TPTP Problem File: SLH0984^1.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_00229_008395__19852400_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1562 ( 334 unt; 280 typ; 0 def)
% Number of atoms : 5013 (1373 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 17621 ( 271 ~; 46 |; 312 &;14082 @)
% ( 0 <=>;2910 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Number of types : 29 ( 28 usr)
% Number of type conns : 2463 (2463 >; 0 *; 0 +; 0 <<)
% Number of symbols : 255 ( 252 usr; 14 con; 0-5 aty)
% Number of variables : 5318 ( 785 ^;4430 !; 103 ?;5318 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:14:33.177
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Real__Oreal,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (252)
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thf(sy_c_Bochner__Integration_Ointegrable_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Bochner__Integration_Ointegrable_001tf__a_001t__Real__Oreal,type,
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thf(sy_c_Bochner__Integration_Olebesgue__integral_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Bochner__Integration_Olebesgue__integral_001tf__a_001t__Real__Oreal,type,
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thf(sy_c_Complete__Measure_Ocompletion_001t__Real__Oreal,type,
comple3506806835435775778n_real: sigma_measure_real > sigma_measure_real ).
thf(sy_c_Complete__Measure_Ocompletion_001tf__a,type,
comple3428971583294703880tion_a: sigma_measure_a > sigma_measure_a ).
thf(sy_c_Countable__Set_Ocountable_001tf__a,type,
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thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
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thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001t__Real__Oreal_001tf__a,type,
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thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__a_001t__Real__Oreal,type,
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thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__a_001tf__a,type,
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thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__b_001t__Real__Oreal,type,
prob_k7188637637266226338b_real: sigma_measure_a > nat > ( b > sigma_measure_real ) > ( b > a > real ) > set_b > $o ).
thf(sy_c_Definitions_Oprob__space_Ok__wise__indep__vars_001tf__a_001tf__b_001tf__a,type,
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thf(sy_c_Extended__Nonnegative__Real_Oennreal,type,
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thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ocard_001tf__a,type,
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thf(sy_c_Finite__Set_Ocard_001tf__b,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mt__Real__Oreal_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
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thf(sy_c_Finite__Set_Ofinite_001tf__b,type,
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thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
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thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
comp_r6279034453215524981l_real: ( real > extend8495563244428889912nnreal ) > ( real > real ) > real > extend8495563244428889912nnreal ).
thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001tf__a,type,
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thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
comp_real_real_a: ( real > real ) > ( a > real ) > a > real ).
thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001tf__a,type,
comp_a_real_a: ( a > real ) > ( a > a ) > a > real ).
thf(sy_c_Giry__Monad_Osubprob__space_001tf__a,type,
giry_subprob_space_a: sigma_measure_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_M_Eo_J,type,
minus_5834690139603824627real_o: ( ( real > extend8495563244428889912nnreal ) > $o ) > ( ( real > extend8495563244428889912nnreal ) > $o ) > ( real > extend8495563244428889912nnreal ) > $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_It__Extended____Nonnegative____Real__Oennreal_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__b_M_Eo_J,type,
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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,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__a_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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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__Real__Oreal_J,type,
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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,
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thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Real__Oreal_J,type,
times_times_set_real: set_real > set_real > set_real ).
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nonnegative____Real__Oennreal,type,
zero_z7100319975126383169nnreal: extend8495563244428889912nnreal ).
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,
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thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Nat__Onat,type,
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measur1776380161843274167a_real: set_set_a > ( set_a > real ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
measur8202069185322079731_set_a: set_set_a > ( set_a > set_set_a ) > $o ).
thf(sy_c_Measure__Space_Oincreasing_001tf__a_001t__Set__Oset_Itf__a_J,type,
measur7842569353079325843_set_a: set_set_a > ( set_a > set_a ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nonnegative____Real__Oennreal,type,
bot_bo841427958541957580nnreal: extend8495563244428889912nnreal ).
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_Itf__a_J,type,
bot_bot_set_a: set_a ).
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_I_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_M_Eo_J,type,
ord_le8425755247534135084real_o: ( ( real > extend8495563244428889912nnreal ) > $o ) > ( ( real > extend8495563244428889912nnreal ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_Itf__a_Mt__Real__Oreal_J_M_Eo_J,type,
ord_less_eq_a_real_o: ( ( a > real ) > $o ) > ( ( a > real ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Extended____Nonnegative____Real__Oennreal_M_Eo_J,type,
ord_le7025323315894483639real_o: ( extend8495563244428889912nnreal > $o ) > ( extend8495563244428889912nnreal > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
ord_le1618294441215897699nnreal: ( real > extend8495563244428889912nnreal ) > ( real > extend8495563244428889912nnreal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
ord_less_eq_set_a_o: ( set_a > $o ) > ( set_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $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_062_Itf__b_M_Eo_J,type,
ord_less_eq_b_o: ( b > $o ) > ( b > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
ord_le3935885782089961368nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_J,type,
ord_le2462468573666744473nnreal: set_re5328672808648366137nnreal > set_re5328672808648366137nnreal > $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_It__Extended____Nonnegative____Real__Oennreal_J,type,
ord_le6787938422905777998nnreal: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > $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__Real__Oreal_J,type,
ord_less_eq_set_real: set_real > set_real > $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_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__Mass__Function_Opmf_Omeasure__pmf_001t__Real__Oreal,type,
probab3584279362189474492f_real: probab5323077797692357992f_real > sigma_measure_real ).
thf(sy_c_Probability__Mass__Function_Opmf_Omeasure__pmf_001tf__a,type,
probab7257411610070727406_pmf_a: probab3364570286911266904_pmf_a > sigma_measure_a ).
thf(sy_c_Probability__Measure_Oprob__space_001t__Extended____Nonnegative____Real__Oennreal,type,
probab6612481188548237749nnreal: sigma_7234349610311085201nnreal > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001t__Real__Oreal,type,
probab535871623910865577e_real: sigma_measure_real > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001tf__a,type,
probab7247484486040049089pace_a: sigma_measure_a > $o ).
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_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
collec9130413544115709400nnreal: ( ( real > extend8495563244428889912nnreal ) > $o ) > set_re5328672808648366137nnreal ).
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_001t__Extended____Nonnegative____Real__Oennreal,type,
collec6648975593938027277nnreal: ( extend8495563244428889912nnreal > $o ) > set_Ex3793607809372303086nnreal ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
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_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
insert152533262698245683nnreal: ( real > extend8495563244428889912nnreal ) > set_re5328672808648366137nnreal > set_re5328672808648366137nnreal ).
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_001t__Extended____Nonnegative____Real__Oennreal,type,
insert7407984058720857448nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ).
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_Sigma__Algebra_Omeasurable_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
sigma_7926153774531450434nnreal: sigma_7234349610311085201nnreal > sigma_7234349610311085201nnreal > set_Ex7514979451064110021nnreal ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
sigma_7049758200512112822l_real: sigma_7234349610311085201nnreal > sigma_measure_real > set_Ex5658717452565810105l_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
sigma_9017504469962657078nnreal: sigma_measure_real > sigma_7234349610311085201nnreal > set_re5328672808648366137nnreal ).
thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Real__Oreal,type,
sigma_5267869275261027754l_real: sigma_measure_real > sigma_measure_real > set_real_real ).
thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001t__Extended____Nonnegative____Real__Oennreal,type,
sigma_214952329563889126nnreal: sigma_measure_a > sigma_7234349610311085201nnreal > set_a_7161065143582548615nnreal ).
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_Omeasurable_001tf__a_001tf__a,type,
sigma_measurable_a_a: sigma_measure_a > sigma_measure_a > set_a_a ).
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_Ospace_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
member8329810500450651686nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > set_Ex7514979451064110021nnreal > $o ).
thf(sy_c_member_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Real__Oreal_J,type,
member2874014351250825754l_real: ( extend8495563244428889912nnreal > real ) > set_Ex5658717452565810105l_real > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
member2919562650594848410nnreal: ( real > extend8495563244428889912nnreal ) > set_re5328672808648366137nnreal > $o ).
thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
member_real_real: ( real > real ) > set_real_real > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
member298456594901751504nnreal: ( a > extend8495563244428889912nnreal ) > set_a_7161065143582548615nnreal > $o ).
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_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $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__Extended____Nonnegative____Real__Oennreal_J,type,
member603777416030116741nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
member_set_real: set_real > set_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 (1274)
thf(fact_0_a,axiom,
! [I: b,J: b] :
( ( member_b @ I @ i )
=> ( ( member_b @ J @ i )
=> ( bochne2139062162225249880a_real @ m
@ ^ [Omega: a] : ( times_times_real @ ( minus_minus_real @ ( f @ I @ Omega ) @ ( bochne378719280626478695a_real @ m @ ( f @ I ) ) ) @ ( minus_minus_real @ ( f @ J @ Omega ) @ ( bochne378719280626478695a_real @ m @ ( f @ J ) ) ) ) ) ) ) ).
% a
thf(fact_1_covariance__def,axiom,
! [F: a > real,G: a > real] :
( ( probab3938396695707481060a_real @ m @ F @ G )
= ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] : ( times_times_real @ ( minus_minus_real @ ( F @ Omega ) @ ( bochne378719280626478695a_real @ m @ F ) ) @ ( minus_minus_real @ ( G @ Omega ) @ ( bochne378719280626478695a_real @ m @ G ) ) ) ) ) ).
% covariance_def
thf(fact_2_assms_I1_J,axiom,
finite_finite_b @ i ).
% assms(1)
thf(fact_3_prob__space__axioms,axiom,
probab7247484486040049089pace_a @ m ).
% prob_space_axioms
thf(fact_4_prob__space_Ocovariance_Ocong,axiom,
probab3938396695707481060a_real = probab3938396695707481060a_real ).
% prob_space.covariance.cong
thf(fact_5_integral__mult__left__zero,axiom,
! [M: sigma_measure_a,F: a > real,C: real] :
( ( bochne378719280626478695a_real @ M
@ ^ [X: a] : ( times_times_real @ ( F @ X ) @ C ) )
= ( times_times_real @ ( bochne378719280626478695a_real @ M @ F ) @ C ) ) ).
% integral_mult_left_zero
thf(fact_6_integral__mult__right__zero,axiom,
! [M: sigma_measure_a,C: real,F: a > real] :
( ( bochne378719280626478695a_real @ M
@ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) )
= ( times_times_real @ C @ ( bochne378719280626478695a_real @ M @ F ) ) ) ).
% integral_mult_right_zero
thf(fact_7_local_Ointegrable__const,axiom,
! [A: real] :
( bochne2139062162225249880a_real @ m
@ ^ [X: a] : A ) ).
% local.integrable_const
thf(fact_8_set__times__intro,axiom,
! [A: set_Ex3793607809372303086nnreal,C2: set_se4580700918925141924nnreal,B: set_Ex3793607809372303086nnreal,D: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ A @ C2 )
=> ( ( member603777416030116741nnreal @ B @ D )
=> ( member603777416030116741nnreal @ ( times_4022348038934646771nnreal @ A @ B ) @ ( times_4034357736632202409nnreal @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_9_set__times__intro,axiom,
! [A: set_nat,C2: set_set_nat,B: set_nat,D: set_set_nat] :
( ( member_set_nat @ A @ C2 )
=> ( ( member_set_nat @ B @ D )
=> ( member_set_nat @ ( times_times_set_nat @ A @ B ) @ ( times_4850922872519784769et_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_10_set__times__intro,axiom,
! [A: set_real,C2: set_set_real,B: set_real,D: set_set_real] :
( ( member_set_real @ A @ C2 )
=> ( ( member_set_real @ B @ D )
=> ( member_set_real @ ( times_times_set_real @ A @ B ) @ ( times_2589694258209383069t_real @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_11_set__times__intro,axiom,
! [A: real,C2: set_real,B: real,D: set_real] :
( ( member_real @ A @ C2 )
=> ( ( member_real @ B @ D )
=> ( member_real @ ( times_times_real @ A @ B ) @ ( times_times_set_real @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_12_set__times__intro,axiom,
! [A: nat,C2: set_nat,B: nat,D: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( member_nat @ B @ D )
=> ( member_nat @ ( times_times_nat @ A @ B ) @ ( times_times_set_nat @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_13_set__times__intro,axiom,
! [A: extend8495563244428889912nnreal,C2: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal,D: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ A @ C2 )
=> ( ( member7908768830364227535nnreal @ B @ D )
=> ( member7908768830364227535nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ ( times_4022348038934646771nnreal @ C2 @ D ) ) ) ) ).
% set_times_intro
thf(fact_14_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_15_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_16_sum__distrib__left,axiom,
! [R: real,F: b > real,A2: set_b] :
( ( times_times_real @ R @ ( groups8336678772925405937b_real @ F @ A2 ) )
= ( groups8336678772925405937b_real
@ ^ [N: b] : ( times_times_real @ R @ ( F @ N ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_17_sum__distrib__left,axiom,
! [R: real,F: a > real,A2: set_a] :
( ( times_times_real @ R @ ( groups2740460157737275248a_real @ F @ A2 ) )
= ( groups2740460157737275248a_real
@ ^ [N: a] : ( times_times_real @ R @ ( F @ N ) )
@ A2 ) ) ).
% sum_distrib_left
thf(fact_18_sum__distrib__right,axiom,
! [F: b > real,A2: set_b,R: real] :
( ( times_times_real @ ( groups8336678772925405937b_real @ F @ A2 ) @ R )
= ( groups8336678772925405937b_real
@ ^ [N: b] : ( times_times_real @ ( F @ N ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_19_sum__distrib__right,axiom,
! [F: a > real,A2: set_a,R: real] :
( ( times_times_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ R )
= ( groups2740460157737275248a_real
@ ^ [N: a] : ( times_times_real @ ( F @ N ) @ R )
@ A2 ) ) ).
% sum_distrib_right
thf(fact_20_vector__space__over__itself_Oscale__sum__left,axiom,
! [F: b > real,A2: set_b,X2: real] :
( ( times_times_real @ ( groups8336678772925405937b_real @ F @ A2 ) @ X2 )
= ( groups8336678772925405937b_real
@ ^ [A3: b] : ( times_times_real @ ( F @ A3 ) @ X2 )
@ A2 ) ) ).
% vector_space_over_itself.scale_sum_left
thf(fact_21_vector__space__over__itself_Oscale__sum__left,axiom,
! [F: a > real,A2: set_a,X2: real] :
( ( times_times_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ X2 )
= ( groups2740460157737275248a_real
@ ^ [A3: a] : ( times_times_real @ ( F @ A3 ) @ X2 )
@ A2 ) ) ).
% vector_space_over_itself.scale_sum_left
thf(fact_22_vector__space__over__itself_Oscale__sum__right,axiom,
! [A: real,F: b > real,A2: set_b] :
( ( times_times_real @ A @ ( groups8336678772925405937b_real @ F @ A2 ) )
= ( groups8336678772925405937b_real
@ ^ [X: b] : ( times_times_real @ A @ ( F @ X ) )
@ A2 ) ) ).
% vector_space_over_itself.scale_sum_right
thf(fact_23_vector__space__over__itself_Oscale__sum__right,axiom,
! [A: real,F: a > real,A2: set_a] :
( ( times_times_real @ A @ ( groups2740460157737275248a_real @ F @ A2 ) )
= ( groups2740460157737275248a_real
@ ^ [X: a] : ( times_times_real @ A @ ( F @ X ) )
@ A2 ) ) ).
% vector_space_over_itself.scale_sum_right
thf(fact_24_sum__product,axiom,
! [F: b > real,A2: set_b,G: b > real,B2: set_b] :
( ( times_times_real @ ( groups8336678772925405937b_real @ F @ A2 ) @ ( groups8336678772925405937b_real @ G @ B2 ) )
= ( groups8336678772925405937b_real
@ ^ [I2: b] :
( groups8336678772925405937b_real
@ ^ [J2: b] : ( times_times_real @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_25_sum__product,axiom,
! [F: b > real,A2: set_b,G: a > real,B2: set_a] :
( ( times_times_real @ ( groups8336678772925405937b_real @ F @ A2 ) @ ( groups2740460157737275248a_real @ G @ B2 ) )
= ( groups8336678772925405937b_real
@ ^ [I2: b] :
( groups2740460157737275248a_real
@ ^ [J2: a] : ( times_times_real @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_26_sum__product,axiom,
! [F: a > real,A2: set_a,G: b > real,B2: set_b] :
( ( times_times_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ ( groups8336678772925405937b_real @ G @ B2 ) )
= ( groups2740460157737275248a_real
@ ^ [I2: a] :
( groups8336678772925405937b_real
@ ^ [J2: b] : ( times_times_real @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_27_sum__product,axiom,
! [F: a > real,A2: set_a,G: a > real,B2: set_a] :
( ( times_times_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ ( groups2740460157737275248a_real @ G @ B2 ) )
= ( groups2740460157737275248a_real
@ ^ [I2: a] :
( groups2740460157737275248a_real
@ ^ [J2: a] : ( times_times_real @ ( F @ I2 ) @ ( G @ J2 ) )
@ B2 )
@ A2 ) ) ).
% sum_product
thf(fact_28_inf__period_I2_J,axiom,
! [P: real > $o,D: real,Q: real > $o] :
( ! [X3: real,K: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D ) ) ) )
=> ( ! [X3: real,K: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D ) ) ) )
=> ! [X4: real,K2: real] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D ) ) )
| ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_29_Bochner__Integration_Ointegrable__diff,axiom,
! [M: sigma_measure_a,F: a > real,G: a > real] :
( ( bochne2139062162225249880a_real @ M @ F )
=> ( ( bochne2139062162225249880a_real @ M @ G )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) ) ) ) ).
% Bochner_Integration.integrable_diff
thf(fact_30_Bochner__Integration_Ointegrable__sum,axiom,
! [I3: set_nat,M: sigma_measure_a,F: nat > a > real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( F @ I2 @ X )
@ I3 ) ) ) ).
% Bochner_Integration.integrable_sum
thf(fact_31_Bochner__Integration_Ointegrable__sum,axiom,
! [I3: set_re5328672808648366137nnreal,M: sigma_measure_a,F: ( real > extend8495563244428889912nnreal ) > a > real] :
( ! [I4: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] :
( groups8532390058693651307l_real
@ ^ [I2: real > extend8495563244428889912nnreal] : ( F @ I2 @ X )
@ I3 ) ) ) ).
% Bochner_Integration.integrable_sum
thf(fact_32_Bochner__Integration_Ointegrable__sum,axiom,
! [I3: set_a_real,M: sigma_measure_a,F: ( a > real ) > a > real] :
( ! [I4: a > real] :
( ( member_a_real @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] :
( groups6125259628802515085l_real
@ ^ [I2: a > real] : ( F @ I2 @ X )
@ I3 ) ) ) ).
% Bochner_Integration.integrable_sum
thf(fact_33_Bochner__Integration_Ointegrable__sum,axiom,
! [I3: set_Ex3793607809372303086nnreal,M: sigma_measure_a,F: extend8495563244428889912nnreal > a > real] :
( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] :
( groups2265062954415509024l_real
@ ^ [I2: extend8495563244428889912nnreal] : ( F @ I2 @ X )
@ I3 ) ) ) ).
% Bochner_Integration.integrable_sum
thf(fact_34_Bochner__Integration_Ointegrable__sum,axiom,
! [I3: set_real,M: sigma_measure_a,F: real > a > real] :
( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] :
( groups8097168146408367636l_real
@ ^ [I2: real] : ( F @ I2 @ X )
@ I3 ) ) ) ).
% Bochner_Integration.integrable_sum
thf(fact_35_Bochner__Integration_Ointegrable__sum,axiom,
! [I3: set_b,M: sigma_measure_a,F: b > a > real] :
( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] :
( groups8336678772925405937b_real
@ ^ [I2: b] : ( F @ I2 @ X )
@ I3 ) ) ) ).
% Bochner_Integration.integrable_sum
thf(fact_36_Bochner__Integration_Ointegrable__sum,axiom,
! [I3: set_a,M: sigma_measure_a,F: a > a > real] :
( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( F @ I2 @ X )
@ I3 ) ) ) ).
