups8507830703676809646_o_nat @ G @ ( minus_minus_set_o @ A2 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_419_sum_Oremove,axiom,
! [A2: set_nat,X2: nat,G: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups6591440286371151544t_real @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_420_sum_Oremove,axiom,
! [A2: set_nat,X2: nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_421_sum_Oremove,axiom,
! [A2: set_b,X2: b,G: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( member_b @ X2 @ A2 )
=> ( ( groups8336678772925405937b_real @ G @ A2 )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups8336678772925405937b_real @ G @ ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_422_sum_Oremove,axiom,
! [A2: set_a,X2: a,G: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X2 @ A2 )
=> ( ( groups2740460157737275248a_real @ G @ A2 )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups2740460157737275248a_real @ G @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_423_sum_Oremove,axiom,
! [A2: set_set_a,X2: set_a,G: set_a > real] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X2 @ A2 )
=> ( ( groups9174420418583655632a_real @ G @ A2 )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups9174420418583655632a_real @ G @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_424_sum_Oremove,axiom,
! [A2: set_set_a,X2: set_a,G: set_a > nat] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X2 @ A2 )
=> ( ( groups6141743369313575924_a_nat @ G @ A2 )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups6141743369313575924_a_nat @ G @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_425_finite__Diff2,axiom,
! [B: set_b,A2: set_b] :
( ( finite_finite_b @ B )
=> ( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B ) )
= ( finite_finite_b @ A2 ) ) ) ).
% finite_Diff2
thf(fact_426_finite__Diff2,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_427_finite__Diff2,axiom,
! [B: set_o,A2: set_o] :
( ( finite_finite_o @ B )
=> ( ( finite_finite_o @ ( minus_minus_set_o @ A2 @ B ) )
= ( finite_finite_o @ A2 ) ) ) ).
% finite_Diff2
thf(fact_428_finite__Diff2,axiom,
! [B: set_a_real,A2: set_a_real] :
( ( finite_finite_a_real @ B )
=> ( ( finite_finite_a_real @ ( minus_4124197362600706274a_real @ A2 @ B ) )
= ( finite_finite_a_real @ A2 ) ) ) ).
% finite_Diff2
thf(fact_429_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_430_finite__Diff2,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( finite_finite_set_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_431_finite__Diff,axiom,
! [A2: set_b,B: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_432_finite__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_433_finite__Diff,axiom,
! [A2: set_o,B: set_o] :
( ( finite_finite_o @ A2 )
=> ( finite_finite_o @ ( minus_minus_set_o @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_434_finite__Diff,axiom,
! [A2: set_a_real,B: set_a_real] :
( ( finite_finite_a_real @ A2 )
=> ( finite_finite_a_real @ ( minus_4124197362600706274a_real @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_435_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_436_finite__Diff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_437_finite__insert,axiom,
! [A: $o,A2: set_o] :
( ( finite_finite_o @ ( insert_o @ A @ A2 ) )
= ( finite_finite_o @ A2 ) ) ).
% finite_insert
thf(fact_438_finite__insert,axiom,
! [A: a > real,A2: set_a_real] :
( ( finite_finite_a_real @ ( insert_a_real @ A @ A2 ) )
= ( finite_finite_a_real @ A2 ) ) ).
% finite_insert
thf(fact_439_finite__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
= ( finite_finite_set_a @ A2 ) ) ).
% finite_insert
thf(fact_440_finite__insert,axiom,
! [A: b,A2: set_b] :
( ( finite_finite_b @ ( insert_b @ A @ A2 ) )
= ( finite_finite_b @ A2 ) ) ).
% finite_insert
thf(fact_441_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_442_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_443_finite__empty__induct,axiom,
! [A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: b,A6: set_b] :
( ( finite_finite_b @ A6 )
=> ( ( member_b @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_b @ A6 @ ( insert_b @ A5 @ bot_bot_set_b ) ) ) ) ) )
=> ( P @ bot_bot_set_b ) ) ) ) ).
% finite_empty_induct
thf(fact_444_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_445_finite__empty__induct,axiom,
! [A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: $o,A6: set_o] :
( ( finite_finite_o @ A6 )
=> ( ( member_o @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_o @ A6 @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ) )
=> ( P @ bot_bot_set_o ) ) ) ) ).
% finite_empty_induct
thf(fact_446_finite__empty__induct,axiom,
! [A2: set_a_real,P: set_a_real > $o] :
( ( finite_finite_a_real @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: a > real,A6: set_a_real] :
( ( finite_finite_a_real @ A6 )
=> ( ( member_a_real @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_4124197362600706274a_real @ A6 @ ( insert_a_real @ A5 @ bot_bot_set_a_real ) ) ) ) ) )
=> ( P @ bot_bot_set_a_real ) ) ) ) ).
% finite_empty_induct
thf(fact_447_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_448_finite__empty__induct,axiom,
! [A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: set_a,A6: set_set_a] :
( ( finite_finite_set_a @ A6 )
=> ( ( member_set_a @ A5 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ A5 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_449_infinite__coinduct,axiom,
! [X4: set_b > $o,A2: set_b] :
( ( X4 @ A2 )
=> ( ! [A6: set_b] :
( ( X4 @ A6 )
=> ? [X5: b] :
( ( member_b @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_b @ A6 @ ( insert_b @ X5 @ bot_bot_set_b ) ) )
| ~ ( finite_finite_b @ ( minus_minus_set_b @ A6 @ ( insert_b @ X5 @ bot_bot_set_b ) ) ) ) ) )
=> ~ ( finite_finite_b @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_450_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A6: set_a] :
( ( X4 @ A6 )
=> ? [X5: a] :
( ( member_a @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_451_infinite__coinduct,axiom,
! [X4: set_o > $o,A2: set_o] :
( ( X4 @ A2 )
=> ( ! [A6: set_o] :
( ( X4 @ A6 )
=> ? [X5: $o] :
( ( member_o @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_o @ A6 @ ( insert_o @ X5 @ bot_bot_set_o ) ) )
| ~ ( finite_finite_o @ ( minus_minus_set_o @ A6 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ) )
=> ~ ( finite_finite_o @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_452_infinite__coinduct,axiom,
! [X4: set_a_real > $o,A2: set_a_real] :
( ( X4 @ A2 )
=> ( ! [A6: set_a_real] :
( ( X4 @ A6 )
=> ? [X5: a > real] :
( ( member_a_real @ X5 @ A6 )
& ( ( X4 @ ( minus_4124197362600706274a_real @ A6 @ ( insert_a_real @ X5 @ bot_bot_set_a_real ) ) )
| ~ ( finite_finite_a_real @ ( minus_4124197362600706274a_real @ A6 @ ( insert_a_real @ X5 @ bot_bot_set_a_real ) ) ) ) ) )
=> ~ ( finite_finite_a_real @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_453_infinite__coinduct,axiom,
! [X4: set_nat > $o,A2: set_nat] :
( ( X4 @ A2 )
=> ( ! [A6: set_nat] :
( ( X4 @ A6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_454_infinite__coinduct,axiom,
! [X4: set_set_a > $o,A2: set_set_a] :
( ( X4 @ A2 )
=> ( ! [A6: set_set_a] :
( ( X4 @ A6 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ A6 )
& ( ( X4 @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) )
| ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A6 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) ) ) )
=> ~ ( finite_finite_set_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_455_infinite__remove,axiom,
! [S: set_b,A: b] :
( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% infinite_remove
thf(fact_456_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_457_infinite__remove,axiom,
! [S: set_o,A: $o] :
( ~ ( finite_finite_o @ S )
=> ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% infinite_remove
thf(fact_458_infinite__remove,axiom,
! [S: set_a_real,A: a > real] :
( ~ ( finite_finite_a_real @ S )
=> ~ ( finite_finite_a_real @ ( minus_4124197362600706274a_real @ S @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) ) ) ).
% infinite_remove
thf(fact_459_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_460_infinite__remove,axiom,
! [S: set_set_a,A: set_a] :
( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% infinite_remove
thf(fact_461_finite__Collect__disjI,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( finite_finite_o
@ ( collect_o
@ ^ [X: $o] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_o @ ( collect_o @ P ) )
& ( finite_finite_o @ ( collect_o @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_462_finite__Collect__disjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_b @ ( collect_b @ P ) )
& ( finite_finite_b @ ( collect_b @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_463_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_464_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_465_finite__Collect__conjI,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ( finite_finite_o @ ( collect_o @ P ) )
| ( finite_finite_o @ ( collect_o @ Q ) ) )
=> ( finite_finite_o
@ ( collect_o
@ ^ [X: $o] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_466_finite__Collect__conjI,axiom,
! [P: b > $o,Q: b > $o] :
( ( ( finite_finite_b @ ( collect_b @ P ) )
| ( finite_finite_b @ ( collect_b @ Q ) ) )
=> ( finite_finite_b
@ ( collect_b
@ ^ [X: b] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_467_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_468_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_469_sum__diff1__nat,axiom,
! [A: b,A2: set_b,F: b > nat] :
( ( ( member_b @ A @ A2 )
=> ( ( groups7570001007293516437_b_nat @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( minus_minus_nat @ ( groups7570001007293516437_b_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_b @ A @ A2 )
=> ( ( groups7570001007293516437_b_nat @ F @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ bot_bot_set_b ) ) )
= ( groups7570001007293516437_b_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_470_sum__diff1__nat,axiom,
! [A: a,A2: set_a,F: a > nat] :
( ( ( member_a @ A @ A2 )
=> ( ( groups6334556678337121940_a_nat @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( groups6334556678337121940_a_nat @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( groups6334556678337121940_a_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_471_sum__diff1__nat,axiom,
! [A: $o,A2: set_o,F: $o > nat] :
( ( ( member_o @ A @ A2 )
=> ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_o @ A @ A2 )
=> ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( groups8507830703676809646_o_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_472_sum__diff1__nat,axiom,
! [A: a > real,A2: set_a_real,F: ( a > real ) > nat] :
( ( ( member_a_real @ A @ A2 )
=> ( ( groups1701885688937111089al_nat @ F @ ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) )
= ( minus_minus_nat @ ( groups1701885688937111089al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_a_real @ A @ A2 )
=> ( ( groups1701885688937111089al_nat @ F @ ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ A @ bot_bot_set_a_real ) ) )
= ( groups1701885688937111089al_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_473_sum__diff1__nat,axiom,
! [A: nat,A2: set_nat,F: nat > nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_474_sum__diff1__nat,axiom,
! [A: set_a,A2: set_set_a,F: set_a > nat] :
( ( ( member_set_a @ A @ A2 )
=> ( ( groups6141743369313575924_a_nat @ F @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( groups6141743369313575924_a_nat @ F @ A2 ) @ ( F @ A ) ) ) )
& ( ~ ( member_set_a @ A @ A2 )
=> ( ( groups6141743369313575924_a_nat @ F @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= ( groups6141743369313575924_a_nat @ F @ A2 ) ) ) ) ).
% sum_diff1_nat
thf(fact_475_sum_Ocong,axiom,
! [A2: set_b,B: set_b,G: b > real,H: b > real] :
( ( A2 = B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ A2 )
= ( groups8336678772925405937b_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_476_sum_Ocong,axiom,
! [A2: set_a,B: set_a,G: a > real,H: a > real] :
( ( A2 = B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ A2 )
= ( groups2740460157737275248a_real @ H @ B ) ) ) ) ).