% Bochner_Integration.integrable_sum
thf(fact_37_Bochner__Integration_Ointegral__diff,axiom,
! [M: sigma_measure_a,F: a > real,G: a > real] :
( ( bochne2139062162225249880a_real @ M @ F )
=> ( ( bochne2139062162225249880a_real @ M @ G )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) ) )
= ( minus_minus_real @ ( bochne378719280626478695a_real @ M @ F ) @ ( bochne378719280626478695a_real @ M @ G ) ) ) ) ) ).
% Bochner_Integration.integral_diff
thf(fact_38_finite__set__times,axiom,
! [S: set_Ex3793607809372303086nnreal,T: set_Ex3793607809372303086nnreal] :
( ( finite3782138982310603983nnreal @ S )
=> ( ( finite3782138982310603983nnreal @ T )
=> ( finite3782138982310603983nnreal @ ( times_4022348038934646771nnreal @ S @ T ) ) ) ) ).
% finite_set_times
thf(fact_39_finite__set__times,axiom,
! [S: set_nat,T: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( finite_finite_nat @ ( times_times_set_nat @ S @ T ) ) ) ) ).
% finite_set_times
thf(fact_40_finite__set__times,axiom,
! [S: set_real,T: set_real] :
( ( finite_finite_real @ S )
=> ( ( finite_finite_real @ T )
=> ( finite_finite_real @ ( times_times_set_real @ S @ T ) ) ) ) ).
% finite_set_times
thf(fact_41_sum_Oswap__restrict,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_b,G: extend8495563244428889912nnreal > b > real,R2: extend8495563244428889912nnreal > b > $o] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups2265062954415509024l_real
@ ^ [X: extend8495563244428889912nnreal] :
( groups8336678772925405937b_real @ ( G @ X )
@ ( collect_b
@ ^ [Y: b] :
( ( member_b @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [Y: b] :
( groups2265062954415509024l_real
@ ^ [X: extend8495563244428889912nnreal] : ( G @ X @ Y )
@ ( collec6648975593938027277nnreal
@ ^ [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_42_sum_Oswap__restrict,axiom,
! [A2: set_real,B2: set_b,G: real > b > real,R2: real > b > $o] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups8097168146408367636l_real
@ ^ [X: real] :
( groups8336678772925405937b_real @ ( G @ X )
@ ( collect_b
@ ^ [Y: b] :
( ( member_b @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [Y: b] :
( groups8097168146408367636l_real
@ ^ [X: real] : ( G @ X @ Y )
@ ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_43_sum_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_b,G: nat > b > real,R2: nat > b > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups6591440286371151544t_real
@ ^ [X: nat] :
( groups8336678772925405937b_real @ ( G @ X )
@ ( collect_b
@ ^ [Y: b] :
( ( member_b @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [Y: b] :
( groups6591440286371151544t_real
@ ^ [X: nat] : ( G @ X @ Y )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_44_sum_Oswap__restrict,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_a,G: extend8495563244428889912nnreal > a > real,R2: extend8495563244428889912nnreal > a > $o] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups2265062954415509024l_real
@ ^ [X: extend8495563244428889912nnreal] :
( groups2740460157737275248a_real @ ( G @ X )
@ ( collect_a
@ ^ [Y: a] :
( ( member_a @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [Y: a] :
( groups2265062954415509024l_real
@ ^ [X: extend8495563244428889912nnreal] : ( G @ X @ Y )
@ ( collec6648975593938027277nnreal
@ ^ [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_45_sum_Oswap__restrict,axiom,
! [A2: set_real,B2: set_a,G: real > a > real,R2: real > a > $o] :
( ( finite_finite_real @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups8097168146408367636l_real
@ ^ [X: real] :
( groups2740460157737275248a_real @ ( G @ X )
@ ( collect_a
@ ^ [Y: a] :
( ( member_a @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [Y: a] :
( groups8097168146408367636l_real
@ ^ [X: real] : ( G @ X @ Y )
@ ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_46_sum_Oswap__restrict,axiom,
! [A2: set_nat,B2: set_a,G: nat > a > real,R2: nat > a > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( groups6591440286371151544t_real
@ ^ [X: nat] :
( groups2740460157737275248a_real @ ( G @ X )
@ ( collect_a
@ ^ [Y: a] :
( ( member_a @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [Y: a] :
( groups6591440286371151544t_real
@ ^ [X: nat] : ( G @ X @ Y )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_47_sum_Oswap__restrict,axiom,
! [A2: set_b,B2: set_Ex3793607809372303086nnreal,G: b > extend8495563244428889912nnreal > real,R2: b > extend8495563244428889912nnreal > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite3782138982310603983nnreal @ B2 )
=> ( ( groups8336678772925405937b_real
@ ^ [X: b] :
( groups2265062954415509024l_real @ ( G @ X )
@ ( collec6648975593938027277nnreal
@ ^ [Y: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups2265062954415509024l_real
@ ^ [Y: extend8495563244428889912nnreal] :
( groups8336678772925405937b_real
@ ^ [X: b] : ( G @ X @ Y )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_48_sum_Oswap__restrict,axiom,
! [A2: set_b,B2: set_real,G: b > real > real,R2: b > real > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_real @ B2 )
=> ( ( groups8336678772925405937b_real
@ ^ [X: b] :
( groups8097168146408367636l_real @ ( G @ X )
@ ( collect_real
@ ^ [Y: real] :
( ( member_real @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups8097168146408367636l_real
@ ^ [Y: real] :
( groups8336678772925405937b_real
@ ^ [X: b] : ( G @ X @ Y )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_49_sum_Oswap__restrict,axiom,
! [A2: set_b,B2: set_nat,G: b > nat > real,R2: b > nat > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( groups8336678772925405937b_real
@ ^ [X: b] :
( groups6591440286371151544t_real @ ( G @ X )
@ ( collect_nat
@ ^ [Y: nat] :
( ( member_nat @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups6591440286371151544t_real
@ ^ [Y: nat] :
( groups8336678772925405937b_real
@ ^ [X: b] : ( G @ X @ Y )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_50_sum_Oswap__restrict,axiom,
! [A2: set_b,B2: set_b,G: b > b > real,R2: b > b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ( groups8336678772925405937b_real
@ ^ [X: b] :
( groups8336678772925405937b_real @ ( G @ X )
@ ( collect_b
@ ^ [Y: b] :
( ( member_b @ Y @ B2 )
& ( R2 @ X @ Y ) ) ) )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [Y: b] :
( groups8336678772925405937b_real
@ ^ [X: b] : ( G @ X @ Y )
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( R2 @ X @ Y ) ) ) )
@ B2 ) ) ) ) ).
% sum.swap_restrict
thf(fact_51_integral__eq__cases,axiom,
! [M: sigma_measure_a,F: a > real,N2: sigma_measure_a,G: a > real] :
( ( ( bochne2139062162225249880a_real @ M @ F )
= ( bochne2139062162225249880a_real @ N2 @ G ) )
=> ( ( ( bochne2139062162225249880a_real @ M @ F )
=> ( ( bochne2139062162225249880a_real @ N2 @ G )
=> ( ( bochne378719280626478695a_real @ M @ F )
= ( bochne378719280626478695a_real @ N2 @ G ) ) ) )
=> ( ( bochne378719280626478695a_real @ M @ F )
= ( bochne378719280626478695a_real @ N2 @ G ) ) ) ) ).
% integral_eq_cases
thf(fact_52_set__times__elim,axiom,
! [X2: set_Ex3793607809372303086nnreal,A2: set_se4580700918925141924nnreal,B2: set_se4580700918925141924nnreal] :
( ( member603777416030116741nnreal @ X2 @ ( times_4034357736632202409nnreal @ A2 @ B2 ) )
=> ~ ! [A4: set_Ex3793607809372303086nnreal,B3: set_Ex3793607809372303086nnreal] :
( ( X2
= ( times_4022348038934646771nnreal @ A4 @ B3 ) )
=> ( ( member603777416030116741nnreal @ A4 @ A2 )
=> ~ ( member603777416030116741nnreal @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_53_set__times__elim,axiom,
! [X2: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ X2 @ ( times_4850922872519784769et_nat @ A2 @ B2 ) )
=> ~ ! [A4: set_nat,B3: set_nat] :
( ( X2
= ( times_times_set_nat @ A4 @ B3 ) )
=> ( ( member_set_nat @ A4 @ A2 )
=> ~ ( member_set_nat @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_54_set__times__elim,axiom,
! [X2: set_real,A2: set_set_real,B2: set_set_real] :
( ( member_set_real @ X2 @ ( times_2589694258209383069t_real @ A2 @ B2 ) )
=> ~ ! [A4: set_real,B3: set_real] :
( ( X2
= ( times_times_set_real @ A4 @ B3 ) )
=> ( ( member_set_real @ A4 @ A2 )
=> ~ ( member_set_real @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_55_set__times__elim,axiom,
! [X2: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ X2 @ ( times_4022348038934646771nnreal @ A2 @ B2 ) )
=> ~ ! [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( X2
= ( times_1893300245718287421nnreal @ A4 @ B3 ) )
=> ( ( member7908768830364227535nnreal @ A4 @ A2 )
=> ~ ( member7908768830364227535nnreal @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_56_set__times__elim,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ X2 @ ( times_times_set_nat @ A2 @ B2 ) )
=> ~ ! [A4: nat,B3: nat] :
( ( X2
= ( times_times_nat @ A4 @ B3 ) )
=> ( ( member_nat @ A4 @ A2 )
=> ~ ( member_nat @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_57_set__times__elim,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ( member_real @ X2 @ ( times_times_set_real @ A2 @ B2 ) )
=> ~ ! [A4: real,B3: real] :
( ( X2
= ( times_times_real @ A4 @ B3 ) )
=> ( ( member_real @ A4 @ A2 )
=> ~ ( member_real @ B3 @ B2 ) ) ) ) ).
% set_times_elim
thf(fact_58_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ B @ ( times_times_real @ A @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_59_vector__space__over__itself_Oscale__scale,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ A @ ( times_times_real @ B @ X2 ) )
= ( times_times_real @ ( times_times_real @ A @ B ) @ X2 ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_60_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: b > nat,J: nat > b,T2: set_b,H: b > real,G: nat > real] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_b @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( member_nat @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups8336678772925405937b_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_61_sum_Oreindex__bij__witness,axiom,
! [S2: set_Ex3793607809372303086nnreal,I: b > extend8495563244428889912nnreal,J: extend8495563244428889912nnreal > b,T2: set_b,H: b > real,G: extend8495563244428889912nnreal > real] :
( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( member_b @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( member7908768830364227535nnreal @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2265062954415509024l_real @ G @ S2 )
= ( groups8336678772925405937b_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_62_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I: b > real,J: real > b,T2: set_b,H: b > real,G: real > real] :
( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( member_b @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( member_real @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8336678772925405937b_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_63_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: a > nat,J: nat > a,T2: set_a,H: a > real,G: nat > real] :
( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( member_a @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( member_nat @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups2740460157737275248a_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_64_sum_Oreindex__bij__witness,axiom,
! [S2: set_Ex3793607809372303086nnreal,I: a > extend8495563244428889912nnreal,J: extend8495563244428889912nnreal > a,T2: set_a,H: a > real,G: extend8495563244428889912nnreal > real] :
( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( member_a @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( member7908768830364227535nnreal @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2265062954415509024l_real @ G @ S2 )
= ( groups2740460157737275248a_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_65_sum_Oreindex__bij__witness,axiom,
! [S2: set_real,I: a > real,J: real > a,T2: set_a,H: a > real,G: real > real] :
( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( member_a @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: a] :
( ( member_a @ B3 @ T2 )
=> ( member_real @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups2740460157737275248a_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_66_sum_Oreindex__bij__witness,axiom,
! [S2: set_b,I: nat > b,J: b > nat,T2: set_nat,H: nat > real,G: b > real] :
( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( member_nat @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T2 )
=> ( member_b @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_67_sum_Oreindex__bij__witness,axiom,
! [S2: set_b,I: extend8495563244428889912nnreal > b,J: b > extend8495563244428889912nnreal,T2: set_Ex3793607809372303086nnreal,H: extend8495563244428889912nnreal > real,G: b > real] :
( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( member7908768830364227535nnreal @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ T2 )
=> ( member_b @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ S2 )
= ( groups2265062954415509024l_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_68_sum_Oreindex__bij__witness,axiom,
! [S2: set_b,I: real > b,J: b > real,T2: set_real,H: real > real,G: b > real] :
( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( member_real @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T2 )
=> ( member_b @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_69_sum_Oreindex__bij__witness,axiom,
! [S2: set_b,I: b > b,J: b > b,T2: set_b,H: b > real,G: b > real] :
( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( member_b @ ( J @ A4 ) @ T2 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ T2 )
=> ( member_b @ ( I @ B3 ) @ S2 ) )
=> ( ! [A4: b] :
( ( member_b @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ S2 )
= ( groups8336678772925405937b_real @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_70_sum_Oeq__general__inverses,axiom,
! [B2: set_b,K3: b > nat,A2: set_nat,H: nat > b,Gamma: b > real,Phi: nat > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ( ( member_nat @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_71_sum_Oeq__general__inverses,axiom,
! [B2: set_b,K3: b > extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,H: extend8495563244428889912nnreal > b,Gamma: b > real,Phi: extend8495563244428889912nnreal > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ( ( member7908768830364227535nnreal @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2265062954415509024l_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_72_sum_Oeq__general__inverses,axiom,
! [B2: set_b,K3: b > real,A2: set_real,H: real > b,Gamma: b > real,Phi: real > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ( ( member_real @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_73_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K3: a > nat,A2: set_nat,H: nat > a,Gamma: a > real,Phi: nat > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B2 )
=> ( ( member_nat @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_74_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K3: a > extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,H: extend8495563244428889912nnreal > a,Gamma: a > real,Phi: extend8495563244428889912nnreal > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B2 )
=> ( ( member7908768830364227535nnreal @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2265062954415509024l_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_75_sum_Oeq__general__inverses,axiom,
! [B2: set_a,K3: a > real,A2: set_real,H: real > a,Gamma: a > real,Phi: real > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B2 )
=> ( ( member_real @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_76_sum_Oeq__general__inverses,axiom,
! [B2: set_nat,K3: nat > b,A2: set_b,H: b > nat,Gamma: nat > real,Phi: b > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
=> ( ( member_b @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_77_sum_Oeq__general__inverses,axiom,
! [B2: set_Ex3793607809372303086nnreal,K3: extend8495563244428889912nnreal > b,A2: set_b,H: b > extend8495563244428889912nnreal,Gamma: extend8495563244428889912nnreal > real,Phi: b > real] :
( ! [Y2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Y2 @ B2 )
=> ( ( member_b @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member7908768830364227535nnreal @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups2265062954415509024l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_78_sum_Oeq__general__inverses,axiom,
! [B2: set_real,K3: real > b,A2: set_b,H: b > real,Gamma: real > real,Phi: b > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B2 )
=> ( ( member_b @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_79_sum_Oeq__general__inverses,axiom,
! [B2: set_b,K3: b > b,A2: set_b,H: b > b,Gamma: b > real,Phi: b > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ( ( member_b @ ( K3 @ Y2 ) @ A2 )
& ( ( H @ ( K3 @ Y2 ) )
= Y2 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B2 )
& ( ( K3 @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_80_sum_Oeq__general,axiom,
! [B2: set_b,A2: set_nat,H: nat > b,Gamma: b > real,Phi: nat > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_81_sum_Oeq__general,axiom,
! [B2: set_b,A2: set_Ex3793607809372303086nnreal,H: extend8495563244428889912nnreal > b,Gamma: b > real,Phi: extend8495563244428889912nnreal > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ? [X4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: extend8495563244428889912nnreal] :
( ( ( member7908768830364227535nnreal @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2265062954415509024l_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_82_sum_Oeq__general,axiom,
! [B2: set_b,A2: set_real,H: real > b,Gamma: b > real,Phi: real > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_83_sum_Oeq__general,axiom,
! [B2: set_a,A2: set_nat,H: nat > a,Gamma: a > real,Phi: nat > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B2 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups6591440286371151544t_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_84_sum_Oeq__general,axiom,
! [B2: set_a,A2: set_Ex3793607809372303086nnreal,H: extend8495563244428889912nnreal > a,Gamma: a > real,Phi: extend8495563244428889912nnreal > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B2 )
=> ? [X4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: extend8495563244428889912nnreal] :
( ( ( member7908768830364227535nnreal @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups2265062954415509024l_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_85_sum_Oeq__general,axiom,
! [B2: set_a,A2: set_real,H: real > a,Gamma: a > real,Phi: real > real] :
( ! [Y2: a] :
( ( member_a @ Y2 @ B2 )
=> ? [X4: real] :
( ( member_real @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: real] :
( ( ( member_real @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ( member_a @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8097168146408367636l_real @ Phi @ A2 )
= ( groups2740460157737275248a_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_86_sum_Oeq__general,axiom,
! [B2: set_nat,A2: set_b,H: b > nat,Gamma: nat > real,Phi: b > real] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B2 )
=> ? [X4: b] :
( ( member_b @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_nat @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups6591440286371151544t_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_87_sum_Oeq__general,axiom,
! [B2: set_Ex3793607809372303086nnreal,A2: set_b,H: b > extend8495563244428889912nnreal,Gamma: extend8495563244428889912nnreal > real,Phi: b > real] :
( ! [Y2: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Y2 @ B2 )
=> ? [X4: b] :
( ( member_b @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member7908768830364227535nnreal @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups2265062954415509024l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_88_sum_Oeq__general,axiom,
! [B2: set_real,A2: set_b,H: b > real,Gamma: real > real,Phi: b > real] :
( ! [Y2: real] :
( ( member_real @ Y2 @ B2 )
=> ? [X4: b] :
( ( member_b @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_real @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups8097168146408367636l_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_89_sum_Oeq__general,axiom,
! [B2: set_b,A2: set_b,H: b > b,Gamma: b > real,Phi: b > real] :
( ! [Y2: b] :
( ( member_b @ Y2 @ B2 )
=> ? [X4: b] :
( ( member_b @ X4 @ A2 )
& ( ( H @ X4 )
= Y2 )
& ! [Ya: b] :
( ( ( member_b @ Ya @ A2 )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X4 ) ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( member_b @ ( H @ X3 ) @ B2 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups8336678772925405937b_real @ Phi @ A2 )
= ( groups8336678772925405937b_real @ Gamma @ B2 ) ) ) ) ).
% sum.eq_general
thf(fact_90_sum_Ocong,axiom,
! [A2: set_b,B2: set_b,G: b > real,H: b > real] :
( ( A2 = B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ B2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups8336678772925405937b_real @ G @ A2 )
= ( groups8336678772925405937b_real @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_91_sum_Ocong,axiom,
! [A2: set_a,B2: set_a,G: a > real,H: a > real] :
( ( A2 = B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups2740460157737275248a_real @ G @ A2 )
= ( groups2740460157737275248a_real @ H @ B2 ) ) ) ) ).
% sum.cong
thf(fact_92_Bochner__Integration_Ointegral__sum,axiom,
! [I3: set_nat,M: sigma_measure_a,F: nat > a > real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( F @ I2 @ X )
@ I3 ) )
= ( groups6591440286371151544t_real
@ ^ [I2: nat] : ( bochne378719280626478695a_real @ M @ ( F @ I2 ) )
@ I3 ) ) ) ).
% Bochner_Integration.integral_sum
thf(fact_93_Bochner__Integration_Ointegral__sum,axiom,
! [I3: set_re5328672808648366137nnreal,M: sigma_measure_a,F: ( real > extend8495563244428889912nnreal ) > a > real] :
( ! [I4: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] :
( groups8532390058693651307l_real
@ ^ [I2: real > extend8495563244428889912nnreal] : ( F @ I2 @ X )
@ I3 ) )
= ( groups8532390058693651307l_real
@ ^ [I2: real > extend8495563244428889912nnreal] : ( bochne378719280626478695a_real @ M @ ( F @ I2 ) )
@ I3 ) ) ) ).