% sum.cong
thf(fact_477_sum_Oeq__general,axiom,
! [B: set_b,A2: set_nat,H: nat > b,Gamma: b > real,Phi: nat > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_478_sum_Oeq__general,axiom,
! [B: set_b,A2: set_o,H: $o > b,Gamma: b > real,Phi: $o > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_479_sum_Oeq__general,axiom,
! [B: set_a,A2: set_nat,H: nat > a,Gamma: a > real,Phi: nat > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_480_sum_Oeq__general,axiom,
! [B: set_a,A2: set_o,H: $o > a,Gamma: a > real,Phi: $o > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ? [X5: $o] :
( ( member_o @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: $o] :
( ( ( member_o @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_481_sum_Oeq__general,axiom,
! [B: set_nat,A2: set_b,H: b > nat,Gamma: nat > real,Phi: b > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ? [X5: b] :
( ( member_b @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_482_sum_Oeq__general,axiom,
! [B: set_o,A2: set_b,H: b > $o,Gamma: $o > real,Phi: b > real] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ B )
=> ? [X5: b] :
( ( member_b @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_o @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups8691415230153176458o_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_483_sum_Oeq__general,axiom,
! [B: set_b,A2: set_b,H: b > b,Gamma: b > real,Phi: b > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B )
=> ? [X5: b] :
( ( member_b @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_484_sum_Oeq__general,axiom,
! [B: set_a,A2: set_b,H: b > a,Gamma: a > real,Phi: b > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ? [X5: b] :
( ( member_b @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_485_sum_Oeq__general,axiom,
! [B: set_nat,A2: set_a,H: a > nat,Gamma: nat > real,Phi: a > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_486_sum_Oeq__general,axiom,
! [B: set_o,A2: set_a,H: a > $o,Gamma: $o > real,Phi: a > real] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ B )
=> ? [X5: a] :
( ( member_a @ X5 @ A2 )
& ( ( H @ X5 )
= Y2 )
& ! [Ya: a] :
( ( ( member_a @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X5 ) ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_o @ ( H @ X3 ) @ B )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A2 )
= ( groups8691415230153176458o_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general
thf(fact_487_sum_Oeq__general__inverses,axiom,
! [B: set_b,K: b > nat,A2: set_nat,H: nat > b,Gamma: b > real,Phi: nat > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_488_sum_Oeq__general__inverses,axiom,
! [B: set_b,K: b > $o,A2: set_o,H: $o > b,Gamma: b > real,Phi: $o > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B )
=> ( ( member_o @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_489_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > nat,A2: set_nat,H: nat > a,Gamma: a > real,Phi: nat > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( ( member_nat @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_490_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > $o,A2: set_o,H: $o > a,Gamma: a > real,Phi: $o > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( ( member_o @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8691415230153176458o_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_491_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > b,A2: set_b,H: b > nat,Gamma: nat > real,Phi: b > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( ( member_b @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_492_sum_Oeq__general__inverses,axiom,
! [B: set_o,K: $o > b,A2: set_b,H: b > $o,Gamma: $o > real,Phi: b > real] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ B )
=> ( ( member_b @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_o @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups8691415230153176458o_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_493_sum_Oeq__general__inverses,axiom,
! [B: set_b,K: b > b,A2: set_b,H: b > b,Gamma: b > real,Phi: b > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B )
=> ( ( member_b @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_494_sum_Oeq__general__inverses,axiom,
! [B: set_a,K: a > b,A2: set_b,H: b > a,Gamma: a > real,Phi: b > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B )
=> ( ( member_b @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_495_sum_Oeq__general__inverses,axiom,
! [B: set_nat,K: nat > a,A2: set_a,H: a > nat,Gamma: nat > real,Phi: a > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( ( member_a @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_496_sum_Oeq__general__inverses,axiom,
! [B: set_o,K: $o > a,A2: set_a,H: a > $o,Gamma: $o > real,Phi: a > real] :
( ! [Y2: $o] :
( ( member_o @ Y2 @ B )
=> ( ( member_a @ ( K @ Y2 ) @ A2 )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_o @ ( H @ X3 ) @ B )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2740460157737275248a_real @ Phi @ A2 )
= ( groups8691415230153176458o_real @ Gamma @ B ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_497_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: b > nat,J: nat > b,T: set_b,H: b > real,G: nat > real] :
( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( member_b @ ( J @ A5 ) @ T ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( member_nat @ ( I @ B6 ) @ S ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups8336678772925405937b_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_498_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I: b > $o,J: $o > b,T: set_b,H: b > real,G: $o > real] :
( ! [A5: $o] :
( ( member_o @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ S )
=> ( member_b @ ( J @ A5 ) @ T ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( member_o @ ( I @ B6 ) @ S ) )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups8691415230153176458o_real @ G @ S )
= ( groups8336678772925405937b_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_499_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: a > nat,J: nat > a,T: set_a,H: a > real,G: nat > real] :
( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( member_a @ ( J @ A5 ) @ T ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( member_nat @ ( I @ B6 ) @ S ) )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S )
= ( groups2740460157737275248a_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_500_sum_Oreindex__bij__witness,axiom,
! [S: set_o,I: a > $o,J: $o > a,T: set_a,H: a > real,G: $o > real] :
( ! [A5: $o] :
( ( member_o @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ S )
=> ( member_a @ ( J @ A5 ) @ T ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( member_o @ ( I @ B6 ) @ S ) )
=> ( ! [A5: $o] :
( ( member_o @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups8691415230153176458o_real @ G @ S )
= ( groups2740460157737275248a_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_501_sum_Oreindex__bij__witness,axiom,
! [S: set_b,I: nat > b,J: b > nat,T: set_nat,H: nat > real,G: b > real] :
( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( member_nat @ ( J @ A5 ) @ T ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T )
=> ( member_b @ ( I @ B6 ) @ S ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_502_sum_Oreindex__bij__witness,axiom,
! [S: set_b,I: $o > b,J: b > $o,T: set_o,H: $o > real,G: b > real] :
( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( member_o @ ( J @ A5 ) @ T ) )
=> ( ! [B6: $o] :
( ( member_o @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: $o] :
( ( member_o @ B6 @ T )
=> ( member_b @ ( I @ B6 ) @ S ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ S )
= ( groups8691415230153176458o_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_503_sum_Oreindex__bij__witness,axiom,
! [S: set_b,I: b > b,J: b > b,T: set_b,H: b > real,G: b > real] :
( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( member_b @ ( J @ A5 ) @ T ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: b] :
( ( member_b @ B6 @ T )
=> ( member_b @ ( I @ B6 ) @ S ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ S )
= ( groups8336678772925405937b_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_504_sum_Oreindex__bij__witness,axiom,
! [S: set_b,I: a > b,J: b > a,T: set_a,H: a > real,G: b > real] :
( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( member_a @ ( J @ A5 ) @ T ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: a] :
( ( member_a @ B6 @ T )
=> ( member_b @ ( I @ B6 ) @ S ) )
=> ( ! [A5: b] :
( ( member_b @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ S )
= ( groups2740460157737275248a_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_505_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I: nat > a,J: a > nat,T: set_nat,H: nat > real,G: a > real] :
( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( member_nat @ ( J @ A5 ) @ T ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T )
=> ( member_a @ ( I @ B6 ) @ S ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S )
= ( groups6591440286371151544t_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_506_sum_Oreindex__bij__witness,axiom,
! [S: set_a,I: $o > a,J: a > $o,T: set_o,H: $o > real,G: a > real] :
( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( member_o @ ( J @ A5 ) @ T ) )
=> ( ! [B6: $o] :
( ( member_o @ B6 @ T )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: $o] :
( ( member_o @ B6 @ T )
=> ( member_a @ ( I @ B6 ) @ S ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ S )
=> ( ( H @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ S )
= ( groups8691415230153176458o_real @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_507_pigeonhole__infinite__rel,axiom,
! [A2: set_o,B: set_b,R: $o > b > $o] :
( ~ ( finite_finite_o @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_508_pigeonhole__infinite__rel,axiom,
! [A2: set_o,B: set_nat,R: $o > nat > $o] :
( ~ ( finite_finite_o @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_509_pigeonhole__infinite__rel,axiom,
! [A2: set_o,B: set_a,R: $o > a > $o] :
( ~ ( finite_finite_o @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_o
@ ( collect_o
@ ^ [A4: $o] :
( ( member_o @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_510_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_b,R: b > b > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_511_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_nat,R: b > nat > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_512_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B: set_a,R: b > a > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A4: b] :
( ( member_b @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_513_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_b,R: nat > b > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_514_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_515_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B: set_a,R: nat > a > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A4: nat] :
( ( member_nat @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_516_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B: set_b,R: a > b > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A4: a] :
( ( member_a @ A4 @ A2 )
& ( R @ A4 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_517_not__finite__existsD,axiom,
! [P: $o > $o] :
( ~ ( finite_finite_o @ ( collect_o @ P ) )
=> ? [X_1: $o] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_518_not__finite__existsD,axiom,
! [P: b > $o] :
( ~ ( finite_finite_b @ ( collect_b @ P ) )
=> ? [X_1: b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_519_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_520_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_521_sum_Oswap,axiom,
! [G: b > b > real,B: set_b,A2: set_b] :
( ( groups8336678772925405937b_real
@ ^ [I2: b] : ( groups8336678772925405937b_real @ ( G @ I2 ) @ B )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [J2: b] :
( groups8336678772925405937b_real
@ ^ [I2: b] : ( G @ I2 @ J2 )
@ A2 )
@ B ) ) ).
% sum.swap
thf(fact_522_sum_Oswap,axiom,
! [G: b > a > real,B: set_a,A2: set_b] :
( ( groups8336678772925405937b_real
@ ^ [I2: b] : ( groups2740460157737275248a_real @ ( G @ I2 ) @ B )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [J2: a] :
( groups8336678772925405937b_real
@ ^ [I2: b] : ( G @ I2 @ J2 )
@ A2 )
@ B ) ) ).
% sum.swap
thf(fact_523_sum_Oswap,axiom,
! [G: a > b > real,B: set_b,A2: set_a] :
( ( groups2740460157737275248a_real
@ ^ [I2: a] : ( groups8336678772925405937b_real @ ( G @ I2 ) @ B )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [J2: b] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( G @ I2 @ J2 )
@ A2 )
@ B ) ) ).
% sum.swap
thf(fact_524_sum_Oswap,axiom,
! [G: a > a > real,B: set_a,A2: set_a] :
( ( groups2740460157737275248a_real
@ ^ [I2: a] : ( groups2740460157737275248a_real @ ( G @ I2 ) @ B )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [J2: a] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( G @ I2 @ J2 )
@ A2 )
@ B ) ) ).
% sum.swap
thf(fact_525_finite_OemptyI,axiom,
finite_finite_b @ bot_bot_set_b ).
% finite.emptyI
thf(fact_526_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_527_finite_OemptyI,axiom,
finite_finite_o @ bot_bot_set_o ).
% finite.emptyI
thf(fact_528_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_529_infinite__imp__nonempty,axiom,
! [S: set_b] :
( ~ ( finite_finite_b @ S )
=> ( S != bot_bot_set_b ) ) ).
% infinite_imp_nonempty
thf(fact_530_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_531_infinite__imp__nonempty,axiom,
! [S: set_o] :
( ~ ( finite_finite_o @ S )
=> ( S != bot_bot_set_o ) ) ).
% infinite_imp_nonempty
thf(fact_532_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_533_finite_OinsertI,axiom,
! [A2: set_o,A: $o] :
( ( finite_finite_o @ A2 )
=> ( finite_finite_o @ ( insert_o @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_534_finite_OinsertI,axiom,
! [A2: set_a_real,A: a > real] :
( ( finite_finite_a_real @ A2 )
=> ( finite_finite_a_real @ ( insert_a_real @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_535_finite_OinsertI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_536_finite_OinsertI,axiom,
! [A2: set_b,A: b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( insert_b @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_537_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_538_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_539_Diff__infinite__finite,axiom,
! [T: set_b,S: set_b] :
( ( finite_finite_b @ T )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_540_Diff__infinite__finite,axiom,
! [T: set_a,S: set_a] :
( ( finite_finite_a @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_541_Diff__infinite__finite,axiom,
! [T: set_o,S: set_o] :
( ( finite_finite_o @ T )
=> ( ~ ( finite_finite_o @ S )
=> ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_542_Diff__infinite__finite,axiom,
! [T: set_a_real,S: set_a_real] :
( ( finite_finite_a_real @ T )
=> ( ~ ( finite_finite_a_real @ S )
=> ~ ( finite_finite_a_real @ ( minus_4124197362600706274a_real @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_543_Diff__infinite__finite,axiom,
! [T: set_nat,S: set_nat] :
( ( finite_finite_nat @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_544_Diff__infinite__finite,axiom,
! [T: set_set_a,S: set_set_a] :
( ( finite_finite_set_a @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T ) ) ) ) ).
% Diff_infinite_finite
thf(fact_545_sum_Odistrib,axiom,
! [G: b > real,H: b > real,A2: set_b] :
( ( groups8336678772925405937b_real
@ ^ [X: b] : ( plus_plus_real @ ( G @ X ) @ ( H @ X ) )
@ A2 )
= ( plus_plus_real @ ( groups8336678772925405937b_real @ G @ A2 ) @ ( groups8336678772925405937b_real @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_546_sum_Odistrib,axiom,
! [G: a > real,H: a > real,A2: set_a] :
( ( groups2740460157737275248a_real
@ ^ [X: a] : ( plus_plus_real @ ( G @ X ) @ ( H @ X ) )
@ A2 )
= ( plus_plus_real @ ( groups2740460157737275248a_real @ G @ A2 ) @ ( groups2740460157737275248a_real @ H @ A2 ) ) ) ).
% sum.distrib
thf(fact_547_sum__subtractf,axiom,
! [F: b > real,G: b > real,A2: set_b] :
( ( groups8336678772925405937b_real
@ ^ [X: b] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_real @ ( groups8336678772925405937b_real @ F @ A2 ) @ ( groups8336678772925405937b_real @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_548_sum__subtractf,axiom,
! [F: a > real,G: a > real,A2: set_a] :
( ( groups2740460157737275248a_real
@ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ ( groups2740460157737275248a_real @ G @ A2 ) ) ) ).
% sum_subtractf
thf(fact_549_sum_Oswap__restrict,axiom,
! [A2: set_o,B: set_b,G: $o > b > real,R: $o > b > $o] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ( groups8691415230153176458o_real
@ ^ [X: $o] :
( groups8336678772925405937b_real @ ( G @ X )
@ ( collect_b
@ ^ [Y4: b] :
( ( member_b @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [Y4: b] :
( groups8691415230153176458o_real
@ ^ [X: $o] : ( G @ X @ Y4 )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_550_sum_Oswap__restrict,axiom,
! [A2: set_nat,B: set_b,G: nat > b > real,R: nat > b > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ( groups6591440286371151544t_real
@ ^ [X: nat] :
( groups8336678772925405937b_real @ ( G @ X )
@ ( collect_b
@ ^ [Y4: b] :
( ( member_b @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [Y4: b] :
( groups6591440286371151544t_real
@ ^ [X: nat] : ( G @ X @ Y4 )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_551_sum_Oswap__restrict,axiom,
! [A2: set_o,B: set_a,G: $o > a > real,R: $o > a > $o] :
( ( finite_finite_o @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( groups8691415230153176458o_real
@ ^ [X: $o] :
( groups2740460157737275248a_real @ ( G @ X )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [Y4: a] :
( groups8691415230153176458o_real
@ ^ [X: $o] : ( G @ X @ Y4 )
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_552_sum_Oswap__restrict,axiom,
! [A2: set_nat,B: set_a,G: nat > a > real,R: nat > a > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( groups6591440286371151544t_real
@ ^ [X: nat] :
( groups2740460157737275248a_real @ ( G @ X )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [Y4: a] :
( groups6591440286371151544t_real
@ ^ [X: nat] : ( G @ X @ Y4 )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_553_sum_Oswap__restrict,axiom,
! [A2: set_b,B: set_o,G: b > $o > real,R: b > $o > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_o @ B )
=> ( ( groups8336678772925405937b_real
@ ^ [X: b] :
( groups8691415230153176458o_real @ ( G @ X )
@ ( collect_o
@ ^ [Y4: $o] :
( ( member_o @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups8691415230153176458o_real
@ ^ [Y4: $o] :
( groups8336678772925405937b_real
@ ^ [X: b] : ( G @ X @ Y4 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_554_sum_Oswap__restrict,axiom,
! [A2: set_b,B: set_nat,G: b > nat > real,R: b > nat > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( groups8336678772925405937b_real
@ ^ [X: b] :
( groups6591440286371151544t_real @ ( G @ X )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups8336678772925405937b_real
@ ^ [X: b] : ( G @ X @ Y4 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_555_sum_Oswap__restrict,axiom,
! [A2: set_b,B: set_b,G: b > b > real,R: b > b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B )
=> ( ( groups8336678772925405937b_real
@ ^ [X: b] :
( groups8336678772925405937b_real @ ( G @ X )
@ ( collect_b
@ ^ [Y4: b] :
( ( member_b @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [Y4: b] :
( groups8336678772925405937b_real
@ ^ [X: b] : ( G @ X @ Y4 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_556_sum_Oswap__restrict,axiom,
! [A2: set_b,B: set_a,G: b > a > real,R: b > a > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B )
=> ( ( groups8336678772925405937b_real
@ ^ [X: b] :
( groups2740460157737275248a_real @ ( G @ X )
@ ( collect_a
@ ^ [Y4: a] :
( ( member_a @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [Y4: a] :
( groups8336678772925405937b_real
@ ^ [X: b] : ( G @ X @ Y4 )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_557_sum_Oswap__restrict,axiom,
! [A2: set_a,B: set_o,G: a > $o > real,R: a > $o > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_o @ B )
=> ( ( groups2740460157737275248a_real
@ ^ [X: a] :
( groups8691415230153176458o_real @ ( G @ X )
@ ( collect_o
@ ^ [Y4: $o] :
( ( member_o @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups8691415230153176458o_real
@ ^ [Y4: $o] :
( groups2740460157737275248a_real
@ ^ [X: a] : ( G @ X @ Y4 )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_558_sum_Oswap__restrict,axiom,
! [A2: set_a,B: set_nat,G: a > nat > real,R: a > nat > $o] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( groups2740460157737275248a_real
@ ^ [X: a] :
( groups6591440286371151544t_real @ ( G @ X )
@ ( collect_nat
@ ^ [Y4: nat] :
( ( member_nat @ Y4 @ B )
& ( R @ X @ Y4 ) ) ) )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [Y4: nat] :
( groups2740460157737275248a_real
@ ^ [X: a] : ( G @ X @ Y4 )
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( R @ X @ Y4 ) ) ) )
@ B ) ) ) ) ).
% sum.swap_restrict
thf(fact_559_finite_Ocases,axiom,
! [A: set_a_real] :
( ( finite_finite_a_real @ A )
=> ( ( A != bot_bot_set_a_real )
=> ~ ! [A6: set_a_real] :
( ? [A5: a > real] :
( A
= ( insert_a_real @ A5 @ A6 ) )
=> ~ ( finite_finite_a_real @ A6 ) ) ) ) ).
% finite.cases
thf(fact_560_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A6: set_set_a] :
( ? [A5: set_a] :
( A
= ( insert_set_a @ A5 @ A6 ) )
=> ~ ( finite_finite_set_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_561_finite_Ocases,axiom,
! [A: set_b] :
( ( finite_finite_b @ A )
=> ( ( A != bot_bot_set_b )
=> ~ ! [A6: set_b] :
( ? [A5: b] :
( A
= ( insert_b @ A5 @ A6 ) )
=> ~ ( finite_finite_b @ A6 ) ) ) ) ).
% finite.cases
thf(fact_562_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A5: a] :
( A
= ( insert_a @ A5 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_563_finite_Ocases,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ~ ! [A6: set_o] :
( ? [A5: $o] :
( A
= ( insert_o @ A5 @ A6 ) )
=> ~ ( finite_finite_o @ A6 ) ) ) ) ).
% finite.cases
thf(fact_564_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A5: nat] :
( A
= ( insert_nat @ A5 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_565_finite_Osimps,axiom,
( finite_finite_a_real
= ( ^ [A4: set_a_real] :
( ( A4 = bot_bot_set_a_real )
| ? [A3: set_a_real,B4: a > real] :
( ( A4
= ( insert_a_real @ B4 @ A3 ) )
& ( finite_finite_a_real @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_566_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A4: set_set_a] :
( ( A4 = bot_bot_set_set_a )
| ? [A3: set_set_a,B4: set_a] :
( ( A4
= ( insert_set_a @ B4 @ A3 ) )
& ( finite_finite_set_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_567_finite_Osimps,axiom,
( finite_finite_b
= ( ^ [A4: set_b] :
( ( A4 = bot_bot_set_b )
| ? [A3: set_b,B4: b] :
( ( A4
= ( insert_b @ B4 @ A3 ) )
& ( finite_finite_b @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_568_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A3: set_a,B4: a] :
( ( A4
= ( insert_a @ B4 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_569_finite_Osimps,axiom,
( finite_finite_o
= ( ^ [A4: set_o] :
( ( A4 = bot_bot_set_o )
| ? [A3: set_o,B4: $o] :
( ( A4
= ( insert_o @ B4 @ A3 ) )
& ( finite_finite_o @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_570_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B4: nat] :
( ( A4
= ( insert_nat @ B4 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_571_finite__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_572_finite__induct,axiom,
! [F2: set_a_real,P: set_a_real > $o] :
( ( finite_finite_a_real @ F2 )
=> ( ( P @ bot_bot_set_a_real )
=> ( ! [X3: a > real,F3: set_a_real] :
( ( finite_finite_a_real @ F3 )
=> ( ~ ( member_a_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_real @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_573_finite__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_574_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_575_finite__induct,axiom,
! [F2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ~ ( member_o @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_576_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_577_finite__ne__induct,axiom,
! [F2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X3: set_a] : ( P @ ( insert_set_a @ X3 @ bot_bot_set_set_a ) )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_578_finite__ne__induct,axiom,
! [F2: set_a_real,P: set_a_real > $o] :
( ( finite_finite_a_real @ F2 )
=> ( ( F2 != bot_bot_set_a_real )
=> ( ! [X3: a > real] : ( P @ ( insert_a_real @ X3 @ bot_bot_set_a_real ) )
=> ( ! [X3: a > real,F3: set_a_real] :
( ( finite_finite_a_real @ F3 )
=> ( ( F3 != bot_bot_set_a_real )
=> ( ~ ( member_a_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_real @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_579_finite__ne__induct,axiom,
! [F2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( F2 != bot_bot_set_b )
=> ( ! [X3: b] : ( P @ ( insert_b @ X3 @ bot_bot_set_b ) )
=> ( ! [X3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( F3 != bot_bot_set_b )
=> ( ~ ( member_b @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_580_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_581_finite__ne__induct,axiom,
! [F2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( F2 != bot_bot_set_o )
=> ( ! [X3: $o] : ( P @ ( insert_o @ X3 @ bot_bot_set_o ) )
=> ( ! [X3: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ( F3 != bot_bot_set_o )
=> ( ~ ( member_o @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_582_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_583_infinite__finite__induct,axiom,
! [P: set_set_a > $o,A2: set_set_a] :
( ! [A6: set_set_a] :
( ~ ( finite_finite_set_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [X3: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_584_infinite__finite__induct,axiom,
! [P: set_a_real > $o,A2: set_a_real] :
( ! [A6: set_a_real] :
( ~ ( finite_finite_a_real @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a_real )
=> ( ! [X3: a > real,F3: set_a_real] :
( ( finite_finite_a_real @ F3 )
=> ( ~ ( member_a_real @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_real @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_585_infinite__finite__induct,axiom,
! [P: set_b > $o,A2: set_b] :
( ! [A6: set_b] :
( ~ ( finite_finite_b @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_b )
=> ( ! [X3: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ~ ( member_b @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_586_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_587_infinite__finite__induct,axiom,
! [P: set_o > $o,A2: set_o] :
( ! [A6: set_o] :
( ~ ( finite_finite_o @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ~ ( member_o @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_588_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_589_sum_Oinsert__if,axiom,
! [A2: set_o,X2: $o,G: $o > real] :
( ( finite_finite_o @ A2 )
=> ( ( ( member_o @ X2 @ A2 )
=> ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X2 @ A2 ) )
= ( groups8691415230153176458o_real @ G @ A2 ) ) )
& ( ~ ( member_o @ X2 @ A2 )
=> ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups8691415230153176458o_real @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_590_sum_Oinsert__if,axiom,
! [A2: set_nat,X2: nat,G: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ( groups6591440286371151544t_real @ G @ ( insert_nat @ X2 @ A2 ) )
= ( groups6591440286371151544t_real @ G @ A2 ) ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( groups6591440286371151544t_real @ G @ ( insert_nat @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_591_sum_Oinsert__if,axiom,
! [A2: set_o,X2: $o,G: $o > nat] :
( ( finite_finite_o @ A2 )
=> ( ( ( member_o @ X2 @ A2 )
=> ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X2 @ A2 ) )
= ( groups8507830703676809646_o_nat @ G @ A2 ) ) )
& ( ~ ( member_o @ X2 @ A2 )
=> ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups8507830703676809646_o_nat @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_592_sum_Oinsert__if,axiom,
! [A2: set_b,X2: b,G: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ X2 @ A2 )
=> ( ( groups7570001007293516437_b_nat @ G @ ( insert_b @ X2 @ A2 ) )
= ( groups7570001007293516437_b_nat @ G @ A2 ) ) )
& ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups7570001007293516437_b_nat @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups7570001007293516437_b_nat @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_593_sum_Oinsert__if,axiom,
! [A2: set_nat,X2: nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( insert_nat @ X2 @ A2 ) )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( insert_nat @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_594_sum_Oinsert__if,axiom,
! [A2: set_a,X2: a,G: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X2 @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G @ ( insert_a @ X2 @ A2 ) )
= ( groups6334556678337121940_a_nat @ G @ A2 ) ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups6334556678337121940_a_nat @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_595_sum_Oinsert__if,axiom,
! [A2: set_b,X2: b,G: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ X2 @ A2 )
=> ( ( groups8336678772925405937b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( groups8336678772925405937b_real @ G @ A2 ) ) )
& ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups8336678772925405937b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups8336678772925405937b_real @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_596_sum_Oinsert__if,axiom,
! [A2: set_a,X2: a,G: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X2 @ A2 )
=> ( ( groups2740460157737275248a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( groups2740460157737275248a_real @ G @ A2 ) ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups2740460157737275248a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups2740460157737275248a_real @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_597_sum_Oinsert__if,axiom,
! [A2: set_set_a,X2: set_a,G: set_a > real] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( member_set_a @ X2 @ A2 )
=> ( ( groups9174420418583655632a_real @ G @ ( insert_set_a @ X2 @ A2 ) )
= ( groups9174420418583655632a_real @ G @ A2 ) ) )
& ( ~ ( member_set_a @ X2 @ A2 )
=> ( ( groups9174420418583655632a_real @ G @ ( insert_set_a @ X2 @ A2 ) )
= ( plus_plus_real @ ( G @ X2 ) @ ( groups9174420418583655632a_real @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_598_sum_Oinsert__if,axiom,
! [A2: set_set_a,X2: set_a,G: set_a > nat] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( member_set_a @ X2 @ A2 )
=> ( ( groups6141743369313575924_a_nat @ G @ ( insert_set_a @ X2 @ A2 ) )
= ( groups6141743369313575924_a_nat @ G @ A2 ) ) )
& ( ~ ( member_set_a @ X2 @ A2 )
=> ( ( groups6141743369313575924_a_nat @ G @ ( insert_set_a @ X2 @ A2 ) )
= ( plus_plus_nat @ ( G @ X2 ) @ ( groups6141743369313575924_a_nat @ G @ A2 ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_599_sum__clauses_I2_J,axiom,
! [S: set_o,X2: $o,F: $o > real] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ X2 @ S )
=> ( ( groups8691415230153176458o_real @ F @ ( insert_o @ X2 @ S ) )
= ( groups8691415230153176458o_real @ F @ S ) ) )
& ( ~ ( member_o @ X2 @ S )
=> ( ( groups8691415230153176458o_real @ F @ ( insert_o @ X2 @ S ) )
= ( plus_plus_real @ ( F @ X2 ) @ ( groups8691415230153176458o_real @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_600_sum__clauses_I2_J,axiom,
! [S: set_nat,X2: nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real @ F @ ( insert_nat @ X2 @ S ) )
= ( groups6591440286371151544t_real @ F @ S ) ) )
& ( ~ ( member_nat @ X2 @ S )
=> ( ( groups6591440286371151544t_real @ F @ ( insert_nat @ X2 @ S ) )
= ( plus_plus_real @ ( F @ X2 ) @ ( groups6591440286371151544t_real @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_601_sum__clauses_I2_J,axiom,
! [S: set_o,X2: $o,F: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( ( member_o @ X2 @ S )
=> ( ( groups8507830703676809646_o_nat @ F @ ( insert_o @ X2 @ S ) )
= ( groups8507830703676809646_o_nat @ F @ S ) ) )
& ( ~ ( member_o @ X2 @ S )
=> ( ( groups8507830703676809646_o_nat @ F @ ( insert_o @ X2 @ S ) )
= ( plus_plus_nat @ ( F @ X2 ) @ ( groups8507830703676809646_o_nat @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_602_sum__clauses_I2_J,axiom,
! [S: set_b,X2: b,F: b > nat] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ X2 @ S )
=> ( ( groups7570001007293516437_b_nat @ F @ ( insert_b @ X2 @ S ) )
= ( groups7570001007293516437_b_nat @ F @ S ) ) )
& ( ~ ( member_b @ X2 @ S )
=> ( ( groups7570001007293516437_b_nat @ F @ ( insert_b @ X2 @ S ) )
= ( plus_plus_nat @ ( F @ X2 ) @ ( groups7570001007293516437_b_nat @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_603_sum__clauses_I2_J,axiom,
! [S: set_nat,X2: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat @ F @ ( insert_nat @ X2 @ S ) )
= ( groups3542108847815614940at_nat @ F @ S ) ) )
& ( ~ ( member_nat @ X2 @ S )
=> ( ( groups3542108847815614940at_nat @ F @ ( insert_nat @ X2 @ S ) )
= ( plus_plus_nat @ ( F @ X2 ) @ ( groups3542108847815614940at_nat @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_604_sum__clauses_I2_J,axiom,
! [S: set_a,X2: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ X2 @ S )
=> ( ( groups6334556678337121940_a_nat @ F @ ( insert_a @ X2 @ S ) )
= ( groups6334556678337121940_a_nat @ F @ S ) ) )
& ( ~ ( member_a @ X2 @ S )
=> ( ( groups6334556678337121940_a_nat @ F @ ( insert_a @ X2 @ S ) )
= ( plus_plus_nat @ ( F @ X2 ) @ ( groups6334556678337121940_a_nat @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_605_sum__clauses_I2_J,axiom,
! [S: set_b,X2: b,F: b > real] :
( ( finite_finite_b @ S )
=> ( ( ( member_b @ X2 @ S )
=> ( ( groups8336678772925405937b_real @ F @ ( insert_b @ X2 @ S ) )
= ( groups8336678772925405937b_real @ F @ S ) ) )
& ( ~ ( member_b @ X2 @ S )
=> ( ( groups8336678772925405937b_real @ F @ ( insert_b @ X2 @ S ) )
= ( plus_plus_real @ ( F @ X2 ) @ ( groups8336678772925405937b_real @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_606_sum__clauses_I2_J,axiom,
! [S: set_a,X2: a,F: a > real] :
( ( finite_finite_a @ S )
=> ( ( ( member_a @ X2 @ S )
=> ( ( groups2740460157737275248a_real @ F @ ( insert_a @ X2 @ S ) )
= ( groups2740460157737275248a_real @ F @ S ) ) )
& ( ~ ( member_a @ X2 @ S )
=> ( ( groups2740460157737275248a_real @ F @ ( insert_a @ X2 @ S ) )
= ( plus_plus_real @ ( F @ X2 ) @ ( groups2740460157737275248a_real @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_607_sum__clauses_I2_J,axiom,
! [S: set_set_a,X2: set_a,F: set_a > real] :
( ( finite_finite_set_a @ S )
=> ( ( ( member_set_a @ X2 @ S )
=> ( ( groups9174420418583655632a_real @ F @ ( insert_set_a @ X2 @ S ) )
= ( groups9174420418583655632a_real @ F @ S ) ) )
& ( ~ ( member_set_a @ X2 @ S )
=> ( ( groups9174420418583655632a_real @ F @ ( insert_set_a @ X2 @ S ) )
= ( plus_plus_real @ ( F @ X2 ) @ ( groups9174420418583655632a_real @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_608_sum__clauses_I2_J,axiom,
! [S: set_set_a,X2: set_a,F: set_a > nat] :
( ( finite_finite_set_a @ S )
=> ( ( ( member_set_a @ X2 @ S )
=> ( ( groups6141743369313575924_a_nat @ F @ ( insert_set_a @ X2 @ S ) )
= ( groups6141743369313575924_a_nat @ F @ S ) ) )
& ( ~ ( member_set_a @ X2 @ S )
=> ( ( groups6141743369313575924_a_nat @ F @ ( insert_set_a @ X2 @ S ) )
= ( plus_plus_nat @ ( F @ X2 ) @ ( groups6141743369313575924_a_nat @ F @ S ) ) ) ) ) ) ).
% sum_clauses(2)
thf(fact_609_prob__space__completion,axiom,
probab7247484486040049089pace_a @ ( comple3428971583294703880tion_a @ m ) ).