% Bochner_Integration.integral_sum
thf(fact_94_Bochner__Integration_Ointegral__sum,axiom,
! [I3: set_a_real,M: sigma_measure_a,F: ( a > real ) > a > real] :
( ! [I4: a > real] :
( ( member_a_real @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] :
( groups6125259628802515085l_real
@ ^ [I2: a > real] : ( F @ I2 @ X )
@ I3 ) )
= ( groups6125259628802515085l_real
@ ^ [I2: a > real] : ( bochne378719280626478695a_real @ M @ ( F @ I2 ) )
@ I3 ) ) ) ).
% Bochner_Integration.integral_sum
thf(fact_95_Bochner__Integration_Ointegral__sum,axiom,
! [I3: set_Ex3793607809372303086nnreal,M: sigma_measure_a,F: extend8495563244428889912nnreal > a > real] :
( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] :
( groups2265062954415509024l_real
@ ^ [I2: extend8495563244428889912nnreal] : ( F @ I2 @ X )
@ I3 ) )
= ( groups2265062954415509024l_real
@ ^ [I2: extend8495563244428889912nnreal] : ( bochne378719280626478695a_real @ M @ ( F @ I2 ) )
@ I3 ) ) ) ).
% Bochner_Integration.integral_sum
thf(fact_96_Bochner__Integration_Ointegral__sum,axiom,
! [I3: set_real,M: sigma_measure_a,F: real > a > real] :
( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] :
( groups8097168146408367636l_real
@ ^ [I2: real] : ( F @ I2 @ X )
@ I3 ) )
= ( groups8097168146408367636l_real
@ ^ [I2: real] : ( bochne378719280626478695a_real @ M @ ( F @ I2 ) )
@ I3 ) ) ) ).
% Bochner_Integration.integral_sum
thf(fact_97_Bochner__Integration_Ointegral__sum,axiom,
! [I3: set_b,M: sigma_measure_a,F: b > a > real] :
( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] :
( groups8336678772925405937b_real
@ ^ [I2: b] : ( F @ I2 @ X )
@ I3 ) )
= ( groups8336678772925405937b_real
@ ^ [I2: b] : ( bochne378719280626478695a_real @ M @ ( F @ I2 ) )
@ I3 ) ) ) ).
% Bochner_Integration.integral_sum
thf(fact_98_Bochner__Integration_Ointegral__sum,axiom,
! [I3: set_a,M: sigma_measure_a,F: a > a > real] :
( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ M @ ( F @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( F @ I2 @ X )
@ I3 ) )
= ( groups2740460157737275248a_real
@ ^ [I2: a] : ( bochne378719280626478695a_real @ M @ ( F @ I2 ) )
@ I3 ) ) ) ).
% Bochner_Integration.integral_sum
thf(fact_99_prob__space_Ocovariance__def,axiom,
! [M: sigma_measure_a,F: a > real,G: a > real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( probab3938396695707481060a_real @ M @ F @ G )
= ( bochne378719280626478695a_real @ M
@ ^ [Omega: a] : ( times_times_real @ ( minus_minus_real @ ( F @ Omega ) @ ( bochne378719280626478695a_real @ M @ F ) ) @ ( minus_minus_real @ ( G @ Omega ) @ ( bochne378719280626478695a_real @ M @ G ) ) ) ) ) ) ).
% prob_space.covariance_def
thf(fact_100_sum_Oswap,axiom,
! [G: b > b > real,B2: set_b,A2: set_b] :
( ( groups8336678772925405937b_real
@ ^ [I2: b] : ( groups8336678772925405937b_real @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [J2: b] :
( groups8336678772925405937b_real
@ ^ [I2: b] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_101_sum_Oswap,axiom,
! [G: b > a > real,B2: set_a,A2: set_b] :
( ( groups8336678772925405937b_real
@ ^ [I2: b] : ( groups2740460157737275248a_real @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [J2: a] :
( groups8336678772925405937b_real
@ ^ [I2: b] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_102_sum_Oswap,axiom,
! [G: a > b > real,B2: set_b,A2: set_a] :
( ( groups2740460157737275248a_real
@ ^ [I2: a] : ( groups8336678772925405937b_real @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups8336678772925405937b_real
@ ^ [J2: b] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_103_sum_Oswap,axiom,
! [G: a > a > real,B2: set_a,A2: set_a] :
( ( groups2740460157737275248a_real
@ ^ [I2: a] : ( groups2740460157737275248a_real @ ( G @ I2 ) @ B2 )
@ A2 )
= ( groups2740460157737275248a_real
@ ^ [J2: a] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( G @ I2 @ J2 )
@ A2 )
@ B2 ) ) ).
% sum.swap
thf(fact_104_vector__space__over__itself_Oscale__right__diff__distrib,axiom,
! [A: real,X2: real,Y3: real] :
( ( times_times_real @ A @ ( minus_minus_real @ X2 @ Y3 ) )
= ( minus_minus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ A @ Y3 ) ) ) ).
% vector_space_over_itself.scale_right_diff_distrib
thf(fact_105_vector__space__over__itself_Oscale__left__diff__distrib,axiom,
! [A: real,B: real,X2: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ X2 )
= ( minus_minus_real @ ( times_times_real @ A @ X2 ) @ ( times_times_real @ B @ X2 ) ) ) ).
% vector_space_over_itself.scale_left_diff_distrib
thf(fact_106_inf__period_I1_J,axiom,
! [P: real > $o,D: real,Q: real > $o] :
( ! [X3: real,K: real] :
( ( P @ X3 )
= ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D ) ) ) )
=> ( ! [X3: real,K: real] :
( ( Q @ X3 )
= ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D ) ) ) )
=> ! [X4: real,K2: real] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D ) ) )
& ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_107_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_108_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_109_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_110_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_111_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_112_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_113_prob__space__completion,axiom,
probab7247484486040049089pace_a @ ( comple3428971583294703880tion_a @ m ) ).
% prob_space_completion
thf(fact_114_finite__Diff,axiom,
! [A2: set_b,B2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_115_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_116_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_117_finite__Diff2,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( finite_finite_b @ A2 ) ) ) ).
% finite_Diff2
thf(fact_118_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_119_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_120_right__diff__distrib_H,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_121_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_122_left__diff__distrib_H,axiom,
! [B: real,C: real,A: real] :
( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
= ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_123_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_124_mem__Collect__eq,axiom,
! [A: real > extend8495563244428889912nnreal,P: ( real > extend8495563244428889912nnreal ) > $o] :
( ( member2919562650594848410nnreal @ A @ ( collec9130413544115709400nnreal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_125_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_126_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A: extend8495563244428889912nnreal,P: extend8495563244428889912nnreal > $o] :
( ( member7908768830364227535nnreal @ A @ ( collec6648975593938027277nnreal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A2: set_re5328672808648366137nnreal] :
( ( collec9130413544115709400nnreal
@ ^ [X: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_132_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_133_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X: b] : ( member_b @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A2: set_Ex3793607809372303086nnreal] :
( ( collec6648975593938027277nnreal
@ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X: real] : ( member_real @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_138_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_139_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_140_right__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
= ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_141_left__diff__distrib,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
= ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_142_subprob__space__axioms,axiom,
giry_subprob_space_a @ m ).
% subprob_space_axioms
thf(fact_143_measure__pmf_Ocovariance__def,axiom,
! [M: probab3364570286911266904_pmf_a,F: a > real,G: a > real] :
( ( probab3938396695707481060a_real @ ( probab7257411610070727406_pmf_a @ M ) @ F @ G )
= ( bochne378719280626478695a_real @ ( probab7257411610070727406_pmf_a @ M )
@ ^ [Omega: a] : ( times_times_real @ ( minus_minus_real @ ( F @ Omega ) @ ( bochne378719280626478695a_real @ ( probab7257411610070727406_pmf_a @ M ) @ F ) ) @ ( minus_minus_real @ ( G @ Omega ) @ ( bochne378719280626478695a_real @ ( probab7257411610070727406_pmf_a @ M ) @ G ) ) ) ) ) ).
% measure_pmf.covariance_def
thf(fact_144_Diff__infinite__finite,axiom,
! [T2: set_b,S2: set_b] :
( ( finite_finite_b @ T2 )
=> ( ~ ( finite_finite_b @ S2 )
=> ~ ( finite_finite_b @ ( minus_minus_set_b @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_145_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_146_Diff__infinite__finite,axiom,
! [T2: set_a,S2: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_147_pigeonhole__infinite__rel,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_b,R2: extend8495563244428889912nnreal > b > $o] :
( ~ ( finite3782138982310603983nnreal @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B2 )
& ~ ( finite3782138982310603983nnreal
@ ( collec6648975593938027277nnreal
@ ^ [A3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_148_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B2: set_b,R2: real > b > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B2 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_149_pigeonhole__infinite__rel,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_nat,R2: extend8495563244428889912nnreal > nat > $o] :
( ~ ( finite3782138982310603983nnreal @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite3782138982310603983nnreal
@ ( collec6648975593938027277nnreal
@ ^ [A3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_150_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B2: set_nat,R2: real > nat > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_151_pigeonhole__infinite__rel,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_a,R2: extend8495563244428889912nnreal > a > $o] :
( ~ ( finite3782138982310603983nnreal @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite3782138982310603983nnreal
@ ( collec6648975593938027277nnreal
@ ^ [A3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_152_pigeonhole__infinite__rel,axiom,
! [A2: set_real,B2: set_a,R2: real > a > $o] :
( ~ ( finite_finite_real @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite_finite_real
@ ( collect_real
@ ^ [A3: real] :
( ( member_real @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_153_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_b,R2: b > b > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_154_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_nat,R2: b > nat > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_155_pigeonhole__infinite__rel,axiom,
! [A2: set_b,B2: set_a,R2: b > a > $o] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite_finite_b
@ ( collect_b
@ ^ [A3: b] :
( ( member_b @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_156_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_b,R2: nat > b > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_b @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: b] :
( ( member_b @ Xa @ B2 )
& ( R2 @ X3 @ Xa ) ) )
=> ? [X3: b] :
( ( member_b @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A2 )
& ( R2 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_157_not__finite__existsD,axiom,
! [P: b > $o] :
( ~ ( finite_finite_b @ ( collect_b @ P ) )
=> ? [X_1: b] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_158_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_159_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_160_measure__pmf_Ointegrable__const,axiom,
! [M: probab3364570286911266904_pmf_a,A: real] :
( bochne2139062162225249880a_real @ ( probab7257411610070727406_pmf_a @ M )
@ ^ [X: a] : A ) ).
% measure_pmf.integrable_const
thf(fact_161_measure__pmf_Oprob__space__completion,axiom,
! [M: probab3364570286911266904_pmf_a] : ( probab7247484486040049089pace_a @ ( comple3428971583294703880tion_a @ ( probab7257411610070727406_pmf_a @ M ) ) ) ).
% measure_pmf.prob_space_completion
thf(fact_162_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_163_DiffI,axiom,
! [C: real > extend8495563244428889912nnreal,A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ A2 )
=> ( ~ ( member2919562650594848410nnreal @ C @ B2 )
=> ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_164_DiffI,axiom,
! [C: a > real,A2: set_a_real,B2: set_a_real] :
( ( member_a_real @ C @ A2 )
=> ( ~ ( member_a_real @ C @ B2 )
=> ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_165_DiffI,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( ~ ( member_b @ C @ B2 )
=> ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_166_DiffI,axiom,
! [C: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C @ A2 )
=> ( ~ ( member7908768830364227535nnreal @ C @ B2 )
=> ( member7908768830364227535nnreal @ C @ ( minus_104578273773384135nnreal @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_167_DiffI,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ A2 )
=> ( ~ ( member_real @ C @ B2 )
=> ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_168_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_169_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_170_Diff__iff,axiom,
! [C: real > extend8495563244428889912nnreal,A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A2 @ B2 ) )
= ( ( member2919562650594848410nnreal @ C @ A2 )
& ~ ( member2919562650594848410nnreal @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_171_Diff__iff,axiom,
! [C: a > real,A2: set_a_real,B2: set_a_real] :
( ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B2 ) )
= ( ( member_a_real @ C @ A2 )
& ~ ( member_a_real @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_172_Diff__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
& ~ ( member_b @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_173_Diff__iff,axiom,
! [C: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C @ ( minus_104578273773384135nnreal @ A2 @ B2 ) )
= ( ( member7908768830364227535nnreal @ C @ A2 )
& ~ ( member7908768830364227535nnreal @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_174_Diff__iff,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
= ( ( member_real @ C @ A2 )
& ~ ( member_real @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_175_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_176_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_177_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_178_subprob__space__measure__pmf,axiom,
! [X2: probab3364570286911266904_pmf_a] : ( giry_subprob_space_a @ ( probab7257411610070727406_pmf_a @ X2 ) ) ).
% subprob_space_measure_pmf
thf(fact_179_measure__pmf_Osubprob__space__axioms,axiom,
! [M: probab3364570286911266904_pmf_a] : ( giry_subprob_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) ).
% measure_pmf.subprob_space_axioms
thf(fact_180_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_181_prob__space__measure__pmf,axiom,
! [P2: probab3364570286911266904_pmf_a] : ( probab7247484486040049089pace_a @ ( probab7257411610070727406_pmf_a @ P2 ) ) ).
% prob_space_measure_pmf
thf(fact_182_measure__pmf_Oprob__space__axioms,axiom,
! [M: probab3364570286911266904_pmf_a] : ( probab7247484486040049089pace_a @ ( probab7257411610070727406_pmf_a @ M ) ) ).
% measure_pmf.prob_space_axioms
thf(fact_183_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_184_DiffD2,axiom,
! [C: real > extend8495563244428889912nnreal,A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A2 @ B2 ) )
=> ~ ( member2919562650594848410nnreal @ C @ B2 ) ) ).
% DiffD2
thf(fact_185_DiffD2,axiom,
! [C: a > real,A2: set_a_real,B2: set_a_real] :
( ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B2 ) )
=> ~ ( member_a_real @ C @ B2 ) ) ).
% DiffD2
thf(fact_186_DiffD2,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( member_b @ C @ B2 ) ) ).
% DiffD2
thf(fact_187_DiffD2,axiom,
! [C: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C @ ( minus_104578273773384135nnreal @ A2 @ B2 ) )
=> ~ ( member7908768830364227535nnreal @ C @ B2 ) ) ).
% DiffD2
thf(fact_188_DiffD2,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( member_real @ C @ B2 ) ) ).
% DiffD2
thf(fact_189_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_190_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_191_DiffD1,axiom,
! [C: real > extend8495563244428889912nnreal,A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A2 @ B2 ) )
=> ( member2919562650594848410nnreal @ C @ A2 ) ) ).
% DiffD1
thf(fact_192_DiffD1,axiom,
! [C: a > real,A2: set_a_real,B2: set_a_real] :
( ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B2 ) )
=> ( member_a_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_193_DiffD1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ A2 ) ) ).
% DiffD1
thf(fact_194_DiffD1,axiom,
! [C: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C @ ( minus_104578273773384135nnreal @ A2 @ B2 ) )
=> ( member7908768830364227535nnreal @ C @ A2 ) ) ).
% DiffD1
thf(fact_195_DiffD1,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ( member_real @ C @ A2 ) ) ).
% DiffD1
thf(fact_196_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_197_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_198_DiffE,axiom,
! [C: real > extend8495563244428889912nnreal,A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ C @ ( minus_3708639258518406418nnreal @ A2 @ B2 ) )
=> ~ ( ( member2919562650594848410nnreal @ C @ A2 )
=> ( member2919562650594848410nnreal @ C @ B2 ) ) ) ).
% DiffE
thf(fact_199_DiffE,axiom,
! [C: a > real,A2: set_a_real,B2: set_a_real] :
( ( member_a_real @ C @ ( minus_4124197362600706274a_real @ A2 @ B2 ) )
=> ~ ( ( member_a_real @ C @ A2 )
=> ( member_a_real @ C @ B2 ) ) ) ).
% DiffE
thf(fact_200_DiffE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( minus_minus_set_b @ A2 @ B2 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% DiffE
thf(fact_201_DiffE,axiom,
! [C: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ C @ ( minus_104578273773384135nnreal @ A2 @ B2 ) )
=> ~ ( ( member7908768830364227535nnreal @ C @ A2 )
=> ( member7908768830364227535nnreal @ C @ B2 ) ) ) ).
% DiffE
thf(fact_202_DiffE,axiom,
! [C: real,A2: set_real,B2: set_real] :
( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
=> ~ ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% DiffE
thf(fact_203_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_204_minus__set__def,axiom,
( minus_3708639258518406418nnreal
= ( ^ [A5: set_re5328672808648366137nnreal,B4: set_re5328672808648366137nnreal] :
( collec9130413544115709400nnreal
@ ( minus_5834690139603824627real_o
@ ^ [X: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X @ A5 )
@ ^ [X: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_205_minus__set__def,axiom,
( minus_4124197362600706274a_real
= ( ^ [A5: set_a_real,B4: set_a_real] :
( collect_a_real
@ ( minus_minus_a_real_o
@ ^ [X: a > real] : ( member_a_real @ X @ A5 )
@ ^ [X: a > real] : ( member_a_real @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_206_minus__set__def,axiom,
( minus_minus_set_b
= ( ^ [A5: set_b,B4: set_b] :
( collect_b
@ ( minus_minus_b_o
@ ^ [X: b] : ( member_b @ X @ A5 )
@ ^ [X: b] : ( member_b @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_207_minus__set__def,axiom,
( minus_104578273773384135nnreal
= ( ^ [A5: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
( collec6648975593938027277nnreal
@ ( minus_3872285240437276030real_o
@ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ A5 )
@ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_208_minus__set__def,axiom,
( minus_minus_set_real
= ( ^ [A5: set_real,B4: set_real] :
( collect_real
@ ( minus_minus_real_o
@ ^ [X: real] : ( member_real @ X @ A5 )
@ ^ [X: real] : ( member_real @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_209_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_210_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B4: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_211_set__diff__eq,axiom,
( minus_3708639258518406418nnreal
= ( ^ [A5: set_re5328672808648366137nnreal,B4: set_re5328672808648366137nnreal] :
( collec9130413544115709400nnreal
@ ^ [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ A5 )
& ~ ( member2919562650594848410nnreal @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_212_set__diff__eq,axiom,
( minus_4124197362600706274a_real
= ( ^ [A5: set_a_real,B4: set_a_real] :
( collect_a_real
@ ^ [X: a > real] :
( ( member_a_real @ X @ A5 )
& ~ ( member_a_real @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_213_set__diff__eq,axiom,
( minus_minus_set_b
= ( ^ [A5: set_b,B4: set_b] :
( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A5 )
& ~ ( member_b @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_214_set__diff__eq,axiom,
( minus_104578273773384135nnreal
= ( ^ [A5: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
( collec6648975593938027277nnreal
@ ^ [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A5 )
& ~ ( member7908768830364227535nnreal @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_215_set__diff__eq,axiom,
( minus_minus_set_real
= ( ^ [A5: set_real,B4: set_real] :
( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A5 )
& ~ ( member_real @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_216_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A5 )
& ~ ( member_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_217_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B4: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A5 )
& ~ ( member_a @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_218_Bochner__Integration_Ointegral__mult__right,axiom,
! [C: real,M: sigma_measure_a,F: a > real] :
( ( ( C != zero_zero_real )
=> ( bochne2139062162225249880a_real @ M @ F ) )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) )
= ( times_times_real @ C @ ( bochne378719280626478695a_real @ M @ F ) ) ) ) ).
% Bochner_Integration.integral_mult_right
thf(fact_219_Bochner__Integration_Ointegral__mult__left,axiom,
! [C: real,M: sigma_measure_a,F: a > real] :
( ( ( C != zero_zero_real )
=> ( bochne2139062162225249880a_real @ M @ F ) )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [X: a] : ( times_times_real @ ( F @ X ) @ C ) )
= ( times_times_real @ ( bochne378719280626478695a_real @ M @ F ) @ C ) ) ) ).
% Bochner_Integration.integral_mult_left
thf(fact_220_sum__delta__notmem_I1_J,axiom,
! [X2: b,S: set_b,P: b > real,Q: b > real] :
( ~ ( member_b @ X2 @ S )
=> ( ( groups8336678772925405937b_real
@ ^ [Y: b] : ( if_real @ ( Y = X2 ) @ ( P @ X2 ) @ ( Q @ Y ) )
@ S )
= ( groups8336678772925405937b_real @ Q @ S ) ) ) ).
% sum_delta_notmem(1)
thf(fact_221_sum__delta__notmem_I1_J,axiom,
! [X2: a,S: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [Y: a] : ( if_real @ ( Y = X2 ) @ ( P @ X2 ) @ ( Q @ Y ) )
@ S )
= ( groups2740460157737275248a_real @ Q @ S ) ) ) ).
% sum_delta_notmem(1)
thf(fact_222_sum__delta__notmem_I2_J,axiom,
! [X2: b,S: set_b,P: b > real,Q: b > real] :
( ~ ( member_b @ X2 @ S )
=> ( ( groups8336678772925405937b_real
@ ^ [Y: b] : ( if_real @ ( X2 = Y ) @ ( P @ X2 ) @ ( Q @ Y ) )
@ S )
= ( groups8336678772925405937b_real @ Q @ S ) ) ) ).
% sum_delta_notmem(2)
thf(fact_223_sum__delta__notmem_I2_J,axiom,
! [X2: a,S: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [Y: a] : ( if_real @ ( X2 = Y ) @ ( P @ X2 ) @ ( Q @ Y ) )
@ S )
= ( groups2740460157737275248a_real @ Q @ S ) ) ) ).
% sum_delta_notmem(2)
thf(fact_224_sum__delta__notmem_I3_J,axiom,
! [X2: b,S: set_b,P: b > real,Q: b > real] :
( ~ ( member_b @ X2 @ S )
=> ( ( groups8336678772925405937b_real
@ ^ [Y: b] : ( if_real @ ( Y = X2 ) @ ( P @ Y ) @ ( Q @ Y ) )
@ S )
= ( groups8336678772925405937b_real @ Q @ S ) ) ) ).
% sum_delta_notmem(3)
thf(fact_225_sum__delta__notmem_I3_J,axiom,
! [X2: a,S: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [Y: a] : ( if_real @ ( Y = X2 ) @ ( P @ Y ) @ ( Q @ Y ) )
@ S )
= ( groups2740460157737275248a_real @ Q @ S ) ) ) ).
% sum_delta_notmem(3)
thf(fact_226_sum__delta__notmem_I4_J,axiom,
! [X2: b,S: set_b,P: b > real,Q: b > real] :
( ~ ( member_b @ X2 @ S )
=> ( ( groups8336678772925405937b_real
@ ^ [Y: b] : ( if_real @ ( X2 = Y ) @ ( P @ Y ) @ ( Q @ Y ) )
@ S )
= ( groups8336678772925405937b_real @ Q @ S ) ) ) ).
% sum_delta_notmem(4)
thf(fact_227_sum__delta__notmem_I4_J,axiom,
! [X2: a,S: set_a,P: a > real,Q: a > real] :
( ~ ( member_a @ X2 @ S )
=> ( ( groups2740460157737275248a_real
@ ^ [Y: a] : ( if_real @ ( X2 = Y ) @ ( P @ Y ) @ ( Q @ Y ) )
@ S )
= ( groups2740460157737275248a_real @ Q @ S ) ) ) ).
% sum_delta_notmem(4)
thf(fact_228_finite__set__sum,axiom,
! [A2: set_b,B2: b > set_nat] :
( ( finite_finite_b @ A2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( finite_finite_nat @ ( B2 @ X3 ) ) )
=> ( finite_finite_nat @ ( groups1208830032083771979et_nat @ B2 @ A2 ) ) ) ) ).
% finite_set_sum
thf(fact_229_finite__set__sum,axiom,
! [A2: set_nat,B2: nat > set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( finite_finite_nat @ ( B2 @ X3 ) ) )
=> ( finite_finite_nat @ ( groups2637260376230714770et_nat @ B2 @ A2 ) ) ) ) ).
% finite_set_sum
thf(fact_230_finite__set__sum,axiom,
! [A2: set_a,B2: a > set_nat] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( finite_finite_nat @ ( B2 @ X3 ) ) )
=> ( finite_finite_nat @ ( groups7609827518827096650et_nat @ B2 @ A2 ) ) ) ) ).
% finite_set_sum
thf(fact_231_mult__commute__abs,axiom,
! [C: real] :
( ( ^ [X: real] : ( times_times_real @ X @ C ) )
= ( times_times_real @ C ) ) ).
% mult_commute_abs
thf(fact_232_mult__commute__abs,axiom,
! [C: nat] :
( ( ^ [X: nat] : ( times_times_nat @ X @ C ) )
= ( times_times_nat @ C ) ) ).
% mult_commute_abs
thf(fact_233_mult__commute__abs,axiom,
! [C: extend8495563244428889912nnreal] :
( ( ^ [X: extend8495563244428889912nnreal] : ( times_1893300245718287421nnreal @ X @ C ) )
= ( times_1893300245718287421nnreal @ C ) ) ).
% mult_commute_abs
thf(fact_234_mult__commute__abs,axiom,
! [C: set_Ex3793607809372303086nnreal] :
( ( ^ [X: set_Ex3793607809372303086nnreal] : ( times_4022348038934646771nnreal @ X @ C ) )
= ( times_4022348038934646771nnreal @ C ) ) ).
% mult_commute_abs
thf(fact_235_mult__commute__abs,axiom,
! [C: set_nat] :
( ( ^ [X: set_nat] : ( times_times_set_nat @ X @ C ) )
= ( times_times_set_nat @ C ) ) ).
% mult_commute_abs
thf(fact_236_mult__commute__abs,axiom,
! [C: set_real] :
( ( ^ [X: set_real] : ( times_times_set_real @ X @ C ) )
= ( times_times_set_real @ C ) ) ).
% mult_commute_abs
thf(fact_237_indep__var__lebesgue__integral,axiom,
! [X1: a > real,X22: a > real] :
( ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
=> ( ( bochne2139062162225249880a_real @ m @ X1 )
=> ( ( bochne2139062162225249880a_real @ m @ X22 )
=> ( ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) )
= ( times_times_real @ ( bochne378719280626478695a_real @ m @ X1 ) @ ( bochne378719280626478695a_real @ m @ X22 ) ) ) ) ) ) ).
% indep_var_lebesgue_integral
thf(fact_238_sum__diff,axiom,
! [A2: set_nat,B2: set_nat,F: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).
% sum_diff
thf(fact_239_sum__diff,axiom,
! [A2: set_set_a,B2: set_set_a,F: set_a > real] :
( ( finite_finite_set_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( groups9174420418583655632a_real @ F @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( minus_minus_real @ ( groups9174420418583655632a_real @ F @ A2 ) @ ( groups9174420418583655632a_real @ F @ B2 ) ) ) ) ) ).
% sum_diff
thf(fact_240_sum__diff,axiom,
! [A2: set_b,B2: set_b,F: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( groups8336678772925405937b_real @ F @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( minus_minus_real @ ( groups8336678772925405937b_real @ F @ A2 ) @ ( groups8336678772925405937b_real @ F @ B2 ) ) ) ) ) ).
% sum_diff
thf(fact_241_sum__diff,axiom,
! [A2: set_a,B2: set_a,F: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( groups2740460157737275248a_real @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_real @ ( groups2740460157737275248a_real @ F @ A2 ) @ ( groups2740460157737275248a_real @ F @ B2 ) ) ) ) ) ).
% sum_diff
thf(fact_242_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_243_subsetI,axiom,
! [A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ! [X3: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X3 @ A2 )
=> ( member2919562650594848410nnreal @ X3 @ B2 ) )
=> ( ord_le2462468573666744473nnreal @ A2 @ B2 ) ) ).
% subsetI
thf(fact_244_subsetI,axiom,
! [A2: set_a_real,B2: set_a_real] :
( ! [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
=> ( member_a_real @ X3 @ B2 ) )
=> ( ord_le3334967407727675675a_real @ A2 @ B2 ) ) ).
% subsetI
thf(fact_245_subsetI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( member_b @ X3 @ B2 ) )
=> ( ord_less_eq_set_b @ A2 @ B2 ) ) ).
% subsetI
thf(fact_246_subsetI,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( member7908768830364227535nnreal @ X3 @ B2 ) )
=> ( ord_le6787938422905777998nnreal @ A2 @ B2 ) ) ).
% subsetI
thf(fact_247_subsetI,axiom,
! [A2: set_real,B2: set_real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( member_real @ X3 @ B2 ) )
=> ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% subsetI
thf(fact_248_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_249_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_250_subset__antisym,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_251_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_252_indep__var__integrable,axiom,
! [X1: a > real,X22: a > real] :
( ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
=> ( ( bochne2139062162225249880a_real @ m @ X1 )
=> ( ( bochne2139062162225249880a_real @ m @ X22 )
=> ( bochne2139062162225249880a_real @ m
@ ^ [Omega: a] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) ) ) ) ) ).
% indep_var_integrable
thf(fact_253_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_254_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_255_mult__zero__left,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_left
thf(fact_256_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_257_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_258_mult__zero__right,axiom,
! [A: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ A @ zero_z7100319975126383169nnreal )
= zero_z7100319975126383169nnreal ) ).
% mult_zero_right
thf(fact_259_mult__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_260_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_261_mult__eq__0__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
= ( ( A = zero_z7100319975126383169nnreal )
| ( B = zero_z7100319975126383169nnreal ) ) ) ).
% mult_eq_0_iff
thf(fact_262_mult__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_263_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_264_mult__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_265_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_266_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X2: real] :
( ( ( times_times_real @ A @ X2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_267_vector__space__over__itself_Oscale__zero__left,axiom,
! [X2: real] :
( ( times_times_real @ zero_zero_real @ X2 )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_268_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_269_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X2: real,Y3: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y3 ) )
= ( ( X2 = Y3 )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_270_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X2: real,B: real] :
( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
= ( ( A = B )
| ( X2 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_271_set__times__mono2,axiom,
! [C2: set_Ex3793607809372303086nnreal,D: set_Ex3793607809372303086nnreal,E: set_Ex3793607809372303086nnreal,F2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ C2 @ D )
=> ( ( ord_le6787938422905777998nnreal @ E @ F2 )
=> ( ord_le6787938422905777998nnreal @ ( times_4022348038934646771nnreal @ C2 @ E ) @ ( times_4022348038934646771nnreal @ D @ F2 ) ) ) ) ).
% set_times_mono2
thf(fact_272_set__times__mono2,axiom,
! [C2: set_nat,D: set_nat,E: set_nat,F2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ( ord_less_eq_set_nat @ E @ F2 )
=> ( ord_less_eq_set_nat @ ( times_times_set_nat @ C2 @ E ) @ ( times_times_set_nat @ D @ F2 ) ) ) ) ).
% set_times_mono2
thf(fact_273_set__times__mono2,axiom,
! [C2: set_real,D: set_real,E: set_real,F2: set_real] :
( ( ord_less_eq_set_real @ C2 @ D )
=> ( ( ord_less_eq_set_real @ E @ F2 )
=> ( ord_less_eq_set_real @ ( times_times_set_real @ C2 @ E ) @ ( times_times_set_real @ D @ F2 ) ) ) ) ).
% set_times_mono2
thf(fact_274_sum_Oneutral__const,axiom,
! [A2: set_b] :
( ( groups8336678772925405937b_real
@ ^ [Uu: b] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_275_sum_Oneutral__const,axiom,
! [A2: set_a] :
( ( groups2740460157737275248a_real
@ ^ [Uu: a] : zero_zero_real
@ A2 )
= zero_zero_real ) ).
% sum.neutral_const
thf(fact_276_finite__Collect__subsets,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ( finite_finite_set_b
@ ( collect_set_b
@ ^ [B4: set_b] : ( ord_less_eq_set_b @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_277_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_278_finite__Collect__subsets,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite7209287970140883943_set_a
@ ( collect_set_set_a
@ ^ [B4: set_set_a] : ( ord_le3724670747650509150_set_a @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_279_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B4: set_a] : ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_280_integrable__zero,axiom,
! [M: sigma_measure_a] :
( bochne2139062162225249880a_real @ M
@ ^ [X: a] : zero_zero_real ) ).
% integrable_zero
thf(fact_281_integral__zero,axiom,
! [M: sigma_measure_a] :
( ( bochne378719280626478695a_real @ M
@ ^ [X: a] : zero_zero_real )
= zero_zero_real ) ).
% integral_zero
thf(fact_282_integrable__mult__left__iff,axiom,
! [M: sigma_measure_a,C: real,F: a > real] :
( ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) )
= ( ( C = zero_zero_real )
| ( bochne2139062162225249880a_real @ M @ F ) ) ) ).
% integrable_mult_left_iff
thf(fact_283_integrable__mult__right__iff,axiom,
! [M: sigma_measure_a,F: a > real,C: real] :
( ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : ( times_times_real @ ( F @ X ) @ C ) )
= ( ( C = zero_zero_real )
| ( bochne2139062162225249880a_real @ M @ F ) ) ) ).
% integrable_mult_right_iff
thf(fact_284_sum__eq__0__iff,axiom,
! [F2: set_b,F: b > nat] :
( ( finite_finite_b @ F2 )
=> ( ( ( groups7570001007293516437_b_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X: b] :
( ( member_b @ X @ F2 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_285_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ F2 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_286_sum__eq__0__iff,axiom,
! [F2: set_a,F: a > nat] :
( ( finite_finite_a @ F2 )
=> ( ( ( groups6334556678337121940_a_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X: a] :
( ( member_a @ X @ F2 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_287_sum__eq__0__iff,axiom,
! [F2: set_b,F: b > extend8495563244428889912nnreal] :
( ( finite_finite_b @ F2 )
=> ( ( ( groups9167310395270569469nnreal @ F @ F2 )
= zero_z7100319975126383169nnreal )
= ( ! [X: b] :
( ( member_b @ X @ F2 )
=> ( ( F @ X )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_288_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups4868793261593263428nnreal @ F @ F2 )
= zero_z7100319975126383169nnreal )
= ( ! [X: nat] :
( ( member_nat @ X @ F2 )
=> ( ( F @ X )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_289_sum__eq__0__iff,axiom,
! [F2: set_a,F: a > extend8495563244428889912nnreal] :
( ( finite_finite_a @ F2 )
=> ( ( ( groups3047462441131878908nnreal @ F @ F2 )
= zero_z7100319975126383169nnreal )
= ( ! [X: a] :
( ( member_a @ X @ F2 )
=> ( ( F @ X )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_290_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > real] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_291_sum_Oinfinite,axiom,
! [A2: set_b,G: b > nat] :
( ~ ( finite_finite_b @ A2 )
=> ( ( groups7570001007293516437_b_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_292_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_293_sum_Oinfinite,axiom,
! [A2: set_a,G: a > nat] :
( ~ ( finite_finite_a @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G @ A2 )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_294_sum_Oinfinite,axiom,
! [A2: set_b,G: b > extend8495563244428889912nnreal] :
( ~ ( finite_finite_b @ A2 )
=> ( ( groups9167310395270569469nnreal @ G @ A2 )
= zero_z7100319975126383169nnreal ) ) ).
% sum.infinite
thf(fact_295_sum_Oinfinite,axiom,
! [A2: set_nat,G: nat > extend8495563244428889912nnreal] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( groups4868793261593263428nnreal @ G @ A2 )
= zero_z7100319975126383169nnreal ) ) ).
% sum.infinite
thf(fact_296_sum_Oinfinite,axiom,
! [A2: set_a,G: a > extend8495563244428889912nnreal] :
( ~ ( finite_finite_a @ A2 )
=> ( ( groups3047462441131878908nnreal @ G @ A2 )
= zero_z7100319975126383169nnreal ) ) ).
% sum.infinite
thf(fact_297_sum_Oinfinite,axiom,
! [A2: set_b,G: b > real] :
( ~ ( finite_finite_b @ A2 )
=> ( ( groups8336678772925405937b_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_298_sum_Oinfinite,axiom,
! [A2: set_a,G: a > real] :
( ~ ( finite_finite_a @ A2 )
=> ( ( groups2740460157737275248a_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.infinite
thf(fact_299_sum_Odelta,axiom,
! [S2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ S2 )
=> ( ( ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups2265062954415509024l_real
@ ^ [K4: extend8495563244428889912nnreal] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups2265062954415509024l_real
@ ^ [K4: extend8495563244428889912nnreal] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_300_sum_Odelta,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K4: real] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K4: real] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_301_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K4: nat] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K4: nat] : ( if_real @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta
thf(fact_302_sum_Odelta,axiom,
! [S2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ S2 )
=> ( ( ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups419121665405347140al_nat
@ ^ [K4: extend8495563244428889912nnreal] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups419121665405347140al_nat
@ ^ [K4: extend8495563244428889912nnreal] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_303_sum_Odelta,axiom,
! [S2: set_real,A: real,B: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K4: real] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K4: real] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_304_sum_Odelta,axiom,
! [S2: set_b,A: b,B: b > nat] :
( ( finite_finite_b @ S2 )
=> ( ( ( member_b @ A @ S2 )
=> ( ( groups7570001007293516437_b_nat
@ ^ [K4: b] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_b @ A @ S2 )
=> ( ( groups7570001007293516437_b_nat
@ ^ [K4: b] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_305_sum_Odelta,axiom,
! [S2: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_306_sum_Odelta,axiom,
! [S2: set_a,A: a,B: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K4: a] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K4: a] : ( if_nat @ ( K4 = A ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_307_sum_Odelta,axiom,
! [S2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ S2 )
=> ( ( ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups4193414088831287468nnreal
@ ^ [K4: extend8495563244428889912nnreal] : ( if_Ext9135588136721118450nnreal @ ( K4 = A ) @ ( B @ K4 ) @ zero_z7100319975126383169nnreal )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups4193414088831287468nnreal
@ ^ [K4: extend8495563244428889912nnreal] : ( if_Ext9135588136721118450nnreal @ ( K4 = A ) @ ( B @ K4 ) @ zero_z7100319975126383169nnreal )
@ S2 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% sum.delta
thf(fact_308_sum_Odelta,axiom,
! [S2: set_real,A: real,B: real > extend8495563244428889912nnreal] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups4232809223866053280nnreal
@ ^ [K4: real] : ( if_Ext9135588136721118450nnreal @ ( K4 = A ) @ ( B @ K4 ) @ zero_z7100319975126383169nnreal )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups4232809223866053280nnreal
@ ^ [K4: real] : ( if_Ext9135588136721118450nnreal @ ( K4 = A ) @ ( B @ K4 ) @ zero_z7100319975126383169nnreal )
@ S2 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% sum.delta
thf(fact_309_sum_Odelta_H,axiom,
! [S2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ S2 )
=> ( ( ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups2265062954415509024l_real
@ ^ [K4: extend8495563244428889912nnreal] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups2265062954415509024l_real
@ ^ [K4: extend8495563244428889912nnreal] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_310_sum_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > real] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K4: real] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups8097168146408367636l_real
@ ^ [K4: real] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_311_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > real] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K4: nat] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups6591440286371151544t_real
@ ^ [K4: nat] : ( if_real @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_real )
@ S2 )
= zero_zero_real ) ) ) ) ).