% prob_space_completion
thf(fact_610_ball__insert,axiom,
! [A: b,B: set_b,P: b > $o] :
( ( ! [X: b] :
( ( member_b @ X @ ( insert_b @ A @ B ) )
=> ( P @ X ) ) )
= ( ( P @ A )
& ! [X: b] :
( ( member_b @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_insert
thf(fact_611_ball__insert,axiom,
! [A: a,B: set_a,P: a > $o] :
( ( ! [X: a] :
( ( member_a @ X @ ( insert_a @ A @ B ) )
=> ( P @ X ) ) )
= ( ( P @ A )
& ! [X: a] :
( ( member_a @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_insert
thf(fact_612_ball__insert,axiom,
! [A: $o,B: set_o,P: $o > $o] :
( ( ! [X: $o] :
( ( member_o @ X @ ( insert_o @ A @ B ) )
=> ( P @ X ) ) )
= ( ( P @ A )
& ! [X: $o] :
( ( member_o @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_insert
thf(fact_613_ball__insert,axiom,
! [A: nat,B: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( insert_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ( P @ A )
& ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_insert
thf(fact_614_ball__insert,axiom,
! [A: a > real,B: set_a_real,P: ( a > real ) > $o] :
( ( ! [X: a > real] :
( ( member_a_real @ X @ ( insert_a_real @ A @ B ) )
=> ( P @ X ) ) )
= ( ( P @ A )
& ! [X: a > real] :
( ( member_a_real @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_insert
thf(fact_615_ball__insert,axiom,
! [A: set_a,B: set_set_a,P: set_a > $o] :
( ( ! [X: set_a] :
( ( member_set_a @ X @ ( insert_set_a @ A @ B ) )
=> ( P @ X ) ) )
= ( ( P @ A )
& ! [X: set_a] :
( ( member_set_a @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_insert
thf(fact_616_sumset__empty_I2_J,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% sumset_empty(2)
thf(fact_617_sumset__empty_I1_J,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% sumset_empty(1)
thf(fact_618_set__plus__intro,axiom,
! [A: real,C3: set_real,B2: real,D2: set_real] :
( ( member_real @ A @ C3 )
=> ( ( member_real @ B2 @ D2 )
=> ( member_real @ ( plus_plus_real @ A @ B2 ) @ ( plus_plus_set_real @ C3 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_619_set__plus__intro,axiom,
! [A: nat,C3: set_nat,B2: nat,D2: set_nat] :
( ( member_nat @ A @ C3 )
=> ( ( member_nat @ B2 @ D2 )
=> ( member_nat @ ( plus_plus_nat @ A @ B2 ) @ ( plus_plus_set_nat @ C3 @ D2 ) ) ) ) ).
% set_plus_intro
thf(fact_620_is__singletonI,axiom,
! [X2: a > real] : ( is_singleton_a_real @ ( insert_a_real @ X2 @ bot_bot_set_a_real ) ) ).
% is_singletonI
thf(fact_621_is__singletonI,axiom,
! [X2: set_a] : ( is_singleton_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ).
% is_singletonI
thf(fact_622_is__singletonI,axiom,
! [X2: b] : ( is_singleton_b @ ( insert_b @ X2 @ bot_bot_set_b ) ) ).
% is_singletonI
thf(fact_623_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_624_is__singletonI,axiom,
! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_625_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_626_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_627_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_628_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_629_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_630_is__singletonI_H,axiom,
! [A2: set_set_a] :
( ( A2 != bot_bot_set_set_a )
=> ( ! [X3: set_a,Y2: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ( member_set_a @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_set_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_631_is__singletonI_H,axiom,
! [A2: set_a_real] :
( ( A2 != bot_bot_set_a_real )
=> ( ! [X3: a > real,Y2: a > real] :
( ( member_a_real @ X3 @ A2 )
=> ( ( member_a_real @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_a_real @ A2 ) ) ) ).
% is_singletonI'
thf(fact_632_is__singletonI_H,axiom,
! [A2: set_b] :
( ( A2 != bot_bot_set_b )
=> ( ! [X3: b,Y2: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_b @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_b @ A2 ) ) ) ).
% is_singletonI'
thf(fact_633_is__singletonI_H,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
=> ( ! [X3: a,Y2: a] :
( ( member_a @ X3 @ A2 )
=> ( ( member_a @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_a @ A2 ) ) ) ).
% is_singletonI'
thf(fact_634_is__singletonI_H,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X3: $o,Y2: $o] :
( ( member_o @ X3 @ A2 )
=> ( ( member_o @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_o @ A2 ) ) ) ).
% is_singletonI'
thf(fact_635_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_636_set__plus__elim,axiom,
! [X2: real,A2: set_real,B: set_real] :
( ( member_real @ X2 @ ( plus_plus_set_real @ A2 @ B ) )
=> ~ ! [A5: real,B6: real] :
( ( X2
= ( plus_plus_real @ A5 @ B6 ) )
=> ( ( member_real @ A5 @ A2 )
=> ~ ( member_real @ B6 @ B ) ) ) ) ).
% set_plus_elim
thf(fact_637_set__plus__elim,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ X2 @ ( plus_plus_set_nat @ A2 @ B ) )
=> ~ ! [A5: nat,B6: nat] :
( ( X2
= ( plus_plus_nat @ A5 @ B6 ) )
=> ( ( member_nat @ A5 @ A2 )
=> ~ ( member_nat @ B6 @ B ) ) ) ) ).
% set_plus_elim
thf(fact_638_finite__set__plus,axiom,
! [S2: set_nat,T2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( finite_finite_nat @ ( plus_plus_set_nat @ S2 @ T2 ) ) ) ) ).
% finite_set_plus
thf(fact_639_is__singleton__def,axiom,
( is_singleton_a_real
= ( ^ [A3: set_a_real] :
? [X: a > real] :
( A3
= ( insert_a_real @ X @ bot_bot_set_a_real ) ) ) ) ).
% is_singleton_def
thf(fact_640_is__singleton__def,axiom,
( is_singleton_set_a
= ( ^ [A3: set_set_a] :
? [X: set_a] :
( A3
= ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_641_is__singleton__def,axiom,
( is_singleton_b
= ( ^ [A3: set_b] :
? [X: b] :
( A3
= ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% is_singleton_def
thf(fact_642_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
? [X: a] :
( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_643_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
? [X: $o] :
( A3
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_644_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X: nat] :
( A3
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_645_is__singletonE,axiom,
! [A2: set_a_real] :
( ( is_singleton_a_real @ A2 )
=> ~ ! [X3: a > real] :
( A2
!= ( insert_a_real @ X3 @ bot_bot_set_a_real ) ) ) ).
% is_singletonE
thf(fact_646_is__singletonE,axiom,
! [A2: set_set_a] :
( ( is_singleton_set_a @ A2 )
=> ~ ! [X3: set_a] :
( A2
!= ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ).
% is_singletonE
thf(fact_647_is__singletonE,axiom,
! [A2: set_b] :
( ( is_singleton_b @ A2 )
=> ~ ! [X3: b] :
( A2
!= ( insert_b @ X3 @ bot_bot_set_b ) ) ) ).
% is_singletonE
thf(fact_648_is__singletonE,axiom,
! [A2: set_a] :
( ( is_singleton_a @ A2 )
=> ~ ! [X3: a] :
( A2
!= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_649_is__singletonE,axiom,
! [A2: set_o] :
( ( is_singleton_o @ A2 )
=> ~ ! [X3: $o] :
( A2
!= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_650_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X3: nat] :
( A2
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_651_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_652_prob__space_Oprob__space__completion,axiom,
! [M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( probab7247484486040049089pace_a @ ( comple3428971583294703880tion_a @ M ) ) ) ).
% prob_space.prob_space_completion
thf(fact_653_finite__transitivity__chain,axiom,
! [A2: set_set_a,R: set_a > set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ! [X3: set_a] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: set_a,Y2: set_a,Z2: set_a] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ? [Y5: set_a] :
( ( member_set_a @ Y5 @ A2 )
& ( R @ X3 @ Y5 ) ) )
=> ( A2 = bot_bot_set_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_654_finite__transitivity__chain,axiom,
! [A2: set_a_real,R: ( a > real ) > ( a > real ) > $o] :
( ( finite_finite_a_real @ A2 )
=> ( ! [X3: a > real] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: a > real,Y2: a > real,Z2: a > real] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
=> ? [Y5: a > real] :
( ( member_a_real @ Y5 @ A2 )
& ( R @ X3 @ Y5 ) ) )
=> ( A2 = bot_bot_set_a_real ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_655_finite__transitivity__chain,axiom,
! [A2: set_b,R: b > b > $o] :
( ( finite_finite_b @ A2 )
=> ( ! [X3: b] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: b,Y2: b,Z2: b] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Y5: b] :
( ( member_b @ Y5 @ A2 )
& ( R @ X3 @ Y5 ) ) )
=> ( A2 = bot_bot_set_b ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_656_finite__transitivity__chain,axiom,
! [A2: set_a,R: a > a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: a,Y2: a,Z2: a] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Y5: a] :
( ( member_a @ Y5 @ A2 )
& ( R @ X3 @ Y5 ) ) )
=> ( A2 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_657_finite__transitivity__chain,axiom,
! [A2: set_o,R: $o > $o > $o] :
( ( finite_finite_o @ A2 )
=> ( ! [X3: $o] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: $o,Y2: $o,Z2: $o] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ? [Y5: $o] :
( ( member_o @ Y5 @ A2 )
& ( R @ X3 @ Y5 ) ) )
=> ( A2 = bot_bot_set_o ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_658_finite__transitivity__chain,axiom,
! [A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z2: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A2 )
& ( R @ X3 @ Y5 ) ) )
=> ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_659_add__diff__add,axiom,
! [A: real,C: real,B2: real,D: real] :
( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) )
= ( plus_plus_real @ ( minus_minus_real @ A @ B2 ) @ ( minus_minus_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_660_remove__def,axiom,
( remove_b
= ( ^ [X: b,A3: set_b] : ( minus_minus_set_b @ A3 @ ( insert_b @ X @ bot_bot_set_b ) ) ) ) ).
% remove_def
thf(fact_661_remove__def,axiom,
( remove_a
= ( ^ [X: a,A3: set_a] : ( minus_minus_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_662_remove__def,axiom,
( remove_o
= ( ^ [X: $o,A3: set_o] : ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% remove_def
thf(fact_663_remove__def,axiom,
( remove_a_real
= ( ^ [X: a > real,A3: set_a_real] : ( minus_4124197362600706274a_real @ A3 @ ( insert_a_real @ X @ bot_bot_set_a_real ) ) ) ) ).
% remove_def
thf(fact_664_remove__def,axiom,
( remove_nat
= ( ^ [X: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_665_remove__def,axiom,
( remove_set_a
= ( ^ [X: set_a,A3: set_set_a] : ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) ) ).
% remove_def
thf(fact_666_subprob__space__axioms,axiom,
giry_subprob_space_a @ m ).
% subprob_space_axioms
thf(fact_667_is__singleton__the__elem,axiom,
( is_singleton_a_real
= ( ^ [A3: set_a_real] :
( A3
= ( insert_a_real @ ( the_elem_a_real @ A3 ) @ bot_bot_set_a_real ) ) ) ) ).
% is_singleton_the_elem
thf(fact_668_is__singleton__the__elem,axiom,
( is_singleton_set_a
= ( ^ [A3: set_set_a] :
( A3
= ( insert_set_a @ ( the_elem_set_a @ A3 ) @ bot_bot_set_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_669_is__singleton__the__elem,axiom,
( is_singleton_b
= ( ^ [A3: set_b] :
( A3
= ( insert_b @ ( the_elem_b @ A3 ) @ bot_bot_set_b ) ) ) ) ).
% is_singleton_the_elem
thf(fact_670_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A3: set_a] :
( A3
= ( insert_a @ ( the_elem_a @ A3 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_671_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
( A3
= ( insert_o @ ( the_elem_o @ A3 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_672_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_673_Set_Ois__empty__def,axiom,
( is_empty_b
= ( ^ [A3: set_b] : ( A3 = bot_bot_set_b ) ) ) ).
% Set.is_empty_def
thf(fact_674_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A3: set_a] : ( A3 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_675_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A3: set_o] : ( A3 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_676_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_677_subprob__not__empty,axiom,
( ( sigma_space_a @ m )
!= bot_bot_set_a ) ).
% subprob_not_empty
thf(fact_678_member__remove,axiom,
! [X2: set_a,Y3: set_a,A2: set_set_a] :
( ( member_set_a @ X2 @ ( remove_set_a @ Y3 @ A2 ) )
= ( ( member_set_a @ X2 @ A2 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_679_member__remove,axiom,
! [X2: nat,Y3: nat,A2: set_nat] :
( ( member_nat @ X2 @ ( remove_nat @ Y3 @ A2 ) )
= ( ( member_nat @ X2 @ A2 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_680_member__remove,axiom,
! [X2: a > real,Y3: a > real,A2: set_a_real] :
( ( member_a_real @ X2 @ ( remove_a_real @ Y3 @ A2 ) )
= ( ( member_a_real @ X2 @ A2 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_681_member__remove,axiom,
! [X2: b,Y3: b,A2: set_b] :
( ( member_b @ X2 @ ( remove_b @ Y3 @ A2 ) )
= ( ( member_b @ X2 @ A2 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_682_member__remove,axiom,
! [X2: a,Y3: a,A2: set_a] :
( ( member_a @ X2 @ ( remove_a @ Y3 @ A2 ) )
= ( ( member_a @ X2 @ A2 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_683_member__remove,axiom,
! [X2: $o,Y3: $o,A2: set_o] :
( ( member_o @ X2 @ ( remove_o @ Y3 @ A2 ) )
= ( ( member_o @ X2 @ A2 )
& ( X2 != Y3 ) ) ) ).
% member_remove
thf(fact_684_the__elem__eq,axiom,
! [X2: a > real] :
( ( the_elem_a_real @ ( insert_a_real @ X2 @ bot_bot_set_a_real ) )
= X2 ) ).
% the_elem_eq
thf(fact_685_the__elem__eq,axiom,
! [X2: set_a] :
( ( the_elem_set_a @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_686_the__elem__eq,axiom,
! [X2: b] :
( ( the_elem_b @ ( insert_b @ X2 @ bot_bot_set_b ) )
= X2 ) ).
% the_elem_eq
thf(fact_687_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_688_the__elem__eq,axiom,
! [X2: $o] :
( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% the_elem_eq
thf(fact_689_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_690_prob__space_Onot__empty,axiom,
! [M: sigma_measure_b] :
( ( probab7247484486040049090pace_b @ M )
=> ( ( sigma_space_b @ M )
!= bot_bot_set_b ) ) ).
% prob_space.not_empty
thf(fact_691_prob__space_Onot__empty,axiom,
! [M: sigma_measure_o] :
( ( probab1190487603588612059pace_o @ M )
=> ( ( sigma_space_o @ M )
!= bot_bot_set_o ) ) ).
% prob_space.not_empty
thf(fact_692_prob__space_Onot__empty,axiom,
! [M: sigma_measure_nat] :
( ( probab2904919403188438605ce_nat @ M )
=> ( ( sigma_space_nat @ M )
!= bot_bot_set_nat ) ) ).
% prob_space.not_empty
thf(fact_693_prob__space_Onot__empty,axiom,
! [M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( sigma_space_a @ M )
!= bot_bot_set_a ) ) ).
% prob_space.not_empty
thf(fact_694_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_695_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_696_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_697_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_698_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_699_space__completion,axiom,
! [M: sigma_measure_a] :
( ( sigma_space_a @ ( comple3428971583294703880tion_a @ M ) )
= ( sigma_space_a @ M ) ) ).
% space_completion
thf(fact_700_space__bot,axiom,
( ( sigma_space_b @ bot_bo2108912055686869392sure_b )
= bot_bot_set_b ) ).