% sum.delta'
thf(fact_312_sum_Odelta_H,axiom,
! [S2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ S2 )
=> ( ( ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups419121665405347140al_nat
@ ^ [K4: extend8495563244428889912nnreal] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups419121665405347140al_nat
@ ^ [K4: extend8495563244428889912nnreal] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_313_sum_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > nat] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K4: real] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups1935376822645274424al_nat
@ ^ [K4: real] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_314_sum_Odelta_H,axiom,
! [S2: set_b,A: b,B: b > nat] :
( ( finite_finite_b @ S2 )
=> ( ( ( member_b @ A @ S2 )
=> ( ( groups7570001007293516437_b_nat
@ ^ [K4: b] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_b @ A @ S2 )
=> ( ( groups7570001007293516437_b_nat
@ ^ [K4: b] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_315_sum_Odelta_H,axiom,
! [S2: set_nat,A: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_nat @ A @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_316_sum_Odelta_H,axiom,
! [S2: set_a,A: a,B: a > nat] :
( ( finite_finite_a @ S2 )
=> ( ( ( member_a @ A @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K4: a] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_a @ A @ S2 )
=> ( ( groups6334556678337121940_a_nat
@ ^ [K4: a] : ( if_nat @ ( A = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_317_sum_Odelta_H,axiom,
! [S2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ S2 )
=> ( ( ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups4193414088831287468nnreal
@ ^ [K4: extend8495563244428889912nnreal] : ( if_Ext9135588136721118450nnreal @ ( A = K4 ) @ ( B @ K4 ) @ zero_z7100319975126383169nnreal )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member7908768830364227535nnreal @ A @ S2 )
=> ( ( groups4193414088831287468nnreal
@ ^ [K4: extend8495563244428889912nnreal] : ( if_Ext9135588136721118450nnreal @ ( A = K4 ) @ ( B @ K4 ) @ zero_z7100319975126383169nnreal )
@ S2 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% sum.delta'
thf(fact_318_sum_Odelta_H,axiom,
! [S2: set_real,A: real,B: real > extend8495563244428889912nnreal] :
( ( finite_finite_real @ S2 )
=> ( ( ( member_real @ A @ S2 )
=> ( ( groups4232809223866053280nnreal
@ ^ [K4: real] : ( if_Ext9135588136721118450nnreal @ ( A = K4 ) @ ( B @ K4 ) @ zero_z7100319975126383169nnreal )
@ S2 )
= ( B @ A ) ) )
& ( ~ ( member_real @ A @ S2 )
=> ( ( groups4232809223866053280nnreal
@ ^ [K4: real] : ( if_Ext9135588136721118450nnreal @ ( A = K4 ) @ ( B @ K4 ) @ zero_z7100319975126383169nnreal )
@ S2 )
= zero_z7100319975126383169nnreal ) ) ) ) ).
% sum.delta'
thf(fact_319_integrable__mult__left,axiom,
! [C: real,M: sigma_measure_a,F: a > real] :
( ( ( C != zero_zero_real )
=> ( bochne2139062162225249880a_real @ M @ F ) )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : ( times_times_real @ ( F @ X ) @ C ) ) ) ).
% integrable_mult_left
thf(fact_320_integrable__mult__right,axiom,
! [C: real,M: sigma_measure_a,F: a > real] :
( ( ( C != zero_zero_real )
=> ( bochne2139062162225249880a_real @ M @ F ) )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : ( times_times_real @ C @ ( F @ X ) ) ) ) ).
% integrable_mult_right
thf(fact_321_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_322_in__mono,axiom,
! [A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal,X2: real > extend8495563244428889912nnreal] :
( ( ord_le2462468573666744473nnreal @ A2 @ B2 )
=> ( ( member2919562650594848410nnreal @ X2 @ A2 )
=> ( member2919562650594848410nnreal @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_323_in__mono,axiom,
! [A2: set_a_real,B2: set_a_real,X2: a > real] :
( ( ord_le3334967407727675675a_real @ A2 @ B2 )
=> ( ( member_a_real @ X2 @ A2 )
=> ( member_a_real @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_324_in__mono,axiom,
! [A2: set_b,B2: set_b,X2: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ X2 @ A2 )
=> ( member_b @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_325_in__mono,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( member7908768830364227535nnreal @ X2 @ A2 )
=> ( member7908768830364227535nnreal @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_326_in__mono,axiom,
! [A2: set_real,B2: set_real,X2: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ X2 @ A2 )
=> ( member_real @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_327_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_328_in__mono,axiom,
! [A2: set_a,B2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_329_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_330_subsetD,axiom,
! [A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal,C: real > extend8495563244428889912nnreal] :
( ( ord_le2462468573666744473nnreal @ A2 @ B2 )
=> ( ( member2919562650594848410nnreal @ C @ A2 )
=> ( member2919562650594848410nnreal @ C @ B2 ) ) ) ).
% subsetD
thf(fact_331_subsetD,axiom,
! [A2: set_a_real,B2: set_a_real,C: a > real] :
( ( ord_le3334967407727675675a_real @ A2 @ B2 )
=> ( ( member_a_real @ C @ A2 )
=> ( member_a_real @ C @ B2 ) ) ) ).
% subsetD
thf(fact_332_subsetD,axiom,
! [A2: set_b,B2: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% subsetD
thf(fact_333_subsetD,axiom,
! [A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ( member7908768830364227535nnreal @ C @ A2 )
=> ( member7908768830364227535nnreal @ C @ B2 ) ) ) ).
% subsetD
thf(fact_334_subsetD,axiom,
! [A2: set_real,B2: set_real,C: real] :
( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ( member_real @ C @ A2 )
=> ( member_real @ C @ B2 ) ) ) ).
% subsetD
thf(fact_335_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_336_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_337_equalityE,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ~ ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_338_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_339_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_340_subset__eq,axiom,
( ord_le2462468573666744473nnreal
= ( ^ [A5: set_re5328672808648366137nnreal,B4: set_re5328672808648366137nnreal] :
! [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ A5 )
=> ( member2919562650594848410nnreal @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_341_subset__eq,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A5: set_a_real,B4: set_a_real] :
! [X: a > real] :
( ( member_a_real @ X @ A5 )
=> ( member_a_real @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_342_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B4: set_b] :
! [X: b] :
( ( member_b @ X @ A5 )
=> ( member_b @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_343_subset__eq,axiom,
( ord_le6787938422905777998nnreal
= ( ^ [A5: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A5 )
=> ( member7908768830364227535nnreal @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_344_subset__eq,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B4: set_real] :
! [X: real] :
( ( member_real @ X @ A5 )
=> ( member_real @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_345_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B4: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A5 )
=> ( member_set_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_346_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_347_equalityD1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_348_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_349_equalityD2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( A2 = B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_350_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_351_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A5 )
=> ( member_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_352_subset__iff,axiom,
( ord_le2462468573666744473nnreal
= ( ^ [A5: set_re5328672808648366137nnreal,B4: set_re5328672808648366137nnreal] :
! [T3: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ T3 @ A5 )
=> ( member2919562650594848410nnreal @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_353_subset__iff,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A5: set_a_real,B4: set_a_real] :
! [T3: a > real] :
( ( member_a_real @ T3 @ A5 )
=> ( member_a_real @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_354_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B4: set_b] :
! [T3: b] :
( ( member_b @ T3 @ A5 )
=> ( member_b @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_355_subset__iff,axiom,
( ord_le6787938422905777998nnreal
= ( ^ [A5: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
! [T3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ T3 @ A5 )
=> ( member7908768830364227535nnreal @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_356_subset__iff,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B4: set_real] :
! [T3: real] :
( ( member_real @ T3 @ A5 )
=> ( member_real @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_357_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B4: set_set_a] :
! [T3: set_a] :
( ( member_set_a @ T3 @ A5 )
=> ( member_set_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_358_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A5 )
=> ( member_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_359_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_360_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_361_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_362_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_363_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_364_subset__trans,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_365_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_366_set__eq__subset,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A5: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_367_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_368_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_369_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_370_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_371_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_372_mult__mono,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_373_mult__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ C @ D2 )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_374_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_375_mult__mono_H,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_376_mult__mono_H,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ C @ D2 )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ A )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_377_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_378_split__mult__pos__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_379_mult__left__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_380_mult__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_381_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_382_mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_383_mult__left__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C @ A ) @ ( times_1893300245718287421nnreal @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_384_mult__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_385_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_386_mult__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_387_mult__right__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ A @ C ) @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_388_mult__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_389_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_390_split__mult__neg__le,axiom,
! [A: real,B: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_391_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_392_mult__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_393_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_394_mult__nonneg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_395_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_396_mult__nonpos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_397_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_398_mult__nonneg__nonpos2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_399_zero__le__mult__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_400_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_401_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_402_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ C )
=> ( ord_le3935885782089961368nnreal @ ( times_1893300245718287421nnreal @ C @ A ) @ ( times_1893300245718287421nnreal @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_403_sum__nonpos,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_404_sum__nonpos,axiom,
! [A2: set_b,F: b > nat] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups7570001007293516437_b_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_405_sum__nonpos,axiom,
! [A2: set_a,F: a > nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_406_sum__nonpos,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > nat] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups419121665405347140al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_407_sum__nonpos,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_408_sum__nonpos,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_409_sum__nonpos,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > real] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups2265062954415509024l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_410_sum__nonpos,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% sum_nonpos
thf(fact_411_sum__nonpos,axiom,
! [A2: set_nat,F: nat > extend8495563244428889912nnreal] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ zero_z7100319975126383169nnreal ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4868793261593263428nnreal @ F @ A2 ) @ zero_z7100319975126383169nnreal ) ) ).
% sum_nonpos
thf(fact_412_sum__nonpos,axiom,
! [A2: set_b,F: b > extend8495563244428889912nnreal] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ zero_z7100319975126383169nnreal ) )
=> ( ord_le3935885782089961368nnreal @ ( groups9167310395270569469nnreal @ F @ A2 ) @ zero_z7100319975126383169nnreal ) ) ).
% sum_nonpos
thf(fact_413_sum__nonneg,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_414_sum__nonneg,axiom,
! [A2: set_b,F: b > nat] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups7570001007293516437_b_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_415_sum__nonneg,axiom,
! [A2: set_a,F: a > nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_416_sum__nonneg,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > nat] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups419121665405347140al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_417_sum__nonneg,axiom,
! [A2: set_real,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_418_sum__nonneg,axiom,
! [A2: set_nat,F: nat > real] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_419_sum__nonneg,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > real] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups2265062954415509024l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_420_sum__nonneg,axiom,
! [A2: set_real,F: real > real] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_421_sum__nonneg,axiom,
! [A2: set_nat,F: nat > extend8495563244428889912nnreal] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X3 ) ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( groups4868793261593263428nnreal @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_422_sum__nonneg,axiom,
! [A2: set_b,F: b > extend8495563244428889912nnreal] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X3 ) ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( groups9167310395270569469nnreal @ F @ A2 ) ) ) ).
% sum_nonneg
thf(fact_423_Collect__subset,axiom,
! [A2: set_re5328672808648366137nnreal,P: ( real > extend8495563244428889912nnreal ) > $o] :
( ord_le2462468573666744473nnreal
@ ( collec9130413544115709400nnreal
@ ^ [X: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_424_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_425_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_426_Collect__subset,axiom,
! [A2: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > $o] :
( ord_le6787938422905777998nnreal
@ ( collec6648975593938027277nnreal
@ ^ [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_427_Collect__subset,axiom,
! [A2: set_real,P: real > $o] :
( ord_less_eq_set_real
@ ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_428_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_429_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_430_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_431_sum__mono2,axiom,
! [B2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ B2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ B2 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups419121665405347140al_nat @ F @ A2 ) @ ( groups419121665405347140al_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_432_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: real > nat] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_433_sum__mono2,axiom,
! [B2: set_b,A2: set_b,F: b > nat] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ B2 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups7570001007293516437_b_nat @ F @ A2 ) @ ( groups7570001007293516437_b_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_434_sum__mono2,axiom,
! [B2: set_nat,A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_435_sum__mono2,axiom,
! [B2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ B2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ B2 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
=> ( ord_less_eq_real @ ( groups2265062954415509024l_real @ F @ A2 ) @ ( groups2265062954415509024l_real @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_436_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: real > real] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_437_sum__mono2,axiom,
! [B2: set_nat,A2: set_nat,F: nat > real] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_438_sum__mono2,axiom,
! [B2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ B2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ B2 )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ B2 @ A2 ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ B3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4193414088831287468nnreal @ F @ A2 ) @ ( groups4193414088831287468nnreal @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_439_sum__mono2,axiom,
! [B2: set_real,A2: set_real,F: real > extend8495563244428889912nnreal] :
( ( finite_finite_real @ B2 )
=> ( ( ord_less_eq_set_real @ A2 @ B2 )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A2 ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ B3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4232809223866053280nnreal @ F @ A2 ) @ ( groups4232809223866053280nnreal @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_440_sum__mono2,axiom,
! [B2: set_b,A2: set_b,F: b > extend8495563244428889912nnreal] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ B2 @ A2 ) )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ B3 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups9167310395270569469nnreal @ F @ A2 ) @ ( groups9167310395270569469nnreal @ F @ B2 ) ) ) ) ) ).
% sum_mono2
thf(fact_441_sum__nonneg__eq__0__iff,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups419121665405347140al_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A2 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_442_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: real > nat] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X: real] :
( ( member_real @ X @ A2 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_443_sum__nonneg__eq__0__iff,axiom,
! [A2: set_b,F: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups7570001007293516437_b_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X: b] :
( ( member_b @ X @ A2 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_444_sum__nonneg__eq__0__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_445_sum__nonneg__eq__0__iff,axiom,
! [A2: set_a,F: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F @ A2 )
= zero_zero_nat )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_446_sum__nonneg__eq__0__iff,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups2265062954415509024l_real @ F @ A2 )
= zero_zero_real )
= ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A2 )
=> ( ( F @ X )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_447_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: real > real] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ A2 )
= zero_zero_real )
= ( ! [X: real] :
( ( member_real @ X @ A2 )
=> ( ( F @ X )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_448_sum__nonneg__eq__0__iff,axiom,
! [A2: set_nat,F: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
=> ( ( ( groups6591440286371151544t_real @ F @ A2 )
= zero_zero_real )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ( F @ X )
= zero_zero_real ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_449_sum__nonneg__eq__0__iff,axiom,
! [A2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X3 ) ) )
=> ( ( ( groups4193414088831287468nnreal @ F @ A2 )
= zero_z7100319975126383169nnreal )
= ( ! [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A2 )
=> ( ( F @ X )
= zero_z7100319975126383169nnreal ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_450_sum__nonneg__eq__0__iff,axiom,
! [A2: set_real,F: real > extend8495563244428889912nnreal] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ X3 ) ) )
=> ( ( ( groups4232809223866053280nnreal @ F @ A2 )
= zero_z7100319975126383169nnreal )
= ( ! [X: real] :
( ( member_real @ X @ A2 )
=> ( ( F @ X )
= zero_z7100319975126383169nnreal ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_451_sum__le__included,axiom,
! [S: set_b,T: set_b,G: b > nat,I: b > b,F: b > nat] :
( ( finite_finite_b @ S )
=> ( ( finite_finite_b @ T )
=> ( ! [X3: b] :
( ( member_b @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ S )
=> ? [Xa: b] :
( ( member_b @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups7570001007293516437_b_nat @ F @ S ) @ ( groups7570001007293516437_b_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_452_sum__le__included,axiom,
! [S: set_b,T: set_nat,G: nat > nat,I: nat > b,F: b > nat] :
( ( finite_finite_b @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups7570001007293516437_b_nat @ F @ S ) @ ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_453_sum__le__included,axiom,
! [S: set_b,T: set_a,G: a > nat,I: a > b,F: b > nat] :
( ( finite_finite_b @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ S )
=> ? [Xa: a] :
( ( member_a @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups7570001007293516437_b_nat @ F @ S ) @ ( groups6334556678337121940_a_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_454_sum__le__included,axiom,
! [S: set_nat,T: set_b,G: b > nat,I: b > nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_b @ T )
=> ( ! [X3: b] :
( ( member_b @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: b] :
( ( member_b @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S ) @ ( groups7570001007293516437_b_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_455_sum__le__included,axiom,
! [S: set_nat,T: set_nat,G: nat > nat,I: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S ) @ ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_456_sum__le__included,axiom,
! [S: set_nat,T: set_a,G: a > nat,I: a > nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: a] :
( ( member_a @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S ) @ ( groups6334556678337121940_a_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_457_sum__le__included,axiom,
! [S: set_a,T: set_b,G: b > nat,I: b > a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_b @ T )
=> ( ! [X3: b] :
( ( member_b @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ S )
=> ? [Xa: b] :
( ( member_b @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ S ) @ ( groups7570001007293516437_b_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_458_sum__le__included,axiom,
! [S: set_a,T: set_nat,G: nat > nat,I: nat > a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ S ) @ ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_459_sum__le__included,axiom,
! [S: set_a,T: set_a,G: a > nat,I: a > a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_a @ T )
=> ( ! [X3: a] :
( ( member_a @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X3 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ S )
=> ? [Xa: a] :
( ( member_a @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ S ) @ ( groups6334556678337121940_a_nat @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_460_sum__le__included,axiom,
! [S: set_nat,T: set_nat,G: nat > real,I: nat > nat,F: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ S ) @ ( groups6591440286371151544t_real @ G @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_461_finite__has__minimal2,axiom,
! [A2: set_re5328672808648366137nnreal,A: real > extend8495563244428889912nnreal] :
( ( finite7684081742213514138nnreal @ A2 )
=> ( ( member2919562650594848410nnreal @ A @ A2 )
=> ? [X3: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X3 @ A2 )
& ( ord_le1618294441215897699nnreal @ X3 @ A )
& ! [Xa: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ Xa @ A2 )
=> ( ( ord_le1618294441215897699nnreal @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_462_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_463_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_464_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_465_finite__has__minimal2,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( member7908768830364227535nnreal @ A @ A2 )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
& ( ord_le3935885782089961368nnreal @ X3 @ A )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_466_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_467_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_468_finite__has__maximal2,axiom,
! [A2: set_re5328672808648366137nnreal,A: real > extend8495563244428889912nnreal] :
( ( finite7684081742213514138nnreal @ A2 )
=> ( ( member2919562650594848410nnreal @ A @ A2 )
=> ? [X3: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X3 @ A2 )
& ( ord_le1618294441215897699nnreal @ A @ X3 )
& ! [Xa: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ Xa @ A2 )
=> ( ( ord_le1618294441215897699nnreal @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_469_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_470_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_471_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_472_finite__has__maximal2,axiom,
! [A2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( member7908768830364227535nnreal @ A @ A2 )
=> ? [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
& ( ord_le3935885782089961368nnreal @ A @ X3 )
& ! [Xa: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ Xa @ A2 )
=> ( ( ord_le3935885782089961368nnreal @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_473_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_474_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_475_rev__finite__subset,axiom,
! [B2: set_b,A2: set_b] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_476_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_477_rev__finite__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_478_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_479_infinite__super,axiom,
! [S2: set_b,T2: set_b] :
( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ~ ( finite_finite_b @ S2 )
=> ~ ( finite_finite_b @ T2 ) ) ) ).
% infinite_super
thf(fact_480_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_481_infinite__super,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_482_infinite__super,axiom,
! [S2: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_483_finite__subset,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( finite_finite_b @ B2 )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_subset
thf(fact_484_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_485_finite__subset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_486_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_487_sum__nonneg__leq__bound,axiom,
! [S: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > nat,B2: nat,I: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ S )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups419121665405347140al_nat @ F @ S )
= B2 )
=> ( ( member7908768830364227535nnreal @ I @ S )
=> ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_488_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: real > nat,B2: nat,I: real] :
( ( finite_finite_real @ S )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S )
= B2 )
=> ( ( member_real @ I @ S )
=> ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_489_sum__nonneg__leq__bound,axiom,
! [S: set_b,F: b > nat,B2: nat,I: b] :
( ( finite_finite_b @ S )
=> ( ! [I4: b] :
( ( member_b @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups7570001007293516437_b_nat @ F @ S )
= B2 )
=> ( ( member_b @ I @ S )
=> ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_490_sum__nonneg__leq__bound,axiom,
! [S: set_nat,F: nat > nat,B2: nat,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S )
= B2 )
=> ( ( member_nat @ I @ S )
=> ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_491_sum__nonneg__leq__bound,axiom,
! [S: set_a,F: a > nat,B2: nat,I: a] :
( ( finite_finite_a @ S )
=> ( ! [I4: a] :
( ( member_a @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F @ S )
= B2 )
=> ( ( member_a @ I @ S )
=> ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_492_sum__nonneg__leq__bound,axiom,
! [S: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > real,B2: real,I: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ S )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups2265062954415509024l_real @ F @ S )
= B2 )
=> ( ( member7908768830364227535nnreal @ I @ S )
=> ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_493_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: real > real,B2: real,I: real] :
( ( finite_finite_real @ S )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S )
= B2 )
=> ( ( member_real @ I @ S )
=> ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_494_sum__nonneg__leq__bound,axiom,
! [S: set_nat,F: nat > real,B2: real,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups6591440286371151544t_real @ F @ S )
= B2 )
=> ( ( member_nat @ I @ S )
=> ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_495_sum__nonneg__leq__bound,axiom,
! [S: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,I: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ S )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ S )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups4193414088831287468nnreal @ F @ S )
= B2 )
=> ( ( member7908768830364227535nnreal @ I @ S )
=> ( ord_le3935885782089961368nnreal @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_496_sum__nonneg__leq__bound,axiom,
! [S: set_real,F: real > extend8495563244428889912nnreal,B2: extend8495563244428889912nnreal,I: real] :
( ( finite_finite_real @ S )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups4232809223866053280nnreal @ F @ S )
= B2 )
=> ( ( member_real @ I @ S )
=> ( ord_le3935885782089961368nnreal @ ( F @ I ) @ B2 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_497_sum__nonneg__0,axiom,
! [S: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > nat,I: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ S )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups419121665405347140al_nat @ F @ S )
= zero_zero_nat )
=> ( ( member7908768830364227535nnreal @ I @ S )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_498_sum__nonneg__0,axiom,
! [S: set_real,F: real > nat,I: real] :
( ( finite_finite_real @ S )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups1935376822645274424al_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_real @ I @ S )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_499_sum__nonneg__0,axiom,
! [S: set_b,F: b > nat,I: b] :
( ( finite_finite_b @ S )
=> ( ! [I4: b] :
( ( member_b @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups7570001007293516437_b_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_b @ I @ S )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_500_sum__nonneg__0,axiom,
! [S: set_nat,F: nat > nat,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_nat @ I @ S )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_501_sum__nonneg__0,axiom,
! [S: set_a,F: a > nat,I: a] :
( ( finite_finite_a @ S )
=> ( ! [I4: a] :
( ( member_a @ I4 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
=> ( ( ( groups6334556678337121940_a_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_a @ I @ S )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_502_sum__nonneg__0,axiom,
! [S: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > real,I: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ S )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups2265062954415509024l_real @ F @ S )
= zero_zero_real )
=> ( ( member7908768830364227535nnreal @ I @ S )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_503_sum__nonneg__0,axiom,
! [S: set_real,F: real > real,I: real] :
( ( finite_finite_real @ S )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups8097168146408367636l_real @ F @ S )
= zero_zero_real )
=> ( ( member_real @ I @ S )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_504_sum__nonneg__0,axiom,
! [S: set_nat,F: nat > real,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ S )
=> ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
=> ( ( ( groups6591440286371151544t_real @ F @ S )
= zero_zero_real )
=> ( ( member_nat @ I @ S )
=> ( ( F @ I )
= zero_zero_real ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_505_sum__nonneg__0,axiom,
! [S: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,I: extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ S )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ S )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups4193414088831287468nnreal @ F @ S )
= zero_z7100319975126383169nnreal )
=> ( ( member7908768830364227535nnreal @ I @ S )
=> ( ( F @ I )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_506_sum__nonneg__0,axiom,
! [S: set_real,F: real > extend8495563244428889912nnreal,I: real] :
( ( finite_finite_real @ S )
=> ( ! [I4: real] :
( ( member_real @ I4 @ S )
=> ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ ( F @ I4 ) ) )
=> ( ( ( groups4232809223866053280nnreal @ F @ S )
= zero_z7100319975126383169nnreal )
=> ( ( member_real @ I @ S )
=> ( ( F @ I )
= zero_z7100319975126383169nnreal ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_507_mult__not__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_508_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_509_mult__not__zero,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
!= zero_z7100319975126383169nnreal )
=> ( ( A != zero_z7100319975126383169nnreal )
& ( B != zero_z7100319975126383169nnreal ) ) ) ).
% mult_not_zero
thf(fact_510_divisors__zero,axiom,
! [A: real,B: real] :
( ( ( times_times_real @ A @ B )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_511_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_512_divisors__zero,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
=> ( ( A = zero_z7100319975126383169nnreal )
| ( B = zero_z7100319975126383169nnreal ) ) ) ).
% divisors_zero
thf(fact_513_no__zero__divisors,axiom,
! [A: real,B: real] :
( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_514_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_515_no__zero__divisors,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( A != zero_z7100319975126383169nnreal )
=> ( ( B != zero_z7100319975126383169nnreal )
=> ( ( times_1893300245718287421nnreal @ A @ B )
!= zero_z7100319975126383169nnreal ) ) ) ).
% no_zero_divisors
thf(fact_516_mult__left__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_517_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_518_mult__right__cancel,axiom,
! [C: real,A: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_519_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_520_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X2: real,Y3: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ A @ Y3 ) )
=> ( X2 = Y3 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_521_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X2: real,A: real,B: real] :
( ( X2 != zero_zero_real )
=> ( ( ( times_times_real @ A @ X2 )
= ( times_times_real @ B @ X2 ) )
=> ( A = B ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_522_Diff__mono,axiom,
! [A2: set_set_a,C2: set_set_a,D: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( ord_le3724670747650509150_set_a @ D @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ ( minus_5736297505244876581_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_523_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_524_Diff__subset,axiom,
! [A2: set_set_a,B2: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_525_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_526_double__diff,axiom,
! [A2: set_set_a,B2: set_set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ C2 )
=> ( ( minus_5736297505244876581_set_a @ B2 @ ( minus_5736297505244876581_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_527_double__diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_528_sum_Oneutral,axiom,
! [A2: set_b,G: b > real] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups8336678772925405937b_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_529_sum_Oneutral,axiom,
! [A2: set_a,G: a > real] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups2740460157737275248a_real @ G @ A2 )
= zero_zero_real ) ) ).
% sum.neutral
thf(fact_530_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > real,A2: set_nat] :
( ( ( groups6591440286371151544t_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_531_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: extend8495563244428889912nnreal > real,A2: set_Ex3793607809372303086nnreal] :
( ( ( groups2265062954415509024l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_532_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > real,A2: set_real] :
( ( ( groups8097168146408367636l_real @ G @ A2 )
!= zero_zero_real )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_real ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_533_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A2: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_534_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: b > nat,A2: set_b] :
( ( ( groups7570001007293516437_b_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: b] :
( ( member_b @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_535_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: a > nat,A2: set_a] :
( ( ( groups6334556678337121940_a_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_536_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: extend8495563244428889912nnreal > nat,A2: set_Ex3793607809372303086nnreal] :
( ( ( groups419121665405347140al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_537_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: real > nat,A2: set_real] :
( ( ( groups1935376822645274424al_nat @ G @ A2 )
!= zero_zero_nat )
=> ~ ! [A4: real] :
( ( member_real @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_538_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > extend8495563244428889912nnreal,A2: set_nat] :
( ( ( groups4868793261593263428nnreal @ G @ A2 )
!= zero_z7100319975126383169nnreal )
=> ~ ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_539_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: b > extend8495563244428889912nnreal,A2: set_b] :
( ( ( groups9167310395270569469nnreal @ G @ A2 )
!= zero_z7100319975126383169nnreal )
=> ~ ! [A4: b] :
( ( member_b @ A4 @ A2 )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_540_sum_Omono__neutral__cong__right,axiom,
! [T2: set_Ex3793607809372303086nnreal,S2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ T2 )
=> ( ( ord_le6787938422905777998nnreal @ S2 @ T2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( minus_104578273773384135nnreal @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups2265062954415509024l_real @ G @ T2 )
= ( groups2265062954415509024l_real @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_541_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ T2 )
= ( groups8097168146408367636l_real @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_542_sum_Omono__neutral__cong__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > real,H: nat > real] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ T2 )
= ( groups6591440286371151544t_real @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_543_sum_Omono__neutral__cong__right,axiom,
! [T2: set_Ex3793607809372303086nnreal,S2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > nat,H: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ T2 )
=> ( ( ord_le6787938422905777998nnreal @ S2 @ T2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( minus_104578273773384135nnreal @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups419121665405347140al_nat @ G @ T2 )
= ( groups419121665405347140al_nat @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_544_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ T2 )
= ( groups1935376822645274424al_nat @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_545_sum_Omono__neutral__cong__right,axiom,
! [T2: set_b,S2: set_b,G: b > nat,H: b > nat] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7570001007293516437_b_nat @ G @ T2 )
= ( groups7570001007293516437_b_nat @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_546_sum_Omono__neutral__cong__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ T2 )
= ( groups3542108847815614940at_nat @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_547_sum_Omono__neutral__cong__right,axiom,
! [T2: set_Ex3793607809372303086nnreal,S2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ T2 )
=> ( ( ord_le6787938422905777998nnreal @ S2 @ T2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( minus_104578273773384135nnreal @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups4193414088831287468nnreal @ G @ T2 )
= ( groups4193414088831287468nnreal @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_548_sum_Omono__neutral__cong__right,axiom,
! [T2: set_real,S2: set_real,G: real > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups4232809223866053280nnreal @ G @ T2 )
= ( groups4232809223866053280nnreal @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_549_sum_Omono__neutral__cong__right,axiom,
! [T2: set_b,S2: set_b,G: b > extend8495563244428889912nnreal,H: b > extend8495563244428889912nnreal] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups9167310395270569469nnreal @ G @ T2 )
= ( groups9167310395270569469nnreal @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_550_sum_Omono__neutral__cong__left,axiom,
! [T2: set_Ex3793607809372303086nnreal,S2: set_Ex3793607809372303086nnreal,H: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ T2 )
=> ( ( ord_le6787938422905777998nnreal @ S2 @ T2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( minus_104578273773384135nnreal @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_zero_real ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups2265062954415509024l_real @ G @ S2 )
= ( groups2265062954415509024l_real @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_551_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > real,G: real > real] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_zero_real ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_552_sum_Omono__neutral__cong__left,axiom,
! [T2: set_nat,S2: set_nat,H: nat > real,G: nat > real] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_zero_real ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_553_sum_Omono__neutral__cong__left,axiom,
! [T2: set_Ex3793607809372303086nnreal,S2: set_Ex3793607809372303086nnreal,H: extend8495563244428889912nnreal > nat,G: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ T2 )
=> ( ( ord_le6787938422905777998nnreal @ S2 @ T2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( minus_104578273773384135nnreal @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups419121665405347140al_nat @ G @ S2 )
= ( groups419121665405347140al_nat @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_554_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > nat,G: real > nat] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat @ G @ S2 )
= ( groups1935376822645274424al_nat @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_555_sum_Omono__neutral__cong__left,axiom,
! [T2: set_b,S2: set_b,H: b > nat,G: b > nat] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7570001007293516437_b_nat @ G @ S2 )
= ( groups7570001007293516437_b_nat @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_556_sum_Omono__neutral__cong__left,axiom,
! [T2: set_nat,S2: set_nat,H: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_zero_nat ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_557_sum_Omono__neutral__cong__left,axiom,
! [T2: set_Ex3793607809372303086nnreal,S2: set_Ex3793607809372303086nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ T2 )
=> ( ( ord_le6787938422905777998nnreal @ S2 @ T2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ ( minus_104578273773384135nnreal @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups4193414088831287468nnreal @ G @ S2 )
= ( groups4193414088831287468nnreal @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_558_sum_Omono__neutral__cong__left,axiom,
! [T2: set_real,S2: set_real,H: real > extend8495563244428889912nnreal,G: real > extend8495563244428889912nnreal] :
( ( finite_finite_real @ T2 )
=> ( ( ord_less_eq_set_real @ S2 @ T2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ ( minus_minus_set_real @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups4232809223866053280nnreal @ G @ S2 )
= ( groups4232809223866053280nnreal @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_559_sum_Omono__neutral__cong__left,axiom,
! [T2: set_b,S2: set_b,H: b > extend8495563244428889912nnreal,G: b > extend8495563244428889912nnreal] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( H @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [X3: b] :
( ( member_b @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups9167310395270569469nnreal @ G @ S2 )
= ( groups9167310395270569469nnreal @ H @ T2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_560_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > real] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ T2 )
= ( groups6591440286371151544t_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_561_sum_Omono__neutral__right,axiom,
! [T2: set_b,S2: set_b,G: b > nat] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups7570001007293516437_b_nat @ G @ T2 )
= ( groups7570001007293516437_b_nat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_562_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ T2 )
= ( groups3542108847815614940at_nat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_563_sum_Omono__neutral__right,axiom,
! [T2: set_b,S2: set_b,G: b > extend8495563244428889912nnreal] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( groups9167310395270569469nnreal @ G @ T2 )
= ( groups9167310395270569469nnreal @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_564_sum_Omono__neutral__right,axiom,
! [T2: set_nat,S2: set_nat,G: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( groups4868793261593263428nnreal @ G @ T2 )
= ( groups4868793261593263428nnreal @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_565_sum_Omono__neutral__right,axiom,
! [T2: set_a,S2: set_a,G: a > nat] :
( ( finite_finite_a @ T2 )
=> ( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups6334556678337121940_a_nat @ G @ T2 )
= ( groups6334556678337121940_a_nat @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_566_sum_Omono__neutral__right,axiom,
! [T2: set_a,S2: set_a,G: a > extend8495563244428889912nnreal] :
( ( finite_finite_a @ T2 )
=> ( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( groups3047462441131878908nnreal @ G @ T2 )
= ( groups3047462441131878908nnreal @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_567_sum_Omono__neutral__right,axiom,
! [T2: set_b,S2: set_b,G: b > real] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups8336678772925405937b_real @ G @ T2 )
= ( groups8336678772925405937b_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_568_sum_Omono__neutral__right,axiom,
! [T2: set_a,S2: set_a,G: a > real] :
( ( finite_finite_a @ T2 )
=> ( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups2740460157737275248a_real @ G @ T2 )
= ( groups2740460157737275248a_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_569_sum_Omono__neutral__right,axiom,
! [T2: set_set_a,S2: set_set_a,G: set_a > real] :
( ( finite_finite_set_a @ T2 )
=> ( ( ord_le3724670747650509150_set_a @ S2 @ T2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( minus_5736297505244876581_set_a @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups9174420418583655632a_real @ G @ T2 )
= ( groups9174420418583655632a_real @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_570_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > real] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups6591440286371151544t_real @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_571_sum_Omono__neutral__left,axiom,
! [T2: set_b,S2: set_b,G: b > nat] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups7570001007293516437_b_nat @ G @ S2 )
= ( groups7570001007293516437_b_nat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_572_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ S2 )
= ( groups3542108847815614940at_nat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_573_sum_Omono__neutral__left,axiom,
! [T2: set_b,S2: set_b,G: b > extend8495563244428889912nnreal] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( groups9167310395270569469nnreal @ G @ S2 )
= ( groups9167310395270569469nnreal @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_574_sum_Omono__neutral__left,axiom,
! [T2: set_nat,S2: set_nat,G: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ T2 )
=> ( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( groups4868793261593263428nnreal @ G @ S2 )
= ( groups4868793261593263428nnreal @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_575_sum_Omono__neutral__left,axiom,
! [T2: set_a,S2: set_a,G: a > nat] :
( ( finite_finite_a @ T2 )
=> ( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_nat ) )
=> ( ( groups6334556678337121940_a_nat @ G @ S2 )
= ( groups6334556678337121940_a_nat @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_576_sum_Omono__neutral__left,axiom,
! [T2: set_a,S2: set_a,G: a > extend8495563244428889912nnreal] :
( ( finite_finite_a @ T2 )
=> ( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( groups3047462441131878908nnreal @ G @ S2 )
= ( groups3047462441131878908nnreal @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_577_sum_Omono__neutral__left,axiom,
! [T2: set_b,S2: set_b,G: b > real] :
( ( finite_finite_b @ T2 )
=> ( ( ord_less_eq_set_b @ S2 @ T2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ ( minus_minus_set_b @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups8336678772925405937b_real @ G @ S2 )
= ( groups8336678772925405937b_real @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_578_sum_Omono__neutral__left,axiom,
! [T2: set_a,S2: set_a,G: a > real] :
( ( finite_finite_a @ T2 )
=> ( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups2740460157737275248a_real @ G @ S2 )
= ( groups2740460157737275248a_real @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_579_sum_Omono__neutral__left,axiom,
! [T2: set_set_a,S2: set_set_a,G: set_a > real] :
( ( finite_finite_set_a @ T2 )
=> ( ( ord_le3724670747650509150_set_a @ S2 @ T2 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ ( minus_5736297505244876581_set_a @ T2 @ S2 ) )
=> ( ( G @ X3 )
= zero_zero_real ) )
=> ( ( groups9174420418583655632a_real @ G @ S2 )
= ( groups9174420418583655632a_real @ G @ T2 ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_580_sum_Osame__carrierI,axiom,
! [C2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C2 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ( ( groups2265062954415509024l_real @ G @ C2 )
= ( groups2265062954415509024l_real @ H @ C2 ) )
=> ( ( groups2265062954415509024l_real @ G @ A2 )
= ( groups2265062954415509024l_real @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_581_sum_Osame__carrierI,axiom,
! [C2: set_real,A2: set_real,B2: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A2 @ C2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ C2 )
= ( groups8097168146408367636l_real @ H @ C2 ) )
=> ( ( groups8097168146408367636l_real @ G @ A2 )
= ( groups8097168146408367636l_real @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_582_sum_Osame__carrierI,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > real,H: nat > real] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ( ( groups6591440286371151544t_real @ G @ C2 )
= ( groups6591440286371151544t_real @ H @ C2 ) )
=> ( ( groups6591440286371151544t_real @ G @ A2 )
= ( groups6591440286371151544t_real @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_583_sum_Osame__carrierI,axiom,
! [C2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > nat,H: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C2 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_nat ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups419121665405347140al_nat @ G @ C2 )
= ( groups419121665405347140al_nat @ H @ C2 ) )
=> ( ( groups419121665405347140al_nat @ G @ A2 )
= ( groups419121665405347140al_nat @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_584_sum_Osame__carrierI,axiom,
! [C2: set_real,A2: set_real,B2: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A2 @ C2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ C2 )
= ( groups1935376822645274424al_nat @ H @ C2 ) )
=> ( ( groups1935376822645274424al_nat @ G @ A2 )
= ( groups1935376822645274424al_nat @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_585_sum_Osame__carrierI,axiom,
! [C2: set_b,A2: set_b,B2: set_b,G: b > nat,H: b > nat] :
( ( finite_finite_b @ C2 )
=> ( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ! [A4: b] :
( ( member_b @ A4 @ ( minus_minus_set_b @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_nat ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups7570001007293516437_b_nat @ G @ C2 )
= ( groups7570001007293516437_b_nat @ H @ C2 ) )
=> ( ( groups7570001007293516437_b_nat @ G @ A2 )
= ( groups7570001007293516437_b_nat @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_586_sum_Osame__carrierI,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G @ C2 )
= ( groups3542108847815614940at_nat @ H @ C2 ) )
=> ( ( groups3542108847815614940at_nat @ G @ A2 )
= ( groups3542108847815614940at_nat @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_587_sum_Osame__carrierI,axiom,
! [C2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C2 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( ( groups4193414088831287468nnreal @ G @ C2 )
= ( groups4193414088831287468nnreal @ H @ C2 ) )
=> ( ( groups4193414088831287468nnreal @ G @ A2 )
= ( groups4193414088831287468nnreal @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_588_sum_Osame__carrierI,axiom,
! [C2: set_real,A2: set_real,B2: set_real,G: real > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A2 @ C2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( ( groups4232809223866053280nnreal @ G @ C2 )