% space_bot
thf(fact_701_space__bot,axiom,
( ( sigma_space_a @ bot_bo2108912051383640591sure_a )
= bot_bot_set_a ) ).
% space_bot
thf(fact_702_space__bot,axiom,
( ( sigma_space_o @ bot_bo5758314138661044393sure_o )
= bot_bot_set_o ) ).
% space_bot
thf(fact_703_space__bot,axiom,
( ( sigma_space_nat @ bot_bo6718502177978453909re_nat )
= bot_bot_set_nat ) ).
% space_bot
thf(fact_704_subprob__space_Osubprob__not__empty,axiom,
! [M: sigma_measure_b] :
( ( giry_subprob_space_b @ M )
=> ( ( sigma_space_b @ M )
!= bot_bot_set_b ) ) ).
% subprob_space.subprob_not_empty
thf(fact_705_subprob__space_Osubprob__not__empty,axiom,
! [M: sigma_measure_o] :
( ( giry_subprob_space_o @ M )
=> ( ( sigma_space_o @ M )
!= bot_bot_set_o ) ) ).
% subprob_space.subprob_not_empty
thf(fact_706_subprob__space_Osubprob__not__empty,axiom,
! [M: sigma_measure_nat] :
( ( giry_s459323515522551452ce_nat @ M )
=> ( ( sigma_space_nat @ M )
!= bot_bot_set_nat ) ) ).
% subprob_space.subprob_not_empty
thf(fact_707_subprob__space_Osubprob__not__empty,axiom,
! [M: sigma_measure_a] :
( ( giry_subprob_space_a @ M )
=> ( ( sigma_space_a @ M )
!= bot_bot_set_a ) ) ).
% subprob_space.subprob_not_empty
thf(fact_708_Collect__empty__eq__bot,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( P = bot_bot_b_o ) ) ).
% Collect_empty_eq_bot
thf(fact_709_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_710_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_711_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_712_bot__empty__eq,axiom,
( bot_bot_set_a_o
= ( ^ [X: set_a] : ( member_set_a @ X @ bot_bot_set_set_a ) ) ) ).
% bot_empty_eq
thf(fact_713_bot__empty__eq,axiom,
( bot_bot_a_real_o
= ( ^ [X: a > real] : ( member_a_real @ X @ bot_bot_set_a_real ) ) ) ).
% bot_empty_eq
thf(fact_714_bot__empty__eq,axiom,
( bot_bot_b_o
= ( ^ [X: b] : ( member_b @ X @ bot_bot_set_b ) ) ) ).
% bot_empty_eq
thf(fact_715_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X: a] : ( member_a @ X @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_716_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_717_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_718_prob__space__imp__subprob__space,axiom,
! [M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( giry_subprob_space_a @ M ) ) ).
% prob_space_imp_subprob_space
thf(fact_719_space__empty__eq__bot,axiom,
! [A: sigma_measure_b] :
( ( ( sigma_space_b @ A )
= bot_bot_set_b )
= ( A = bot_bo2108912055686869392sure_b ) ) ).
% space_empty_eq_bot
thf(fact_720_space__empty__eq__bot,axiom,
! [A: sigma_measure_a] :
( ( ( sigma_space_a @ A )
= bot_bot_set_a )
= ( A = bot_bo2108912051383640591sure_a ) ) ).
% space_empty_eq_bot
thf(fact_721_space__empty__eq__bot,axiom,
! [A: sigma_measure_o] :
( ( ( sigma_space_o @ A )
= bot_bot_set_o )
= ( A = bot_bo5758314138661044393sure_o ) ) ).
% space_empty_eq_bot
thf(fact_722_space__empty__eq__bot,axiom,
! [A: sigma_measure_nat] :
( ( ( sigma_space_nat @ A )
= bot_bot_set_nat )
= ( A = bot_bo6718502177978453909re_nat ) ) ).
% space_empty_eq_bot
thf(fact_723_prob__space__uniform__count__measure,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ( ( A2 != bot_bot_set_b )
=> ( probab7247484486040049090pace_b @ ( nonneg7367794086797660665sure_b @ A2 ) ) ) ) ).
% prob_space_uniform_count_measure
thf(fact_724_prob__space__uniform__count__measure,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ( probab1190487603588612059pace_o @ ( nonneg5198678888045619090sure_o @ A2 ) ) ) ) ).
% prob_space_uniform_count_measure
thf(fact_725_prob__space__uniform__count__measure,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( probab2904919403188438605ce_nat @ ( nonneg7031465154080143958re_nat @ A2 ) ) ) ) ).
% prob_space_uniform_count_measure
thf(fact_726_prob__space__uniform__count__measure,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( A2 != bot_bot_set_a )
=> ( probab7247484486040049089pace_a @ ( nonneg7367794086797660664sure_a @ A2 ) ) ) ) ).
% prob_space_uniform_count_measure
thf(fact_727_sum__delta__notmem_I4_J,axiom,
! [X2: b,S2: set_b,P: b > real,Q: b > real] :
( ~ ( member_b @ X2 @ S2 )
=> ( ( groups8336678772925405937b_real
@ ^ [Y4: b] : ( if_real @ ( X2 = Y4 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups8336678772925405937b_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(4)
thf(fact_728_sum__delta__notmem_I4_J,axiom,
! [X2: a,S2: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups2740460157737275248a_real
@ ^ [Y4: a] : ( if_real @ ( X2 = Y4 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups2740460157737275248a_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(4)
thf(fact_729_sum__delta__notmem_I1_J,axiom,
! [X2: b,S2: set_b,P: b > real,Q: b > real] :
( ~ ( member_b @ X2 @ S2 )
=> ( ( groups8336678772925405937b_real
@ ^ [Y4: b] : ( if_real @ ( Y4 = X2 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups8336678772925405937b_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(1)
thf(fact_730_sum__delta__notmem_I1_J,axiom,
! [X2: a,S2: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups2740460157737275248a_real
@ ^ [Y4: a] : ( if_real @ ( Y4 = X2 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups2740460157737275248a_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(1)
thf(fact_731_sum__delta__notmem_I2_J,axiom,
! [X2: b,S2: set_b,P: b > real,Q: b > real] :
( ~ ( member_b @ X2 @ S2 )
=> ( ( groups8336678772925405937b_real
@ ^ [Y4: b] : ( if_real @ ( X2 = Y4 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups8336678772925405937b_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(2)
thf(fact_732_sum__delta__notmem_I2_J,axiom,
! [X2: a,S2: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups2740460157737275248a_real
@ ^ [Y4: a] : ( if_real @ ( X2 = Y4 ) @ ( P @ X2 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups2740460157737275248a_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(2)
thf(fact_733_sum__delta__notmem_I3_J,axiom,
! [X2: b,S2: set_b,P: b > real,Q: b > real] :
( ~ ( member_b @ X2 @ S2 )
=> ( ( groups8336678772925405937b_real
@ ^ [Y4: b] : ( if_real @ ( Y4 = X2 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups8336678772925405937b_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(3)
thf(fact_734_sum__delta__notmem_I3_J,axiom,
! [X2: a,S2: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S2 )
=> ( ( groups2740460157737275248a_real
@ ^ [Y4: a] : ( if_real @ ( Y4 = X2 ) @ ( P @ Y4 ) @ ( Q @ Y4 ) )
@ S2 )
= ( groups2740460157737275248a_real @ Q @ S2 ) ) ) ).
% sum_delta_notmem(3)
thf(fact_735_space__uniform__count__measure__empty__iff,axiom,
! [X4: set_b] :
( ( ( sigma_space_b @ ( nonneg7367794086797660665sure_b @ X4 ) )
= bot_bot_set_b )
= ( X4 = bot_bot_set_b ) ) ).
% space_uniform_count_measure_empty_iff
thf(fact_736_space__uniform__count__measure__empty__iff,axiom,
! [X4: set_a] :
( ( ( sigma_space_a @ ( nonneg7367794086797660664sure_a @ X4 ) )
= bot_bot_set_a )
= ( X4 = bot_bot_set_a ) ) ).
% space_uniform_count_measure_empty_iff
thf(fact_737_space__uniform__count__measure__empty__iff,axiom,
! [X4: set_o] :
( ( ( sigma_space_o @ ( nonneg5198678888045619090sure_o @ X4 ) )
= bot_bot_set_o )
= ( X4 = bot_bot_set_o ) ) ).
% space_uniform_count_measure_empty_iff
thf(fact_738_space__uniform__count__measure__empty__iff,axiom,
! [X4: set_nat] :
( ( ( sigma_space_nat @ ( nonneg7031465154080143958re_nat @ X4 ) )
= bot_bot_set_nat )
= ( X4 = bot_bot_set_nat ) ) ).
% space_uniform_count_measure_empty_iff
thf(fact_739_space__uniform__count__measure,axiom,
! [A2: set_a] :
( ( sigma_space_a @ ( nonneg7367794086797660664sure_a @ A2 ) )
= A2 ) ).
% space_uniform_count_measure
thf(fact_740_affine__hull__finite__step,axiom,
! [S: set_real,A: real,W: real,Y3: real] :
( ( finite_finite_real @ S )
=> ( ( ? [U: real > real] :
( ( ( groups8097168146408367636l_real @ U @ ( insert_real @ A @ S ) )
= W )
& ( ( groups8097168146408367636l_real
@ ^ [X: real] : ( real_V1485227260804924795R_real @ ( U @ X ) @ X )
@ ( insert_real @ A @ S ) )
= Y3 ) ) )
= ( ? [V: real,U: real > real] :
( ( ( groups8097168146408367636l_real @ U @ S )
= ( minus_minus_real @ W @ V ) )
& ( ( groups8097168146408367636l_real
@ ^ [X: real] : ( real_V1485227260804924795R_real @ ( U @ X ) @ X )
@ S )
= ( minus_minus_real @ Y3 @ ( real_V1485227260804924795R_real @ V @ A ) ) ) ) ) ) ) ).
% affine_hull_finite_step
thf(fact_741_finite__set__sum,axiom,
! [A2: set_b,B: b > set_nat] :
( ( finite_finite_b @ A2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( finite_finite_nat @ ( B @ X3 ) ) )
=> ( finite_finite_nat @ ( groups1208830032083771979et_nat @ B @ A2 ) ) ) ) ).
% finite_set_sum
thf(fact_742_finite__set__sum,axiom,
! [A2: set_nat,B: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( finite_finite_nat @ ( B @ X3 ) ) )
=> ( finite_finite_nat @ ( groups2637260376230714770et_nat @ B @ A2 ) ) ) ) ).
% finite_set_sum
thf(fact_743_finite__set__sum,axiom,
! [A2: set_a,B: a > set_nat] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( finite_finite_nat @ ( B @ X3 ) ) )
=> ( finite_finite_nat @ ( groups7609827518827096650et_nat @ B @ A2 ) ) ) ) ).
% finite_set_sum
thf(fact_744_assms_I2_J,axiom,
! [I: b] :
( ( member_b @ I @ i )
=> ( member_a_real @ ( f @ I ) @ ( sigma_9116425665531756122a_real @ m @ borel_5078946678739801102l_real ) ) ) ).
% assms(2)
thf(fact_745_sum__diff__nat,axiom,
! [B: set_b,A2: set_b,F: b > nat] :
( ( finite_finite_b @ B )
=> ( ( ord_less_eq_set_b @ B @ A2 )
=> ( ( groups7570001007293516437_b_nat @ F @ ( minus_minus_set_b @ A2 @ B ) )
= ( minus_minus_nat @ ( groups7570001007293516437_b_nat @ F @ A2 ) @ ( groups7570001007293516437_b_nat @ F @ B ) ) ) ) ) ).
% sum_diff_nat
thf(fact_746_sum__diff__nat,axiom,
! [B: set_o,A2: set_o,F: $o > nat] :
( ( finite_finite_o @ B )
=> ( ( ord_less_eq_set_o @ B @ A2 )
=> ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A2 @ B ) )
= ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A2 ) @ ( groups8507830703676809646_o_nat @ F @ B ) ) ) ) ) ).
% sum_diff_nat
thf(fact_747_sum__diff__nat,axiom,
! [B: set_a_real,A2: set_a_real,F: ( a > real ) > nat] :
( ( finite_finite_a_real @ B )
=> ( ( ord_le3334967407727675675a_real @ B @ A2 )
=> ( ( groups1701885688937111089al_nat @ F @ ( minus_4124197362600706274a_real @ A2 @ B ) )
= ( minus_minus_nat @ ( groups1701885688937111089al_nat @ F @ A2 ) @ ( groups1701885688937111089al_nat @ F @ B ) ) ) ) ) ).
% sum_diff_nat
thf(fact_748_sum__diff__nat,axiom,
! [B: set_nat,A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B ) ) ) ) ) ).
% sum_diff_nat
thf(fact_749_sum__diff__nat,axiom,
! [B: set_set_a,A2: set_set_a,F: set_a > nat] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( ( groups6141743369313575924_a_nat @ F @ ( minus_5736297505244876581_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( groups6141743369313575924_a_nat @ F @ A2 ) @ ( groups6141743369313575924_a_nat @ F @ B ) ) ) ) ) ).
% sum_diff_nat
thf(fact_750_sum__diff__nat,axiom,
! [B: set_a,A2: set_a,F: a > nat] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( groups6334556678337121940_a_nat @ F @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) @ ( groups6334556678337121940_a_nat @ F @ B ) ) ) ) ) ).
% sum_diff_nat
thf(fact_751_subprob__space__null__measure,axiom,
! [M: sigma_measure_b] :
( ( ( sigma_space_b @ M )
!= bot_bot_set_b )
=> ( giry_subprob_space_b @ ( measur3836006170472588155sure_b @ M ) ) ) ).
% subprob_space_null_measure
thf(fact_752_subprob__space__null__measure,axiom,
! [M: sigma_measure_o] :
( ( ( sigma_space_o @ M )
!= bot_bot_set_o )
=> ( giry_subprob_space_o @ ( measur6133975857628879380sure_o @ M ) ) ) ).
% subprob_space_null_measure
thf(fact_753_subprob__space__null__measure,axiom,
! [M: sigma_measure_nat] :
( ( ( sigma_space_nat @ M )
!= bot_bot_set_nat )
=> ( giry_s459323515522551452ce_nat @ ( measur6922722954359385172re_nat @ M ) ) ) ).
% subprob_space_null_measure
thf(fact_754_subprob__space__null__measure,axiom,
! [M: sigma_measure_a] :
( ( ( sigma_space_a @ M )
!= bot_bot_set_a )
=> ( giry_subprob_space_a @ ( measur3836006170472588154sure_a @ M ) ) ) ).
% subprob_space_null_measure
thf(fact_755_subprob__space__null__measure__iff,axiom,
! [M: sigma_measure_b] :
( ( giry_subprob_space_b @ ( measur3836006170472588155sure_b @ M ) )
= ( ( sigma_space_b @ M )
!= bot_bot_set_b ) ) ).
% subprob_space_null_measure_iff
thf(fact_756_subprob__space__null__measure__iff,axiom,
! [M: sigma_measure_o] :
( ( giry_subprob_space_o @ ( measur6133975857628879380sure_o @ M ) )
= ( ( sigma_space_o @ M )
!= bot_bot_set_o ) ) ).