= ( groups4232809223866053280nnreal @ H @ C2 ) )
=> ( ( groups4232809223866053280nnreal @ G @ A2 )
= ( groups4232809223866053280nnreal @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_589_sum_Osame__carrierI,axiom,
! [C2: set_b,A2: set_b,B2: set_b,G: b > extend8495563244428889912nnreal,H: b > extend8495563244428889912nnreal] :
( ( finite_finite_b @ C2 )
=> ( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ! [A4: b] :
( ( member_b @ A4 @ ( minus_minus_set_b @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( ( groups9167310395270569469nnreal @ G @ C2 )
= ( groups9167310395270569469nnreal @ H @ C2 ) )
=> ( ( groups9167310395270569469nnreal @ G @ A2 )
= ( groups9167310395270569469nnreal @ H @ B2 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_590_sum_Osame__carrier,axiom,
! [C2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C2 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ( ( groups2265062954415509024l_real @ G @ A2 )
= ( groups2265062954415509024l_real @ H @ B2 ) )
= ( ( groups2265062954415509024l_real @ G @ C2 )
= ( groups2265062954415509024l_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_591_sum_Osame__carrier,axiom,
! [C2: set_real,A2: set_real,B2: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A2 @ C2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ( ( groups8097168146408367636l_real @ G @ A2 )
= ( groups8097168146408367636l_real @ H @ B2 ) )
= ( ( groups8097168146408367636l_real @ G @ C2 )
= ( groups8097168146408367636l_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_592_sum_Osame__carrier,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > real,H: nat > real] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ( ( groups6591440286371151544t_real @ G @ A2 )
= ( groups6591440286371151544t_real @ H @ B2 ) )
= ( ( groups6591440286371151544t_real @ G @ C2 )
= ( groups6591440286371151544t_real @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_593_sum_Osame__carrier,axiom,
! [C2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > nat,H: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C2 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_nat ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups419121665405347140al_nat @ G @ A2 )
= ( groups419121665405347140al_nat @ H @ B2 ) )
= ( ( groups419121665405347140al_nat @ G @ C2 )
= ( groups419121665405347140al_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_594_sum_Osame__carrier,axiom,
! [C2: set_real,A2: set_real,B2: set_real,G: real > nat,H: real > nat] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A2 @ C2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_nat ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups1935376822645274424al_nat @ G @ A2 )
= ( groups1935376822645274424al_nat @ H @ B2 ) )
= ( ( groups1935376822645274424al_nat @ G @ C2 )
= ( groups1935376822645274424al_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_595_sum_Osame__carrier,axiom,
! [C2: set_b,A2: set_b,B2: set_b,G: b > nat,H: b > nat] :
( ( finite_finite_b @ C2 )
=> ( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ! [A4: b] :
( ( member_b @ A4 @ ( minus_minus_set_b @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_nat ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups7570001007293516437_b_nat @ G @ A2 )
= ( groups7570001007293516437_b_nat @ H @ B2 ) )
= ( ( groups7570001007293516437_b_nat @ G @ C2 )
= ( groups7570001007293516437_b_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_596_sum_Osame__carrier,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_zero_nat ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G @ A2 )
= ( groups3542108847815614940at_nat @ H @ B2 ) )
= ( ( groups3542108847815614940at_nat @ G @ C2 )
= ( groups3542108847815614940at_nat @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_597_sum_Osame__carrier,axiom,
! [C2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
( ( finite3782138982310603983nnreal @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ A2 @ C2 )
=> ( ( ord_le6787938422905777998nnreal @ B2 @ C2 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( ( groups4193414088831287468nnreal @ G @ A2 )
= ( groups4193414088831287468nnreal @ H @ B2 ) )
= ( ( groups4193414088831287468nnreal @ G @ C2 )
= ( groups4193414088831287468nnreal @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_598_sum_Osame__carrier,axiom,
! [C2: set_real,A2: set_real,B2: set_real,G: real > extend8495563244428889912nnreal,H: real > extend8495563244428889912nnreal] :
( ( finite_finite_real @ C2 )
=> ( ( ord_less_eq_set_real @ A2 @ C2 )
=> ( ( ord_less_eq_set_real @ B2 @ C2 )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( ( groups4232809223866053280nnreal @ G @ A2 )
= ( groups4232809223866053280nnreal @ H @ B2 ) )
= ( ( groups4232809223866053280nnreal @ G @ C2 )
= ( groups4232809223866053280nnreal @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_599_sum_Osame__carrier,axiom,
! [C2: set_b,A2: set_b,B2: set_b,G: b > extend8495563244428889912nnreal,H: b > extend8495563244428889912nnreal] :
( ( finite_finite_b @ C2 )
=> ( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ! [A4: b] :
( ( member_b @ A4 @ ( minus_minus_set_b @ C2 @ A2 ) )
=> ( ( G @ A4 )
= zero_z7100319975126383169nnreal ) )
=> ( ! [B3: b] :
( ( member_b @ B3 @ ( minus_minus_set_b @ C2 @ B2 ) )
=> ( ( H @ B3 )
= zero_z7100319975126383169nnreal ) )
=> ( ( ( groups9167310395270569469nnreal @ G @ A2 )
= ( groups9167310395270569469nnreal @ H @ B2 ) )
= ( ( groups9167310395270569469nnreal @ G @ C2 )
= ( groups9167310395270569469nnreal @ H @ C2 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_600_sum__mono,axiom,
! [K5: set_nat,F: nat > nat,G: nat > nat] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K5 ) @ ( groups3542108847815614940at_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_601_sum__mono,axiom,
! [K5: set_b,F: b > nat,G: b > nat] :
( ! [I4: b] :
( ( member_b @ I4 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups7570001007293516437_b_nat @ F @ K5 ) @ ( groups7570001007293516437_b_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_602_sum__mono,axiom,
! [K5: set_a,F: a > nat,G: a > nat] :
( ! [I4: a] :
( ( member_a @ I4 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups6334556678337121940_a_nat @ F @ K5 ) @ ( groups6334556678337121940_a_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_603_sum__mono,axiom,
! [K5: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > nat,G: extend8495563244428889912nnreal > nat] :
( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups419121665405347140al_nat @ F @ K5 ) @ ( groups419121665405347140al_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_604_sum__mono,axiom,
! [K5: set_real,F: real > nat,G: real > nat] :
( ! [I4: real] :
( ( member_real @ I4 @ K5 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_605_sum__mono,axiom,
! [K5: set_nat,F: nat > real,G: nat > real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K5 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K5 ) @ ( groups6591440286371151544t_real @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_606_sum__mono,axiom,
! [K5: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > real,G: extend8495563244428889912nnreal > real] :
( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ K5 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_real @ ( groups2265062954415509024l_real @ F @ K5 ) @ ( groups2265062954415509024l_real @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_607_sum__mono,axiom,
! [K5: set_real,F: real > real,G: real > real] :
( ! [I4: real] :
( ( member_real @ I4 @ K5 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ K5 ) @ ( groups8097168146408367636l_real @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_608_sum__mono,axiom,
! [K5: set_nat,F: nat > extend8495563244428889912nnreal,G: nat > extend8495563244428889912nnreal] :
( ! [I4: nat] :
( ( member_nat @ I4 @ K5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups4868793261593263428nnreal @ F @ K5 ) @ ( groups4868793261593263428nnreal @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_609_sum__mono,axiom,
! [K5: set_b,F: b > extend8495563244428889912nnreal,G: b > extend8495563244428889912nnreal] :
( ! [I4: b] :
( ( member_b @ I4 @ K5 )
=> ( ord_le3935885782089961368nnreal @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( groups9167310395270569469nnreal @ F @ K5 ) @ ( groups9167310395270569469nnreal @ G @ K5 ) ) ) ).
% sum_mono
thf(fact_610_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_611_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_612_lambda__zero,axiom,
( ( ^ [H2: extend8495563244428889912nnreal] : zero_z7100319975126383169nnreal )
= ( times_1893300245718287421nnreal @ zero_z7100319975126383169nnreal ) ) ).
% lambda_zero
thf(fact_613_sum__mono__inv,axiom,
! [F: extend8495563244428889912nnreal > nat,I3: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > nat,I: extend8495563244428889912nnreal] :
( ( ( groups419121665405347140al_nat @ F @ I3 )
= ( groups419121665405347140al_nat @ G @ I3 ) )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member7908768830364227535nnreal @ I @ I3 )
=> ( ( finite3782138982310603983nnreal @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_614_sum__mono__inv,axiom,
! [F: real > nat,I3: set_real,G: real > nat,I: real] :
( ( ( groups1935376822645274424al_nat @ F @ I3 )
= ( groups1935376822645274424al_nat @ G @ I3 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_real @ I @ I3 )
=> ( ( finite_finite_real @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_615_sum__mono__inv,axiom,
! [F: b > nat,I3: set_b,G: b > nat,I: b] :
( ( ( groups7570001007293516437_b_nat @ F @ I3 )
= ( groups7570001007293516437_b_nat @ G @ I3 ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_b @ I @ I3 )
=> ( ( finite_finite_b @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_616_sum__mono__inv,axiom,
! [F: nat > nat,I3: set_nat,G: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I3 )
= ( groups3542108847815614940at_nat @ G @ I3 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_nat @ I @ I3 )
=> ( ( finite_finite_nat @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_617_sum__mono__inv,axiom,
! [F: a > nat,I3: set_a,G: a > nat,I: a] :
( ( ( groups6334556678337121940_a_nat @ F @ I3 )
= ( groups6334556678337121940_a_nat @ G @ I3 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_a @ I @ I3 )
=> ( ( finite_finite_a @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_618_sum__mono__inv,axiom,
! [F: extend8495563244428889912nnreal > real,I3: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > real,I: extend8495563244428889912nnreal] :
( ( ( groups2265062954415509024l_real @ F @ I3 )
= ( groups2265062954415509024l_real @ G @ I3 ) )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member7908768830364227535nnreal @ I @ I3 )
=> ( ( finite3782138982310603983nnreal @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_619_sum__mono__inv,axiom,
! [F: real > real,I3: set_real,G: real > real,I: real] :
( ( ( groups8097168146408367636l_real @ F @ I3 )
= ( groups8097168146408367636l_real @ G @ I3 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_real @ I @ I3 )
=> ( ( finite_finite_real @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_620_sum__mono__inv,axiom,
! [F: nat > real,I3: set_nat,G: nat > real,I: nat] :
( ( ( groups6591440286371151544t_real @ F @ I3 )
= ( groups6591440286371151544t_real @ G @ I3 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_nat @ I @ I3 )
=> ( ( finite_finite_nat @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_621_sum__mono__inv,axiom,
! [F: b > real,I3: set_b,G: b > real,I: b] :
( ( ( groups8336678772925405937b_real @ F @ I3 )
= ( groups8336678772925405937b_real @ G @ I3 ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_b @ I @ I3 )
=> ( ( finite_finite_b @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_622_sum__mono__inv,axiom,
! [F: a > real,I3: set_a,G: a > real,I: a] :
( ( ( groups2740460157737275248a_real @ F @ I3 )
= ( groups2740460157737275248a_real @ G @ I3 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member_a @ I @ I3 )
=> ( ( finite_finite_a @ I3 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_623_sum__diff__nat,axiom,
! [B2: set_b,A2: set_b,F: b > nat] :
( ( finite_finite_b @ B2 )
=> ( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( groups7570001007293516437_b_nat @ F @ ( minus_minus_set_b @ A2 @ B2 ) )
= ( minus_minus_nat @ ( groups7570001007293516437_b_nat @ F @ A2 ) @ ( groups7570001007293516437_b_nat @ F @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_624_sum__diff__nat,axiom,
! [B2: set_nat,A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_625_sum__diff__nat,axiom,
! [B2: set_set_a,A2: set_set_a,F: set_a > nat] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( groups6141743369313575924_a_nat @ F @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( groups6141743369313575924_a_nat @ F @ A2 ) @ ( groups6141743369313575924_a_nat @ F @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_626_sum__diff__nat,axiom,
! [B2: set_a,A2: set_a,F: a > nat] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( groups6334556678337121940_a_nat @ F @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) @ ( groups6334556678337121940_a_nat @ F @ B2 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_627_not__integrable__integral__eq,axiom,
! [M: sigma_measure_a,F: a > real] :
( ~ ( bochne2139062162225249880a_real @ M @ F )
=> ( ( bochne378719280626478695a_real @ M @ F )
= zero_zero_real ) ) ).
% not_integrable_integral_eq
thf(fact_628_sum_Ointer__filter,axiom,
! [A2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > real,P: extend8495563244428889912nnreal > $o] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( groups2265062954415509024l_real @ G
@ ( collec6648975593938027277nnreal
@ ^ [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups2265062954415509024l_real
@ ^ [X: extend8495563244428889912nnreal] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_629_sum_Ointer__filter,axiom,
! [A2: set_real,G: real > real,P: real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( groups8097168146408367636l_real @ G
@ ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups8097168146408367636l_real
@ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_630_sum_Ointer__filter,axiom,
! [A2: set_nat,G: nat > real,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6591440286371151544t_real @ G
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups6591440286371151544t_real
@ ^ [X: nat] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_631_sum_Ointer__filter,axiom,
! [A2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > nat,P: extend8495563244428889912nnreal > $o] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( groups419121665405347140al_nat @ G
@ ( collec6648975593938027277nnreal
@ ^ [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups419121665405347140al_nat
@ ^ [X: extend8495563244428889912nnreal] : ( if_nat @ ( P @ X ) @ ( G @ X ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_632_sum_Ointer__filter,axiom,
! [A2: set_real,G: real > nat,P: real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( groups1935376822645274424al_nat @ G
@ ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups1935376822645274424al_nat
@ ^ [X: real] : ( if_nat @ ( P @ X ) @ ( G @ X ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_633_sum_Ointer__filter,axiom,
! [A2: set_b,G: b > nat,P: b > $o] :
( ( finite_finite_b @ A2 )
=> ( ( groups7570001007293516437_b_nat @ G
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups7570001007293516437_b_nat
@ ^ [X: b] : ( if_nat @ ( P @ X ) @ ( G @ X ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_634_sum_Ointer__filter,axiom,
! [A2: set_nat,G: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( G @ X ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_635_sum_Ointer__filter,axiom,
! [A2: set_a,G: a > nat,P: a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups6334556678337121940_a_nat
@ ^ [X: a] : ( if_nat @ ( P @ X ) @ ( G @ X ) @ zero_zero_nat )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_636_sum_Ointer__filter,axiom,
! [A2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,P: extend8495563244428889912nnreal > $o] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( groups4193414088831287468nnreal @ G
@ ( collec6648975593938027277nnreal
@ ^ [X: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups4193414088831287468nnreal
@ ^ [X: extend8495563244428889912nnreal] : ( if_Ext9135588136721118450nnreal @ ( P @ X ) @ ( G @ X ) @ zero_z7100319975126383169nnreal )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_637_sum_Ointer__filter,axiom,
! [A2: set_real,G: real > extend8495563244428889912nnreal,P: real > $o] :
( ( finite_finite_real @ A2 )
=> ( ( groups4232809223866053280nnreal @ G
@ ( collect_real
@ ^ [X: real] :
( ( member_real @ X @ A2 )
& ( P @ X ) ) ) )
= ( groups4232809223866053280nnreal
@ ^ [X: real] : ( if_Ext9135588136721118450nnreal @ ( P @ X ) @ ( G @ X ) @ zero_z7100319975126383169nnreal )
@ A2 ) ) ) ).
% sum.inter_filter
thf(fact_638_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_Ex3793607809372303086nnreal,T4: set_Ex3793607809372303086nnreal,S2: set_Ex3793607809372303086nnreal,I: extend8495563244428889912nnreal > extend8495563244428889912nnreal,J: extend8495563244428889912nnreal > extend8495563244428889912nnreal,T2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > real,H: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ S3 )
=> ( ( finite3782138982310603983nnreal @ T4 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ S2 @ S3 ) )
=> ( member7908768830364227535nnreal @ ( J @ A4 ) @ ( minus_104578273773384135nnreal @ T2 @ T4 ) ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ T2 @ T4 ) )
=> ( member7908768830364227535nnreal @ ( I @ B3 ) @ ( minus_104578273773384135nnreal @ S2 @ S3 ) ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2265062954415509024l_real @ G @ S2 )
= ( groups2265062954415509024l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_639_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_Ex3793607809372303086nnreal,T4: set_real,S2: set_Ex3793607809372303086nnreal,I: real > extend8495563244428889912nnreal,J: extend8495563244428889912nnreal > real,T2: set_real,G: extend8495563244428889912nnreal > real,H: real > real] :
( ( finite3782138982310603983nnreal @ S3 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ S2 @ S3 ) )
=> ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member7908768830364227535nnreal @ ( I @ B3 ) @ ( minus_104578273773384135nnreal @ S2 @ S3 ) ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2265062954415509024l_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_640_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_real,T4: set_Ex3793607809372303086nnreal,S2: set_real,I: extend8495563244428889912nnreal > real,J: real > extend8495563244428889912nnreal,T2: set_Ex3793607809372303086nnreal,G: real > real,H: extend8495563244428889912nnreal > real] :
( ( finite_finite_real @ S3 )
=> ( ( finite3782138982310603983nnreal @ T4 )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S3 ) )
=> ( member7908768830364227535nnreal @ ( J @ A4 ) @ ( minus_104578273773384135nnreal @ T2 @ T4 ) ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S3 ) ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups2265062954415509024l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_641_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T2: set_real,G: real > real,H: real > real] :
( ( finite_finite_real @ S3 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S3 ) )
=> ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S3 ) ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_642_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_Ex3793607809372303086nnreal,T4: set_nat,S2: set_Ex3793607809372303086nnreal,I: nat > extend8495563244428889912nnreal,J: extend8495563244428889912nnreal > nat,T2: set_nat,G: extend8495563244428889912nnreal > real,H: nat > real] :
( ( finite3782138982310603983nnreal @ S3 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ S2 @ S3 ) )
=> ( member_nat @ ( J @ A4 ) @ ( minus_minus_set_nat @ T2 @ T4 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T2 @ T4 ) )
=> ( member7908768830364227535nnreal @ ( I @ B3 ) @ ( minus_104578273773384135nnreal @ S2 @ S3 ) ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups2265062954415509024l_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_643_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_real,T4: set_nat,S2: set_real,I: nat > real,J: real > nat,T2: set_nat,G: real > real,H: nat > real] :
( ( finite_finite_real @ S3 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ ( minus_minus_set_real @ S2 @ S3 ) )
=> ( member_nat @ ( J @ A4 ) @ ( minus_minus_set_nat @ T2 @ T4 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T2 @ T4 ) )
=> ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S3 ) ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ! [A4: real] :
( ( member_real @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups8097168146408367636l_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_644_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T4: set_Ex3793607809372303086nnreal,S2: set_nat,I: extend8495563244428889912nnreal > nat,J: nat > extend8495563244428889912nnreal,T2: set_Ex3793607809372303086nnreal,G: nat > real,H: extend8495563244428889912nnreal > real] :
( ( finite_finite_nat @ S3 )
=> ( ( finite3782138982310603983nnreal @ T4 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member7908768830364227535nnreal @ ( J @ A4 ) @ ( minus_104578273773384135nnreal @ T2 @ T4 ) ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ T2 @ T4 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups2265062954415509024l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_645_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T4: set_real,S2: set_nat,I: real > nat,J: nat > real,T2: set_real,G: nat > real,H: real > real] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_real @ T4 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T2 @ T4 ) ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ ( minus_minus_set_real @ T2 @ T4 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: real] :
( ( member_real @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups8097168146408367636l_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_646_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_nat,T4: set_nat,S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,G: nat > real,H: nat > real] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T4 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ ( minus_minus_set_nat @ S2 @ S3 ) )
=> ( member_nat @ ( J @ A4 ) @ ( minus_minus_set_nat @ T2 @ T4 ) ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ ( minus_minus_set_nat @ T2 @ T4 ) )
=> ( member_nat @ ( I @ B3 ) @ ( minus_minus_set_nat @ S2 @ S3 ) ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_real ) )
=> ( ! [B3: nat] :
( ( member_nat @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_real ) )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups6591440286371151544t_real @ G @ S2 )
= ( groups6591440286371151544t_real @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_647_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S3: set_Ex3793607809372303086nnreal,T4: set_Ex3793607809372303086nnreal,S2: set_Ex3793607809372303086nnreal,I: extend8495563244428889912nnreal > extend8495563244428889912nnreal,J: extend8495563244428889912nnreal > extend8495563244428889912nnreal,T2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > nat,H: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ S3 )
=> ( ( finite3782138982310603983nnreal @ T4 )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ S2 @ S3 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ ( minus_104578273773384135nnreal @ S2 @ S3 ) )
=> ( member7908768830364227535nnreal @ ( J @ A4 ) @ ( minus_104578273773384135nnreal @ T2 @ T4 ) ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ T2 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ ( minus_104578273773384135nnreal @ T2 @ T4 ) )
=> ( member7908768830364227535nnreal @ ( I @ B3 ) @ ( minus_104578273773384135nnreal @ S2 @ S3 ) ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S3 )
=> ( ( G @ A4 )
= zero_zero_nat ) )
=> ( ! [B3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ B3 @ T4 )
=> ( ( H @ B3 )
= zero_zero_nat ) )
=> ( ! [A4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups419121665405347140al_nat @ G @ S2 )
= ( groups419121665405347140al_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_648_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ( groups6591440286371151544t_real @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X: nat] :
( ( G @ X )
= zero_zero_real ) ) ) )
= ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_649_sum_Osetdiff__irrelevant,axiom,
! [A2: set_b,G: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ( groups7570001007293516437_b_nat @ G
@ ( minus_minus_set_b @ A2
@ ( collect_b
@ ^ [X: b] :
( ( G @ X )
= zero_zero_nat ) ) ) )
= ( groups7570001007293516437_b_nat @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_650_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( groups3542108847815614940at_nat @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X: nat] :
( ( G @ X )
= zero_zero_nat ) ) ) )
= ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_651_sum_Osetdiff__irrelevant,axiom,
! [A2: set_b,G: b > extend8495563244428889912nnreal] :
( ( finite_finite_b @ A2 )
=> ( ( groups9167310395270569469nnreal @ G
@ ( minus_minus_set_b @ A2
@ ( collect_b
@ ^ [X: b] :
( ( G @ X )
= zero_z7100319975126383169nnreal ) ) ) )
= ( groups9167310395270569469nnreal @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_652_sum_Osetdiff__irrelevant,axiom,
! [A2: set_nat,G: nat > extend8495563244428889912nnreal] :
( ( finite_finite_nat @ A2 )
=> ( ( groups4868793261593263428nnreal @ G
@ ( minus_minus_set_nat @ A2
@ ( collect_nat
@ ^ [X: nat] :
( ( G @ X )
= zero_z7100319975126383169nnreal ) ) ) )
= ( groups4868793261593263428nnreal @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_653_sum_Osetdiff__irrelevant,axiom,
! [A2: set_a,G: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( groups6334556678337121940_a_nat @ G
@ ( minus_minus_set_a @ A2
@ ( collect_a
@ ^ [X: a] :
( ( G @ X )
= zero_zero_nat ) ) ) )
= ( groups6334556678337121940_a_nat @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_654_sum_Osetdiff__irrelevant,axiom,
! [A2: set_a,G: a > extend8495563244428889912nnreal] :
( ( finite_finite_a @ A2 )
=> ( ( groups3047462441131878908nnreal @ G
@ ( minus_minus_set_a @ A2
@ ( collect_a
@ ^ [X: a] :
( ( G @ X )
= zero_z7100319975126383169nnreal ) ) ) )
= ( groups3047462441131878908nnreal @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_655_sum_Osetdiff__irrelevant,axiom,
! [A2: set_b,G: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( groups8336678772925405937b_real @ G
@ ( minus_minus_set_b @ A2
@ ( collect_b
@ ^ [X: b] :
( ( G @ X )
= zero_zero_real ) ) ) )
= ( groups8336678772925405937b_real @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_656_sum_Osetdiff__irrelevant,axiom,
! [A2: set_a,G: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( groups2740460157737275248a_real @ G
@ ( minus_minus_set_a @ A2
@ ( collect_a
@ ^ [X: a] :
( ( G @ X )
= zero_zero_real ) ) ) )
= ( groups2740460157737275248a_real @ G @ A2 ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_657_diff__ge__0__iff__ge,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_658_prob__space_Oindep__var__lebesgue__integral,axiom,
! [M: sigma_measure_a,X1: a > real,X22: a > real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe8958435565499147358a_real @ M @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
=> ( ( bochne2139062162225249880a_real @ M @ X1 )
=> ( ( bochne2139062162225249880a_real @ M @ X22 )
=> ( ( bochne378719280626478695a_real @ M
@ ^ [Omega: a] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) )
= ( times_times_real @ ( bochne378719280626478695a_real @ M @ X1 ) @ ( bochne378719280626478695a_real @ M @ X22 ) ) ) ) ) ) ) ).