% subprob_space_null_measure_iff
thf(fact_757_subprob__space__null__measure__iff,axiom,
! [M: sigma_measure_nat] :
( ( giry_s459323515522551452ce_nat @ ( measur6922722954359385172re_nat @ M ) )
= ( ( sigma_space_nat @ M )
!= bot_bot_set_nat ) ) ).
% subprob_space_null_measure_iff
thf(fact_758_subprob__space__null__measure__iff,axiom,
! [M: sigma_measure_a] :
( ( giry_subprob_space_a @ ( measur3836006170472588154sure_a @ M ) )
= ( ( sigma_space_a @ M )
!= bot_bot_set_a ) ) ).
% subprob_space_null_measure_iff
thf(fact_759_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_760_order__refl,axiom,
! [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_761_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_762_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_763_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_764_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_765_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_766_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_767_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_768_subsetI,axiom,
! [A2: set_a_real,B: set_a_real] :
( ! [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
=> ( member_a_real @ X3 @ B ) )
=> ( ord_le3334967407727675675a_real @ A2 @ B ) ) ).
% subsetI
thf(fact_769_subsetI,axiom,
! [A2: set_b,B: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_b @ X3 @ B ) )
=> ( ord_less_eq_set_b @ A2 @ B ) ) ).
% subsetI
thf(fact_770_subsetI,axiom,
! [A2: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A2 )
=> ( member_o @ X3 @ B ) )
=> ( ord_less_eq_set_o @ A2 @ B ) ) ).
% subsetI
thf(fact_771_subsetI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_772_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_773_subset__antisym,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_774_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_775_add__le__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_776_add__le__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
= ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_777_add__le__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_778_add__le__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_779_subset__empty,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_780_subset__empty,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_781_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_782_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_783_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_784_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_785_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_786_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_787_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_788_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_789_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_790_insert__subset,axiom,
! [X2: a > real,A2: set_a_real,B: set_a_real] :
( ( ord_le3334967407727675675a_real @ ( insert_a_real @ X2 @ A2 ) @ B )
= ( ( member_a_real @ X2 @ B )
& ( ord_le3334967407727675675a_real @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_791_insert__subset,axiom,
! [X2: b,A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X2 @ A2 ) @ B )
= ( ( member_b @ X2 @ B )
& ( ord_less_eq_set_b @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_792_insert__subset,axiom,
! [X2: $o,A2: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X2 @ A2 ) @ B )
= ( ( member_o @ X2 @ B )
& ( ord_less_eq_set_o @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_793_insert__subset,axiom,
! [X2: set_a,A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ A2 ) @ B )
= ( ( member_set_a @ X2 @ B )
& ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_794_insert__subset,axiom,
! [X2: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A2 ) @ B )
= ( ( member_a @ X2 @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_795_space__null__measure,axiom,
! [M: sigma_measure_a] :
( ( sigma_space_a @ ( measur3836006170472588154sure_a @ M ) )
= ( sigma_space_a @ M ) ) ).
% space_null_measure
thf(fact_796_finite__Collect__subsets,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_set_b
@ ( collect_set_b
@ ^ [B3: set_b] : ( ord_less_eq_set_b @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_797_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_798_finite__Collect__subsets,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite7209287970140883943_set_a
@ ( collect_set_set_a
@ ^ [B3: set_set_a] : ( ord_le3724670747650509150_set_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_799_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_800_singleton__insert__inj__eq,axiom,
! [B2: a > real,A: a > real,A2: set_a_real] :
( ( ( insert_a_real @ B2 @ bot_bot_set_a_real )
= ( insert_a_real @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le3334967407727675675a_real @ A2 @ ( insert_a_real @ B2 @ bot_bot_set_a_real ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_801_singleton__insert__inj__eq,axiom,
! [B2: b,A: b,A2: set_b] :
( ( ( insert_b @ B2 @ bot_bot_set_b )
= ( insert_b @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_802_singleton__insert__inj__eq,axiom,
! [B2: $o,A: $o,A2: set_o] :
( ( ( insert_o @ B2 @ bot_bot_set_o )
= ( insert_o @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_803_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_804_singleton__insert__inj__eq,axiom,
! [B2: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B2 @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_805_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A2: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_806_singleton__insert__inj__eq_H,axiom,
! [A: a > real,A2: set_a_real,B2: a > real] :
( ( ( insert_a_real @ A @ A2 )
= ( insert_a_real @ B2 @ bot_bot_set_a_real ) )
= ( ( A = B2 )
& ( ord_le3334967407727675675a_real @ A2 @ ( insert_a_real @ B2 @ bot_bot_set_a_real ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_807_singleton__insert__inj__eq_H,axiom,
! [A: b,A2: set_b,B2: b] :
( ( ( insert_b @ A @ A2 )
= ( insert_b @ B2 @ bot_bot_set_b ) )
= ( ( A = B2 )
& ( ord_less_eq_set_b @ A2 @ ( insert_b @ B2 @ bot_bot_set_b ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_808_singleton__insert__inj__eq_H,axiom,
! [A: $o,A2: set_o,B2: $o] :
( ( ( insert_o @ A @ A2 )
= ( insert_o @ B2 @ bot_bot_set_o ) )
= ( ( A = B2 )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_809_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_810_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B2: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B2 @ bot_bot_set_set_a ) )
= ( ( A = B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_811_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B2: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_812_Diff__eq__empty__iff,axiom,
! [A2: set_b,B: set_b] :
( ( ( minus_minus_set_b @ A2 @ B )
= bot_bot_set_b )
= ( ord_less_eq_set_b @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_813_Diff__eq__empty__iff,axiom,
! [A2: set_o,B: set_o] :
( ( ( minus_minus_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_814_Diff__eq__empty__iff,axiom,
! [A2: set_a_real,B: set_a_real] :
( ( ( minus_4124197362600706274a_real @ A2 @ B )
= bot_bot_set_a_real )
= ( ord_le3334967407727675675a_real @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_815_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_816_Diff__eq__empty__iff,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A2 @ B )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_817_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_818_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_819_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_820_le__cases3,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_821_le__cases3,axiom,
! [X2: real,Y3: real,Z3: real] :
( ( ( ord_less_eq_real @ X2 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_822_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_823_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_set_a,Z: set_set_a] : ( Y = Z ) )
= ( ^ [X: set_set_a,Y4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y4 )
& ( ord_le3724670747650509150_set_a @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_824_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [X: real,Y4: real] :
( ( ord_less_eq_real @ X @ Y4 )
& ( ord_less_eq_real @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_825_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
= ( ^ [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_826_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_827_ord__eq__le__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( A = B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_828_ord__eq__le__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_829_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_830_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_831_ord__le__eq__trans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_832_ord__le__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_833_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_834_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_835_order__antisym,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_836_order__antisym,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_837_order__antisym,axiom,
! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_838_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_839_order_Otrans,axiom,
! [A: set_set_a,B2: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_840_order_Otrans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_841_order_Otrans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_842_order__trans,axiom,
! [X2: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_843_order__trans,axiom,
! [X2: set_set_a,Y3: set_set_a,Z3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z3 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_844_order__trans,axiom,
! [X2: real,Y3: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z3 )
=> ( ord_less_eq_real @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_845_order__trans,axiom,
! [X2: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_846_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_847_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A5: real,B6: real] :
( ( ord_less_eq_real @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: real,B6: real] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_848_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_849_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_set_a,Z: set_set_a] : ( Y = Z ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A4 )
& ( ord_le3724670747650509150_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_850_dual__order_Oeq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A4 )
& ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_851_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A4 )
& ( ord_less_eq_set_a @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_852_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_853_dual__order_Oantisym,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_854_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_855_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_856_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_857_dual__order_Otrans,axiom,
! [B2: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B2 )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_858_dual__order_Otrans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_859_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_860_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_861_antisym,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_862_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_863_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_864_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_865_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_set_a,Z: set_set_a] : ( Y = Z ) )
= ( ^ [A4: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_866_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: real,Z: real] : ( Y = Z ) )
= ( ^ [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
& ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_867_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_868_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_869_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_870_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_871_order__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_872_order__subst1,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_873_order__subst1,axiom,
! [A: real,F: set_a > real,B2: set_a,C: set_a] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_874_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_875_order__subst1,axiom,
! [A: set_a,F: real > set_a,B2: real,C: real] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_876_order__subst1,axiom,
! [A: nat,F: set_set_a > nat,B2: set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_877_order__subst1,axiom,
! [A: set_set_a,F: nat > set_set_a,B2: nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_878_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_879_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_880_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_881_order__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_882_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_883_order__subst2,axiom,
! [A: real,B2: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_884_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_885_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_886_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_887_order__subst2,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_888_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_889_order__eq__refl,axiom,
! [X2: set_set_a,Y3: set_set_a] :
( ( X2 = Y3 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_890_order__eq__refl,axiom,
! [X2: real,Y3: real] :
( ( X2 = Y3 )
=> ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_891_order__eq__refl,axiom,
! [X2: set_a,Y3: set_a] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_892_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_893_linorder__linear,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
| ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_894_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_895_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_896_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_897_ord__eq__le__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_898_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_899_ord__eq__le__subst,axiom,
! [A: set_a,F: real > set_a,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_900_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_901_ord__eq__le__subst,axiom,
! [A: real,F: set_a > real,B2: set_a,C: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_902_ord__eq__le__subst,axiom,
! [A: set_set_a,F: nat > set_set_a,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_903_ord__eq__le__subst,axiom,
! [A: nat,F: set_set_a > nat,B2: set_set_a,C: set_set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C )
=> ( ! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_904_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_905_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_906_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_907_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_908_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_909_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > set_a,C: set_a] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_910_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_911_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > real,C: real] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_912_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_set_a,C: set_set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3724670747650509150_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_913_ord__le__eq__subst,axiom,
! [A: set_set_a,B2: set_set_a,F: set_set_a > nat,C: nat] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_set_a,Y2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_914_linorder__le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_915_linorder__le__cases,axiom,
! [X2: real,Y3: real] :
( ~ ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_916_order__antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_917_order__antisym__conv,axiom,
! [Y3: set_set_a,X2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ Y3 @ X2 )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_918_order__antisym__conv,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq_real @ Y3 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_919_order__antisym__conv,axiom,
! [Y3: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_920_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_921_in__mono,axiom,
! [A2: set_a_real,B: set_a_real,X2: a > real] :
( ( ord_le3334967407727675675a_real @ A2 @ B )
=> ( ( member_a_real @ X2 @ A2 )
=> ( member_a_real @ X2 @ B ) ) ) ).
% in_mono
thf(fact_922_in__mono,axiom,
! [A2: set_b,B: set_b,X2: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_b @ X2 @ A2 )
=> ( member_b @ X2 @ B ) ) ) ).
% in_mono
thf(fact_923_in__mono,axiom,
! [A2: set_o,B: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ X2 @ A2 )
=> ( member_o @ X2 @ B ) ) ) ).
% in_mono
thf(fact_924_in__mono,axiom,
! [A2: set_set_a,B: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_925_in__mono,axiom,
! [A2: set_a,B: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B ) ) ) ).
% in_mono
thf(fact_926_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_927_subsetD,axiom,
! [A2: set_a_real,B: set_a_real,C: a > real] :
( ( ord_le3334967407727675675a_real @ A2 @ B )
=> ( ( member_a_real @ C @ A2 )
=> ( member_a_real @ C @ B ) ) ) ).
% subsetD
thf(fact_928_subsetD,axiom,
! [A2: set_b,B: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B ) ) ) ).
% subsetD
thf(fact_929_subsetD,axiom,
! [A2: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( member_o @ C @ A2 )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_930_subsetD,axiom,
! [A2: set_set_a,B: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B ) ) ) ).
% subsetD
thf(fact_931_subsetD,axiom,
! [A2: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_932_equalityE,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ~ ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_933_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_934_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_935_subset__eq,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A3: set_a_real,B3: set_a_real] :
! [X: a > real] :
( ( member_a_real @ X @ A3 )
=> ( member_a_real @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_936_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
! [X: b] :
( ( member_b @ X @ A3 )
=> ( member_b @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_937_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [X: $o] :
( ( member_o @ X @ A3 )
=> ( member_o @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_938_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A3 )
=> ( member_set_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_939_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_940_equalityD1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_941_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_942_equalityD2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( A2 = B )
=> ( ord_le3724670747650509150_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_943_equalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% equalityD2
thf(fact_944_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A3 )
=> ( member_nat @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_945_subset__iff,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A3: set_a_real,B3: set_a_real] :
! [T3: a > real] :
( ( member_a_real @ T3 @ A3 )
=> ( member_a_real @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_946_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B3: set_b] :
! [T3: b] :
( ( member_b @ T3 @ A3 )
=> ( member_b @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_947_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [T3: $o] :
( ( member_o @ T3 @ A3 )
=> ( member_o @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_948_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A3: set_set_a,B3: set_set_a] :
! [T3: set_a] :
( ( member_set_a @ T3 @ A3 )
=> ( member_set_a @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_949_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A3 )
=> ( member_a @ T3 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_950_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_951_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_952_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_953_Collect__mono,axiom,
! [P: $o > $o,Q: $o > $o] :
( ! [X3: $o] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) ) ) ).
% Collect_mono
thf(fact_954_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_955_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_956_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_957_subset__trans,axiom,
! [A2: set_set_a,B: set_set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C3 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_958_subset__trans,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_959_set__eq__subset,axiom,
( ( ^ [Y: set_set_a,Z: set_set_a] : ( Y = Z ) )
= ( ^ [A3: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
& ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_960_set__eq__subset,axiom,
( ( ^ [Y: set_a,Z: set_a] : ( Y = Z ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_961_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_962_Collect__mono__iff,axiom,
! [P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ ( collect_o @ P ) @ ( collect_o @ Q ) )
= ( ! [X: $o] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_963_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X: b] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_964_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_965_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_966_Collect__subset,axiom,
! [A2: set_a_real,P: ( a > real ) > $o] :
( ord_le3334967407727675675a_real
@ ( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_967_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_968_Collect__subset,axiom,
! [A2: set_o,P: $o > $o] :
( ord_less_eq_set_o
@ ( collect_o
@ ^ [X: $o] :
( ( member_o @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_969_Collect__subset,axiom,
! [A2: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_970_Collect__subset,axiom,
! [A2: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_971_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_972_measurable__completion,axiom,
! [F: a > real,M: sigma_measure_a,N2: sigma_measure_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( comple3428971583294703880tion_a @ M ) @ N2 ) ) ) ).
% measurable_completion
thf(fact_973_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_974_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_975_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_976_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_977_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_978_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_979_add__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_980_add__mono,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_981_add__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_982_add__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_983_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_984_add__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_985_add__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_986_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C5: nat] :
( B4
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_987_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_988_add__le__imp__le__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B2 ) )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_989_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_990_add__le__imp__le__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B2 @ C ) )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_991_diff__mono,axiom,
! [A: real,B2: real,D: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ D @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_992_diff__left__mono,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_993_diff__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_994_diff__eq__diff__less__eq,axiom,
! [A: real,B2: real,C: real,D: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C @ D ) )
=> ( ( ord_less_eq_real @ A @ B2 )
= ( ord_less_eq_real @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_995_bot_Oextremum,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% bot.extremum
thf(fact_996_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_997_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_998_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_999_bot_Oextremum,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% bot.extremum
thf(fact_1000_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_1001_bot_Oextremum__unique,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_1002_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_1003_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_1004_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_1005_bot_Oextremum__unique,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_1006_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_1007_bot_Oextremum__uniqueI,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
=> ( A = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_1008_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_1009_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1010_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_1011_bot_Oextremum__uniqueI,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
=> ( A = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1012_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_1013_finite__has__minimal2,axiom,
! [A2: set_a_real,A: a > real] :
( ( finite_finite_a_real @ A2 )
=> ( ( member_a_real @ A @ A2 )
=> ? [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
& ( ord_less_eq_a_real @ X3 @ A )
& ! [Xa: a > real] :
( ( member_a_real @ Xa @ A2 )
=> ( ( ord_less_eq_a_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1014_finite__has__minimal2,axiom,
! [A2: set_o,A: $o] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ A @ A2 )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( ord_less_eq_o @ X3 @ A )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( ord_less_eq_o @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1015_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1016_finite__has__minimal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
& ( ord_le3724670747650509150_set_a @ X3 @ A )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1017_finite__has__minimal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1018_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1019_finite__has__maximal2,axiom,
! [A2: set_a_real,A: a > real] :
( ( finite_finite_a_real @ A2 )
=> ( ( member_a_real @ A @ A2 )
=> ? [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
& ( ord_less_eq_a_real @ A @ X3 )
& ! [Xa: a > real] :
( ( member_a_real @ Xa @ A2 )
=> ( ( ord_less_eq_a_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1020_finite__has__maximal2,axiom,
! [A2: set_o,A: $o] :
( ( finite_finite_o @ A2 )
=> ( ( member_o @ A @ A2 )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ( ord_less_eq_o @ A @ X3 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( ord_less_eq_o @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1021_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1022_finite__has__maximal2,axiom,
! [A2: set_set_set_a,A: set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( member_set_set_a @ A @ A2 )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
& ( ord_le3724670747650509150_set_a @ A @ X3 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1023_finite__has__maximal2,axiom,
! [A2: set_real,A: real] :
( ( finite_finite_real @ A2 )
=> ( ( member_real @ A @ A2 )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ( ord_less_eq_real @ A @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1024_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1025_rev__finite__subset,axiom,
! [B: set_b,A2: set_b] :
( ( finite_finite_b @ B )
=> ( ( ord_less_eq_set_b @ A2 @ B )
=> ( finite_finite_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1026_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_1027_rev__finite__subset,axiom,
! [B: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1028_rev__finite__subset,axiom,
! [B: set_a,A2: set_a] :
( ( finite_finite_a @ B )
=> ( ( ord_less_eq_set_a @ A2 @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1029_infinite__super,axiom,
! [S: set_b,T: set_b] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ T ) ) ) ).
% infinite_super
thf(fact_1030_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_1031_infinite__super,axiom,
! [S: set_set_a,T: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S @ T )
=> ( ~ ( finite_finite_set_a @ S )
=> ~ ( finite_finite_set_a @ T ) ) ) ).
% infinite_super
thf(fact_1032_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_1033_finite__subset,axiom,
! [A2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( finite_finite_b @ B )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_subset
thf(fact_1034_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_1035_finite__subset,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( finite_finite_set_a @ B )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_1036_finite__subset,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( finite_finite_a @ B )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_1037_insert__mono,axiom,
! [C3: set_b,D2: set_b,A: b] :
( ( ord_less_eq_set_b @ C3 @ D2 )
=> ( ord_less_eq_set_b @ ( insert_b @ A @ C3 ) @ ( insert_b @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1038_insert__mono,axiom,
! [C3: set_o,D2: set_o,A: $o] :
( ( ord_less_eq_set_o @ C3 @ D2 )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C3 ) @ ( insert_o @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1039_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_1040_insert__mono,axiom,
! [C3: set_a_real,D2: set_a_real,A: a > real] :
( ( ord_le3334967407727675675a_real @ C3 @ D2 )
=> ( ord_le3334967407727675675a_real @ ( insert_a_real @ A @ C3 ) @ ( insert_a_real @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1041_insert__mono,axiom,
! [C3: set_set_a,D2: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C3 ) @ ( insert_set_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1042_insert__mono,axiom,
! [C3: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1043_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_1044_subset__insert,axiom,
! [X2: a > real,A2: set_a_real,B: set_a_real] :
( ~ ( member_a_real @ X2 @ A2 )
=> ( ( ord_le3334967407727675675a_real @ A2 @ ( insert_a_real @ X2 @ B ) )
= ( ord_le3334967407727675675a_real @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1045_subset__insert,axiom,
! [X2: b,A2: set_b,B: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X2 @ B ) )
= ( ord_less_eq_set_b @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1046_subset__insert,axiom,
! [X2: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X2 @ A2 )
=> ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X2 @ B ) )
= ( ord_less_eq_set_o @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1047_subset__insert,axiom,
! [X2: set_a,A2: set_set_a,B: set_set_a] :
( ~ ( member_set_a @ X2 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X2 @ B ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1048_subset__insert,axiom,
! [X2: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1049_subset__insertI,axiom,
! [B: set_b,A: b] : ( ord_less_eq_set_b @ B @ ( insert_b @ A @ B ) ) ).
% subset_insertI
thf(fact_1050_subset__insertI,axiom,
! [B: set_o,A: $o] : ( ord_less_eq_set_o @ B @ ( insert_o @ A @ B ) ) ).
% subset_insertI
thf(fact_1051_subset__insertI,axiom,
! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_1052_subset__insertI,axiom,
! [B: set_a_real,A: a > real] : ( ord_le3334967407727675675a_real @ B @ ( insert_a_real @ A @ B ) ) ).
% subset_insertI
thf(fact_1053_subset__insertI,axiom,
! [B: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B @ ( insert_set_a @ A @ B ) ) ).
% subset_insertI
thf(fact_1054_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_1055_subset__insertI2,axiom,
! [A2: set_b,B: set_b,B2: b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ord_less_eq_set_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1056_subset__insertI2,axiom,
! [A2: set_o,B: set_o,B2: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1057_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_1058_subset__insertI2,axiom,
! [A2: set_a_real,B: set_a_real,B2: a > real] :
( ( ord_le3334967407727675675a_real @ A2 @ B )
=> ( ord_le3334967407727675675a_real @ A2 @ ( insert_a_real @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1059_subset__insertI2,axiom,
! [A2: set_set_a,B: set_set_a,B2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1060_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_1061_Diff__mono,axiom,
! [A2: set_b,C3: set_b,D2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A2 @ C3 )
=> ( ( ord_less_eq_set_b @ D2 @ B )
=> ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B ) @ ( minus_minus_set_b @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1062_Diff__mono,axiom,
! [A2: set_o,C3: set_o,D2: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A2 @ C3 )
=> ( ( ord_less_eq_set_o @ D2 @ B )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ B ) @ ( minus_minus_set_o @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1063_Diff__mono,axiom,
! [A2: set_a_real,C3: set_a_real,D2: set_a_real,B: set_a_real] :
( ( ord_le3334967407727675675a_real @ A2 @ C3 )
=> ( ( ord_le3334967407727675675a_real @ D2 @ B )
=> ( ord_le3334967407727675675a_real @ ( minus_4124197362600706274a_real @ A2 @ B ) @ ( minus_4124197362600706274a_real @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1064_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_1065_Diff__mono,axiom,
! [A2: set_set_a,C3: set_set_a,D2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ B )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ ( minus_5736297505244876581_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1066_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1067_Diff__subset,axiom,
! [A2: set_b,B: set_b] : ( ord_less_eq_set_b @ ( minus_minus_set_b @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1068_Diff__subset,axiom,
! [A2: set_o,B: set_o] : ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1069_Diff__subset,axiom,
! [A2: set_a_real,B: set_a_real] : ( ord_le3334967407727675675a_real @ ( minus_4124197362600706274a_real @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1070_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_1071_Diff__subset,axiom,
! [A2: set_set_a,B: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1072_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_1073_double__diff,axiom,
! [A2: set_b,B: set_b,C3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B )
=> ( ( ord_less_eq_set_b @ B @ C3 )
=> ( ( minus_minus_set_b @ B @ ( minus_minus_set_b @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1074_double__diff,axiom,
! [A2: set_o,B: set_o,C3: set_o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ( ord_less_eq_set_o @ B @ C3 )
=> ( ( minus_minus_set_o @ B @ ( minus_minus_set_o @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1075_double__diff,axiom,
! [A2: set_a_real,B: set_a_real,C3: set_a_real] :
( ( ord_le3334967407727675675a_real @ A2 @ B )
=> ( ( ord_le3334967407727675675a_real @ B @ C3 )
=> ( ( minus_4124197362600706274a_real @ B @ ( minus_4124197362600706274a_real @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1076_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_1077_double__diff,axiom,
! [A2: set_set_a,B: set_set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C3 )
=> ( ( minus_5736297505244876581_set_a @ B @ ( minus_5736297505244876581_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1078_double__diff,axiom,
! [A2: set_a,B: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C3 )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_1079_sum__mono,axiom,
! [K3: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K3 ) @ ( groups3542108847815614940at_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1080_sum__mono,axiom,
! [K3: set_b,F: b > nat,G: b > nat] :
( ! [I3: b] :
( ( member_b @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups7570001007293516437_b_nat @ F @ K3 ) @ ( groups7570001007293516437_b_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1081_sum__mono,axiom,
! [K3: set_a,F: a > nat,G: a > nat] :
( ! [I3: a] :
( ( member_a @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ K3 ) @ ( groups6334556678337121940_a_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1082_sum__mono,axiom,
! [K3: set_o,F: $o > nat,G: $o > nat] :
( ! [I3: $o] :
( ( member_o @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K3 ) @ ( groups8507830703676809646_o_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1083_sum__mono,axiom,
! [K3: set_nat,F: nat > real,G: nat > real] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K3 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K3 ) @ ( groups6591440286371151544t_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1084_sum__mono,axiom,
! [K3: set_o,F: $o > real,G: $o > real] :
( ! [I3: $o] :
( ( member_o @ I3 @ K3 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ K3 ) @ ( groups8691415230153176458o_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1085_sum__mono,axiom,
! [K3: set_b,F: b > real,G: b > real] :
( ! [I3: b] :
( ( member_b @ I3 @ K3 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups8336678772925405937b_real @ F @ K3 ) @ ( groups8336678772925405937b_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1086_sum__mono,axiom,
! [K3: set_a,F: a > real,G: a > real] :
( ! [I3: a] :
( ( member_a @ I3 @ K3 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups2740460157737275248a_real @ F @ K3 ) @ ( groups2740460157737275248a_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1087_sum__mono,axiom,
! [K3: set_set_a,F: set_a > nat,G: set_a > nat] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups6141743369313575924_a_nat @ F @ K3 ) @ ( groups6141743369313575924_a_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1088_sum__mono,axiom,
! [K3: set_set_a,F: set_a > real,G: set_a > real] :
( ! [I3: set_a] :
( ( member_set_a @ I3 @ K3 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_real @ ( groups9174420418583655632a_real @ F @ K3 ) @ ( groups9174420418583655632a_real @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_1089_diff__le__eq,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A @ B2 ) @ C )
= ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B2 ) ) ) ).
% diff_le_eq
thf(fact_1090_le__diff__eq,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B2 ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A @ B2 ) @ C ) ) ).
% le_diff_eq
thf(fact_1091_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1092_le__add__diff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1093_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1094_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1095_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1096_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1097_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1098_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1099_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1100_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A )
= C )
= ( B2
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1101_finite__has__minimal,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( ord_less_eq_o @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1102_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1103_finite__has__minimal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1104_finite__has__minimal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1105_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1106_finite__has__maximal,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ? [X3: $o] :
( ( member_o @ X3 @ A2 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A2 )
=> ( ( ord_less_eq_o @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1107_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1108_finite__has__maximal,axiom,
! [A2: set_set_set_a] :
( ( finite7209287970140883943_set_a @ A2 )
=> ( ( A2 != bot_bo3380559777022489994_set_a )
=> ? [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
& ! [Xa: set_set_a] :
( ( member_set_set_a @ Xa @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1109_finite__has__maximal,axiom,
! [A2: set_real] :
( ( finite_finite_real @ A2 )
=> ( ( A2 != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A2 )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1110_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1111_sum__mono__inv,axiom,
! [F: $o > nat,I4: set_o,G: $o > nat,I: $o] :
( ( ( groups8507830703676809646_o_nat @ F @ I4 )
= ( groups8507830703676809646_o_nat @ G @ I4 ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_o @ I @ I4 )
=> ( ( finite_finite_o @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1112_sum__mono__inv,axiom,
! [F: b > nat,I4: set_b,G: b > nat,I: b] :
( ( ( groups7570001007293516437_b_nat @ F @ I4 )
= ( groups7570001007293516437_b_nat @ G @ I4 ) )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_b @ I @ I4 )
=> ( ( finite_finite_b @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1113_sum__mono__inv,axiom,
! [F: nat > nat,I4: set_nat,G: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I4 )
= ( groups3542108847815614940at_nat @ G @ I4 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I @ I4 )
=> ( ( finite_finite_nat @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1114_sum__mono__inv,axiom,
! [F: a > nat,I4: set_a,G: a > nat,I: a] :
( ( ( groups6334556678337121940_a_nat @ F @ I4 )
= ( groups6334556678337121940_a_nat @ G @ I4 ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_a @ I @ I4 )
=> ( ( finite_finite_a @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1115_sum__mono__inv,axiom,
! [F: $o > real,I4: set_o,G: $o > real,I: $o] :
( ( ( groups8691415230153176458o_real @ F @ I4 )
= ( groups8691415230153176458o_real @ G @ I4 ) )
=> ( ! [I3: $o] :
( ( member_o @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_o @ I @ I4 )
=> ( ( finite_finite_o @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1116_sum__mono__inv,axiom,
! [F: nat > real,I4: set_nat,G: nat > real,I: nat] :
( ( ( groups6591440286371151544t_real @ F @ I4 )
= ( groups6591440286371151544t_real @ G @ I4 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I @ I4 )
=> ( ( finite_finite_nat @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1117_sum__mono__inv,axiom,
! [F: b > real,I4: set_b,G: b > real,I: b] :
( ( ( groups8336678772925405937b_real @ F @ I4 )
= ( groups8336678772925405937b_real @ G @ I4 ) )
=> ( ! [I3: b] :
( ( member_b @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_b @ I @ I4 )
=> ( ( finite_finite_b @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1118_sum__mono__inv,axiom,
! [F: a > real,I4: set_a,G: a > real,I: a] :
( ( ( groups2740460157737275248a_real @ F @ I4 )
= ( groups2740460157737275248a_real @ G @ I4 ) )
=> ( ! [I3: a] :
( ( member_a @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_a @ I @ I4 )
=> ( ( finite_finite_a @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1119_sum__mono__inv,axiom,
! [F: set_a > nat,I4: set_set_a,G: set_a > nat,I: set_a] :
( ( ( groups6141743369313575924_a_nat @ F @ I4 )
= ( groups6141743369313575924_a_nat @ G @ I4 ) )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ I4 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_set_a @ I @ I4 )
=> ( ( finite_finite_set_a @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1120_sum__mono__inv,axiom,
! [F: set_a > real,I4: set_set_a,G: set_a > real,I: set_a] :
( ( ( groups9174420418583655632a_real @ F @ I4 )
= ( groups9174420418583655632a_real @ G @ I4 ) )
=> ( ! [I3: set_a] :
( ( member_set_a @ I3 @ I4 )
=> ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_set_a @ I @ I4 )
=> ( ( finite_finite_set_a @ I4 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1121_subset__singletonD,axiom,
! [A2: set_a_real,X2: a > real] :
( ( ord_le3334967407727675675a_real @ A2 @ ( insert_a_real @ X2 @ bot_bot_set_a_real ) )
=> ( ( A2 = bot_bot_set_a_real )
| ( A2
= ( insert_a_real @ X2 @ bot_bot_set_a_real ) ) ) ) ).
% subset_singletonD
thf(fact_1122_subset__singletonD,axiom,
! [A2: set_b,X2: b] :
( ( ord_less_eq_set_b @ A2 @ ( insert_b @ X2 @ bot_bot_set_b ) )
=> ( ( A2 = bot_bot_set_b )
| ( A2
= ( insert_b @ X2 @ bot_bot_set_b ) ) ) ) ).
% subset_singletonD
thf(fact_1123_subset__singletonD,axiom,
! [A2: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X2 @ bot_bot_set_o ) )
=> ( ( A2 = bot_bot_set_o )
| ( A2
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_1124_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_1125_subset__singletonD,axiom,
! [A2: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
=> ( ( A2 = bot_bot_set_set_a )
| ( A2
= ( insert_set_a @ X2 @ bot_bot_set_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1126_subset__singletonD,axiom,
! [A2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1127_subset__singleton__iff,axiom,
! [X4: set_a_real,A: a > real] :
( ( ord_le3334967407727675675a_real @ X4 @ ( insert_a_real @ A @ bot_bot_set_a_real ) )
= ( ( X4 = bot_bot_set_a_real )
| ( X4
= ( insert_a_real @ A @ bot_bot_set_a_real ) ) ) ) ).
% subset_singleton_iff
thf(fact_1128_subset__singleton__iff,axiom,
! [X4: set_b,A: b] :
( ( ord_less_eq_set_b @ X4 @ ( insert_b @ A @ bot_bot_set_b ) )
= ( ( X4 = bot_bot_set_b )
| ( X4
= ( insert_b @ A @ bot_bot_set_b ) ) ) ) ).
% subset_singleton_iff
thf(fact_1129_subset__singleton__iff,axiom,
! [X4: set_o,A: $o] :
( ( ord_less_eq_set_o @ X4 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X4 = bot_bot_set_o )
| ( X4
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_1130_subset__singleton__iff,axiom,
! [X4: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
| ( X4
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1131_subset__singleton__iff,axiom,
! [X4: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ X4 @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( ( X4 = bot_bot_set_set_a )
| ( X4
= ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1132_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1133_subset__Diff__insert,axiom,
! [A2: set_b,B: set_b,X2: b,C3: set_b] :
( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B @ ( insert_b @ X2 @ C3 ) ) )
= ( ( ord_less_eq_set_b @ A2 @ ( minus_minus_set_b @ B @ C3 ) )
& ~ ( member_b @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1134_subset__Diff__insert,axiom,
! [A2: set_o,B: set_o,X2: $o,C3: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B @ ( insert_o @ X2 @ C3 ) ) )
= ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B @ C3 ) )
& ~ ( member_o @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1135_subset__Diff__insert,axiom,
! [A2: set_a_real,B: set_a_real,X2: a > real,C3: set_a_real] :
( ( ord_le3334967407727675675a_real @ A2 @ ( minus_4124197362600706274a_real @ B @ ( insert_a_real @ X2 @ C3 ) ) )
= ( ( ord_le3334967407727675675a_real @ A2 @ ( minus_4124197362600706274a_real @ B @ C3 ) )
& ~ ( member_a_real @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1136_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_1137_subset__Diff__insert,axiom,
! [A2: set_set_a,B: set_set_a,X2: set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ ( insert_set_a @ X2 @ C3 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B @ C3 ) )
& ~ ( member_set_a @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1138_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X2: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X2 @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C3 ) )
& ~ ( member_a @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1139_finite__subset__induct,axiom,
! [F2: set_a_real,A2: set_a_real,P: set_a_real > $o] :
( ( finite_finite_a_real @ F2 )
=> ( ( ord_le3334967407727675675a_real @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a_real )
=> ( ! [A5: a > real,F3: set_a_real] :
( ( finite_finite_a_real @ F3 )
=> ( ( member_a_real @ A5 @ A2 )
=> ( ~ ( member_a_real @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a_real @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1140_finite__subset__induct,axiom,
! [F2: set_b,A2: set_b,P: set_b > $o] :
( ( finite_finite_b @ F2 )
=> ( ( ord_less_eq_set_b @ F2 @ A2 )
=> ( ( P @ bot_bot_set_b )
=> ( ! [A5: b,F3: set_b] :
( ( finite_finite_b @ F3 )
=> ( ( member_b @ A5 @ A2 )
=> ( ~ ( member_b @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_b @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1141_finite__subset__induct,axiom,
! [F2: set_o,A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( ord_less_eq_set_o @ F2 @ A2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A5: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ( member_o @ A5 @ A2 )
=> ( ~ ( member_o @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1142_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 )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1143_finite__subset__induct,axiom,
! [F2: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A5 @ A2 )
=> ( ~ ( member_set_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1144_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A2 )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1145_finite__subset__induct_H,axiom,
! [F2: set_o,A2: set_o,P: set_o > $o] :
( ( finite_finite_o @ F2 )
=> ( ( ord_less_eq_set_o @ F2 @ A2 )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A5: $o,F3: set_o] :
( ( finite_finite_o @ F3 )
=> ( ( member_o @ A5 @ A2 )
=> ( ( ord_less_eq_set_o @ F3 @ A2 )
=> ( ~ ( member_o @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_o @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1146_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 )
=> ( ! [A5: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1147_finite__subset__induct_H,axiom,
! [F2: set_set_a,A2: set_set_a,P: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_set_a )
=> ( ! [A5: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A5 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ~ ( member_set_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_set_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1148_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1149_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1150_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1151_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1152_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1153_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1154_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1155_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1156_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1157_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1158_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1159_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_1160_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1161_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1162_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1163_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1164_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1165_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1166_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_1167_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1168_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1169_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1170_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1171_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1172_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1173_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1174_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1175_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1176_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1177_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1178_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1179_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1180_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1181_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_1182_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_1183_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_1184_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1185_tail__events__sets,axiom,
! [A2: nat > set_set_a] :
( ! [I3: nat] : ( ord_le3724670747650509150_set_a @ ( A2 @ I3 ) @ ( sigma_sets_a @ m ) )
=> ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ m @ A2 ) @ ( sigma_sets_a @ m ) ) ) ).
% tail_events_sets
thf(fact_1186_indep__setD__ev1,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev1
thf(fact_1187_indep__setD__ev2,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
=> ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev2
thf(fact_1188_complete__real,axiom,
! [S: set_real] :
( ? [X5: real] : ( member_real @ X5 @ S )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y2: real] :
( ! [X5: real] :
( ( member_real @ X5 @ S )
=> ( ord_less_eq_real @ X5 @ Y2 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_1189_finite__measure__compl,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) @ ( sigma_measure_a2 @ m @ S ) ) ) ) ).
% finite_measure_compl
thf(fact_1190_sigma__algebra__tail__events,axiom,
! [A2: nat > set_set_a] :
( ! [I3: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A2 @ I3 ) )
=> ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( indepe7538416700049374166_a_nat @ m @ A2 ) ) ) ).
% sigma_algebra_tail_events
thf(fact_1191_bounded__measure,axiom,
! [A2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) ) ).
% bounded_measure
thf(fact_1192_finite__measure__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ).
% finite_measure_mono
thf(fact_1193_finite__measure__eq__sum__singleton,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ! [X3: a] :
( ( member_a @ X3 @ S )
=> ( member_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( sigma_sets_a @ m ) ) )
=> ( ( sigma_measure_a2 @ m @ S )
= ( groups2740460157737275248a_real
@ ^ [X: a] : ( sigma_measure_a2 @ m @ ( insert_a @ X @ bot_bot_set_a ) )
@ S ) ) ) ) ).
% finite_measure_eq_sum_singleton
thf(fact_1194_finite__measure__Diff,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).
% finite_measure_Diff
thf(fact_1195_measure__eq__compl,axiom,
! [S2: set_a,T2: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T2 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S2 ) )
= ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ T2 ) ) )
=> ( ( sigma_measure_a2 @ m @ S2 )
= ( sigma_measure_a2 @ m @ T2 ) ) ) ) ) ).
% measure_eq_compl
thf(fact_1196_measure__increasing,axiom,
measur1776380161843274167a_real @ ( sigma_sets_a @ m ) @ ( sigma_measure_a2 @ m ) ).
% measure_increasing
thf(fact_1197_prob__compl,axiom,
! [A2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ A2 ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_a2 @ m @ A2 ) ) ) ) ).
% prob_compl
thf(fact_1198_covar__indep__eq__zero,axiom,
! [F: a > real,G: a > real] :
( ( bochne2139062162225249880a_real @ m @ F )
=> ( ( bochne2139062162225249880a_real @ m @ G )
=> ( ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ F @ borel_5078946678739801102l_real @ G )
=> ( ( probab3938396695707481060a_real @ m @ F @ G )
= zero_zero_real ) ) ) ) ).
% covar_indep_eq_zero
thf(fact_1199_measure__ge__1__iff,axiom,
! [A2: set_a] :
( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ m @ A2 ) )
= ( ( sigma_measure_a2 @ m @ A2 )
= one_one_real ) ) ).
% measure_ge_1_iff
thf(fact_1200_prob__space,axiom,
( ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) )
= one_one_real ) ).
% prob_space
thf(fact_1201_prob__neg,axiom,
! [P: a > $o] :
( ( member_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) )
@ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ~ ( P @ X ) ) ) )
= ( minus_minus_real @ one_one_real
@ ( sigma_measure_a2 @ m
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ ( sigma_space_a @ m ) )
& ( P @ X ) ) ) ) ) ) ) ).
% prob_neg
thf(fact_1202_prob__le__1,axiom,
! [A2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ one_one_real ) ).
% prob_le_1
thf(fact_1203_subprob__measure__le__1,axiom,
! [X4: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ X4 ) @ one_one_real ) ).
% subprob_measure_le_1
thf(fact_1204_kolmogorov__0__1__law,axiom,
! [A2: nat > set_set_a,X4: set_a] :
( ! [I3: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ m ) @ ( A2 @ I3 ) )
=> ( ( indepe6267730027088848354_a_nat @ m @ A2 @ top_top_set_nat )
=> ( ( member_set_a @ X4 @ ( indepe7538416700049374166_a_nat @ m @ A2 ) )
=> ( ( ( sigma_measure_a2 @ m @ X4 )
= zero_zero_real )
| ( ( sigma_measure_a2 @ m @ X4 )
= one_one_real ) ) ) ) ) ).
% kolmogorov_0_1_law
thf(fact_1205_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1206_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1207_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1208_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1209_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1210_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1211_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1212_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1213_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1214_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1215_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_1216_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_1217_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1218_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1219_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1220_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1221_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1222_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1223_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1224_obtain__positive__integrable__function,axiom,
~ ! [F4: a > real] :
( ( member_a_real @ F4 @ ( sigma_9116425665531756122a_real @ m @ borel_5078946678739801102l_real ) )
=> ( ! [X5: a] : ( ord_less_real @ zero_zero_real @ ( F4 @ X5 ) )
=> ( ! [X5: a] : ( ord_less_eq_real @ ( F4 @ X5 ) @ one_one_real )
=> ~ ( bochne2139062162225249880a_real @ m @ F4 ) ) ) ) ).
% obtain_positive_integrable_function
thf(fact_1225_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1226_indep__set__def,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
= ( indepe7780107833195774214ts_a_o @ m @ ( produc6113963288868236716_set_a @ A2 @ B ) @ top_top_set_o ) ) ).
% indep_set_def
thf(fact_1227_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1228_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X: real,Y4: real] :
( ( ord_less_real @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_1229_measure__exclude,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ A2 )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( ( inf_inf_set_a @ A2 @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ m @ B )
= zero_zero_real ) ) ) ) ) ).
% measure_exclude
thf(fact_1230_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1231_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1232_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_1233_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1234_measure__space__inter,axiom,
! [S2: set_a,T2: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T2 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ T2 )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ S2 @ T2 ) )
= ( sigma_measure_a2 @ m @ S2 ) ) ) ) ) ).
% measure_space_inter
thf(fact_1235_indep__setD,axiom,
! [A2: set_set_a,B: set_set_a,A: set_a,B2: set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
=> ( ( member_set_a @ A @ A2 )
=> ( ( member_set_a @ B2 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A @ B2 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B2 ) ) ) ) ) ) ).
% indep_setD
thf(fact_1236_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1237_finite__measure__Diff_H,axiom,
! [A2: set_a,B: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A2 @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ) ) ).
% finite_measure_Diff'
thf(fact_1238_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1239_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1240_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1241_indep__sets2__eq,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) )
& ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
& ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ! [Y4: set_a] :
( ( member_set_a @ Y4 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ X @ Y4 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ X ) @ ( sigma_measure_a2 @ m @ Y4 ) ) ) ) ) ) ) ).
% indep_sets2_eq
thf(fact_1242_indep__setI,axiom,
! [A2: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ m ) )
=> ( ! [A5: set_a,B6: set_a] :
( ( member_set_a @ A5 @ A2 )
=> ( ( member_set_a @ B6 @ B )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A5 @ B6 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A5 ) @ ( sigma_measure_a2 @ m @ B6 ) ) ) ) )
=> ( indepe2041756565122539606_set_a @ m @ A2 @ B ) ) ) ) ).
% indep_setI
thf(fact_1243_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1244_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1245_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1246_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1247_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1248_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1249_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1250_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1251_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1252_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1253_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K4 )
=> ~ ( P @ I5 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1254_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1255_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1256_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1257_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1258_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1259_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1260_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1261_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1262_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1263_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1264_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_eq_nat @ M3 @ N4 )
& ( M3 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1265_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1266_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
| ( M3 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1267_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1268_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1269_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1270_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ( member_nat @ N5 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1271_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M3: nat] :
? [N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y3: real] :
( ( if_real @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X2: real,Y3: real] :
( ( if_real @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( groups8336678772925405937b_real
@ ^ [I2: b] :
( groups8336678772925405937b_real
@ ^ [J2: b] : ( probab3938396695707481060a_real @ m @ ( f @ I2 ) @ ( f @ J2 ) )
@ i )
@ i )
= ( groups8336678772925405937b_real
@ ^ [I2: b] :
( plus_plus_real @ ( probab3938396695707481060a_real @ m @ ( f @ I2 ) @ ( f @ I2 ) )
@ ( groups8336678772925405937b_real
@ ^ [J2: b] : ( probab3938396695707481060a_real @ m @ ( f @ I2 ) @ ( f @ J2 ) )
@ ( minus_minus_set_b @ i @ ( insert_b @ I2 @ bot_bot_set_b ) ) ) )
@ i ) ) ).
%------------------------------------------------------------------------------