% prob_space.indep_var_lebesgue_integral
thf(fact_659_prob__space_Oindep__var__integrable,axiom,
! [M: sigma_measure_a,X1: a > real,X22: a > real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe8958435565499147358a_real @ M @ borel_5078946678739801102l_real @ X1 @ borel_5078946678739801102l_real @ X22 )
=> ( ( bochne2139062162225249880a_real @ M @ X1 )
=> ( ( bochne2139062162225249880a_real @ M @ X22 )
=> ( bochne2139062162225249880a_real @ M
@ ^ [Omega: a] : ( times_times_real @ ( X1 @ Omega ) @ ( X22 @ Omega ) ) ) ) ) ) ) ).
% prob_space.indep_var_integrable
thf(fact_660_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_661_diff__0__right,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_0_right
thf(fact_662_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_663_diff__zero,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% diff_zero
thf(fact_664_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_665_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_666_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_667_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_668_le__zero__eq,axiom,
! [N3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ N3 @ zero_z7100319975126383169nnreal )
= ( N3 = zero_z7100319975126383169nnreal ) ) ).
% le_zero_eq
thf(fact_669_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_670_indep__var__rv2,axiom,
! [S2: sigma_measure_real,X5: a > real,T2: sigma_measure_real,Y5: a > real] :
( ( indepe8958435565499147358a_real @ m @ S2 @ X5 @ T2 @ Y5 )
=> ( member_a_real @ Y5 @ ( sigma_9116425665531756122a_real @ m @ T2 ) ) ) ).
% indep_var_rv2
thf(fact_671_indep__var__rv1,axiom,
! [S2: sigma_measure_real,X5: a > real,T2: sigma_measure_real,Y5: a > real] :
( ( indepe8958435565499147358a_real @ m @ S2 @ X5 @ T2 @ Y5 )
=> ( member_a_real @ X5 @ ( sigma_9116425665531756122a_real @ m @ S2 ) ) ) ).
% indep_var_rv1
thf(fact_672_sum__subtractf__nat,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_673_sum__subtractf__nat,axiom,
! [A2: set_re5328672808648366137nnreal,G: ( real > extend8495563244428889912nnreal ) > nat,F: ( real > extend8495563244428889912nnreal ) > nat] :
( ! [X3: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3215773089680308239al_nat
@ ^ [X: real > extend8495563244428889912nnreal] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_nat @ ( groups3215773089680308239al_nat @ F @ A2 ) @ ( groups3215773089680308239al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_674_sum__subtractf__nat,axiom,
! [A2: set_a_real,G: ( a > real ) > nat,F: ( a > real ) > nat] :
( ! [X3: a > real] :
( ( member_a_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups1701885688937111089al_nat
@ ^ [X: a > real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_nat @ ( groups1701885688937111089al_nat @ F @ A2 ) @ ( groups1701885688937111089al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_675_sum__subtractf__nat,axiom,
! [A2: set_b,G: b > nat,F: b > nat] :
( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups7570001007293516437_b_nat
@ ^ [X: b] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_nat @ ( groups7570001007293516437_b_nat @ F @ A2 ) @ ( groups7570001007293516437_b_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_676_sum__subtractf__nat,axiom,
! [A2: set_a,G: a > nat,F: a > nat] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups6334556678337121940_a_nat
@ ^ [X: a] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_nat @ ( groups6334556678337121940_a_nat @ F @ A2 ) @ ( groups6334556678337121940_a_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_677_sum__subtractf__nat,axiom,
! [A2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > nat,F: extend8495563244428889912nnreal > nat] :
( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups419121665405347140al_nat
@ ^ [X: extend8495563244428889912nnreal] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_nat @ ( groups419121665405347140al_nat @ F @ A2 ) @ ( groups419121665405347140al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_678_sum__subtractf__nat,axiom,
! [A2: set_real,G: real > nat,F: real > nat] :
( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups1935376822645274424al_nat
@ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
@ A2 )
= ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_679_integrableD_I1_J,axiom,
! [M: sigma_measure_a,F: a > real] :
( ( bochne2139062162225249880a_real @ M @ F )
=> ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% integrableD(1)
thf(fact_680_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_681_less__eq__set__def,axiom,
( ord_le2462468573666744473nnreal
= ( ^ [A5: set_re5328672808648366137nnreal,B4: set_re5328672808648366137nnreal] :
( ord_le8425755247534135084real_o
@ ^ [X: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X @ A5 )
@ ^ [X: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_682_less__eq__set__def,axiom,
( ord_le3334967407727675675a_real
= ( ^ [A5: set_a_real,B4: set_a_real] :
( ord_less_eq_a_real_o
@ ^ [X: a > real] : ( member_a_real @ X @ A5 )
@ ^ [X: a > real] : ( member_a_real @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_683_less__eq__set__def,axiom,
( ord_less_eq_set_b
= ( ^ [A5: set_b,B4: set_b] :
( ord_less_eq_b_o
@ ^ [X: b] : ( member_b @ X @ A5 )
@ ^ [X: b] : ( member_b @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_684_less__eq__set__def,axiom,
( ord_le6787938422905777998nnreal
= ( ^ [A5: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
( ord_le7025323315894483639real_o
@ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ A5 )
@ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_685_less__eq__set__def,axiom,
( ord_less_eq_set_real
= ( ^ [A5: set_real,B4: set_real] :
( ord_less_eq_real_o
@ ^ [X: real] : ( member_real @ X @ A5 )
@ ^ [X: real] : ( member_real @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_686_less__eq__set__def,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B4: set_set_a] :
( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A5 )
@ ^ [X: set_a] : ( member_set_a @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_687_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ A5 )
@ ^ [X: a] : ( member_a @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_688_prob__space_Oindep__var__rv1,axiom,
! [M: sigma_measure_real,S2: sigma_7234349610311085201nnreal,X5: real > extend8495563244428889912nnreal,T2: sigma_7234349610311085201nnreal,Y5: real > extend8495563244428889912nnreal] :
( ( probab535871623910865577e_real @ M )
=> ( ( indepe6767359503340752434nnreal @ M @ S2 @ X5 @ T2 @ Y5 )
=> ( member2919562650594848410nnreal @ X5 @ ( sigma_9017504469962657078nnreal @ M @ S2 ) ) ) ) ).
% prob_space.indep_var_rv1
thf(fact_689_prob__space_Oindep__var__rv1,axiom,
! [M: sigma_measure_a,S2: sigma_measure_real,X5: a > real,T2: sigma_measure_real,Y5: a > real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe8958435565499147358a_real @ M @ S2 @ X5 @ T2 @ Y5 )
=> ( member_a_real @ X5 @ ( sigma_9116425665531756122a_real @ M @ S2 ) ) ) ) ).
% prob_space.indep_var_rv1
thf(fact_690_prob__space_Oindep__var__rv2,axiom,
! [M: sigma_measure_real,S2: sigma_7234349610311085201nnreal,X5: real > extend8495563244428889912nnreal,T2: sigma_7234349610311085201nnreal,Y5: real > extend8495563244428889912nnreal] :
( ( probab535871623910865577e_real @ M )
=> ( ( indepe6767359503340752434nnreal @ M @ S2 @ X5 @ T2 @ Y5 )
=> ( member2919562650594848410nnreal @ Y5 @ ( sigma_9017504469962657078nnreal @ M @ T2 ) ) ) ) ).
% prob_space.indep_var_rv2
thf(fact_691_prob__space_Oindep__var__rv2,axiom,
! [M: sigma_measure_a,S2: sigma_measure_real,X5: a > real,T2: sigma_measure_real,Y5: a > real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe8958435565499147358a_real @ M @ S2 @ X5 @ T2 @ Y5 )
=> ( member_a_real @ Y5 @ ( sigma_9116425665531756122a_real @ M @ T2 ) ) ) ) ).
% prob_space.indep_var_rv2
thf(fact_692_borel__measurable__integrable,axiom,
! [M: sigma_measure_a,F: a > real] :
( ( bochne2139062162225249880a_real @ M @ F )
=> ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_integrable
thf(fact_693_borel__measurable__integrable_H,axiom,
! [M: sigma_measure_real,F: real > real,G: a > real,N2: sigma_measure_a] :
( ( bochne3340023020068487468l_real @ M @ F )
=> ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ N2 @ M ) )
=> ( member_a_real
@ ^ [X: a] : ( F @ ( G @ X ) )
@ ( sigma_9116425665531756122a_real @ N2 @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_integrable'
thf(fact_694_borel__measurable__integrable_H,axiom,
! [M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,N2: sigma_measure_real] :
( ( bochne9025062821074728248l_real @ M @ F )
=> ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ N2 @ M ) )
=> ( member_real_real
@ ^ [X: real] : ( F @ ( G @ X ) )
@ ( sigma_5267869275261027754l_real @ N2 @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_integrable'
thf(fact_695_borel__measurable__integrable_H,axiom,
! [M: sigma_measure_a,F: a > real,G: a > a,N2: sigma_measure_a] :
( ( bochne2139062162225249880a_real @ M @ F )
=> ( ( member_a_a @ G @ ( sigma_measurable_a_a @ N2 @ M ) )
=> ( member_a_real
@ ^ [X: a] : ( F @ ( G @ X ) )
@ ( sigma_9116425665531756122a_real @ N2 @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_integrable'
thf(fact_696_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_697_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_698_zero__reorient,axiom,
! [X2: extend8495563244428889912nnreal] :
( ( zero_z7100319975126383169nnreal = X2 )
= ( X2 = zero_z7100319975126383169nnreal ) ) ).
% zero_reorient
thf(fact_699_mult_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_700_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_701_mult_Oleft__commute,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ B @ ( times_1893300245718287421nnreal @ A @ C ) )
= ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_702_mult_Oleft__commute,axiom,
! [B: set_Ex3793607809372303086nnreal,A: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( times_4022348038934646771nnreal @ B @ ( times_4022348038934646771nnreal @ A @ C ) )
= ( times_4022348038934646771nnreal @ A @ ( times_4022348038934646771nnreal @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_703_mult_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( times_times_set_nat @ B @ ( times_times_set_nat @ A @ C ) )
= ( times_times_set_nat @ A @ ( times_times_set_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_704_mult_Oleft__commute,axiom,
! [B: set_real,A: set_real,C: set_real] :
( ( times_times_set_real @ B @ ( times_times_set_real @ A @ C ) )
= ( times_times_set_real @ A @ ( times_times_set_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_705_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B5: real] : ( times_times_real @ B5 @ A3 ) ) ) ).
% mult.commute
thf(fact_706_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B5: nat] : ( times_times_nat @ B5 @ A3 ) ) ) ).
% mult.commute
thf(fact_707_mult_Ocommute,axiom,
( times_1893300245718287421nnreal
= ( ^ [A3: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] : ( times_1893300245718287421nnreal @ B5 @ A3 ) ) ) ).
% mult.commute
thf(fact_708_mult_Ocommute,axiom,
( times_4022348038934646771nnreal
= ( ^ [A3: set_Ex3793607809372303086nnreal,B5: set_Ex3793607809372303086nnreal] : ( times_4022348038934646771nnreal @ B5 @ A3 ) ) ) ).
% mult.commute
thf(fact_709_mult_Ocommute,axiom,
( times_times_set_nat
= ( ^ [A3: set_nat,B5: set_nat] : ( times_times_set_nat @ B5 @ A3 ) ) ) ).
% mult.commute
thf(fact_710_mult_Ocommute,axiom,
( times_times_set_real
= ( ^ [A3: set_real,B5: set_real] : ( times_times_set_real @ B5 @ A3 ) ) ) ).
% mult.commute
thf(fact_711_mult_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_712_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_713_mult_Oassoc,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C )
= ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).
% mult.assoc
thf(fact_714_mult_Oassoc,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( times_4022348038934646771nnreal @ ( times_4022348038934646771nnreal @ A @ B ) @ C )
= ( times_4022348038934646771nnreal @ A @ ( times_4022348038934646771nnreal @ B @ C ) ) ) ).
% mult.assoc
thf(fact_715_mult_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( times_times_set_nat @ ( times_times_set_nat @ A @ B ) @ C )
= ( times_times_set_nat @ A @ ( times_times_set_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_716_mult_Oassoc,axiom,
! [A: set_real,B: set_real,C: set_real] :
( ( times_times_set_real @ ( times_times_set_real @ A @ B ) @ C )
= ( times_times_set_real @ A @ ( times_times_set_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_717_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_718_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_719_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( times_1893300245718287421nnreal @ ( times_1893300245718287421nnreal @ A @ B ) @ C )
= ( times_1893300245718287421nnreal @ A @ ( times_1893300245718287421nnreal @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_720_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,C: set_Ex3793607809372303086nnreal] :
( ( times_4022348038934646771nnreal @ ( times_4022348038934646771nnreal @ A @ B ) @ C )
= ( times_4022348038934646771nnreal @ A @ ( times_4022348038934646771nnreal @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_721_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( times_times_set_nat @ ( times_times_set_nat @ A @ B ) @ C )
= ( times_times_set_nat @ A @ ( times_times_set_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_722_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: set_real,B: set_real,C: set_real] :
( ( times_times_set_real @ ( times_times_set_real @ A @ B ) @ C )
= ( times_times_set_real @ A @ ( times_times_set_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_723_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C: real,B: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
= ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_724_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_725_diff__eq__diff__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_726_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_727_zero__le,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ zero_z7100319975126383169nnreal @ X2 ) ).
% zero_le
thf(fact_728_diff__eq__diff__less__eq,axiom,
! [A: real,B: real,C: real,D2: real] :
( ( ( minus_minus_real @ A @ B )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_eq_real @ A @ B )
= ( ord_less_eq_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_729_diff__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_730_diff__left__mono,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_731_diff__mono,axiom,
! [A: real,B: real,D2: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ D2 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_732_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A3: real,B5: real] :
( ( minus_minus_real @ A3 @ B5 )
= zero_zero_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_733_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B5 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_734_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: nat > sigma_measure_a,X5: nat > a > a,I3: set_nat,Y5: nat > a > real,N2: nat > sigma_measure_real] :
( ( prob_k6325968634923510307_nat_a @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k6744671002923369159t_real @ m @ K3 @ N2
@ ^ [I2: nat,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_735_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: b > sigma_measure_a,X5: b > a > a,I3: set_b,Y5: b > a > real,N2: b > sigma_measure_real] :
( ( prob_k6574085460301583240_a_b_a @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k7188637637266226338b_real @ m @ K3 @ N2
@ ^ [I2: b,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_736_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: a > sigma_measure_a,X5: a > a > a,I3: set_a,Y5: a > a > real,N2: a > sigma_measure_real] :
( ( prob_k138169005419483465_a_a_a @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k1592419022078095649a_real @ m @ K3 @ N2
@ ^ [I2: a,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_737_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: extend8495563244428889912nnreal > sigma_measure_a,X5: extend8495563244428889912nnreal > a > a,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > a > real,N2: extend8495563244428889912nnreal > sigma_measure_real] :
( ( prob_k282204728494123195real_a @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k3587330016639965743l_real @ m @ K3 @ N2
@ ^ [I2: extend8495563244428889912nnreal,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_738_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: real > sigma_measure_a,X5: real > a > a,I3: set_real,Y5: real > a > real,N2: real > sigma_measure_real] :
( ( prob_k2222437789551045447real_a @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k4854350642655679139l_real @ m @ K3 @ N2
@ ^ [I2: real,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_739_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: nat > sigma_measure_real,X5: nat > a > real,I3: set_nat,Y5: nat > real > extend8495563244428889912nnreal,N2: nat > sigma_7234349610311085201nnreal] :
( ( prob_k6744671002923369159t_real @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k3632087671423262931nnreal @ m @ K3 @ N2
@ ^ [I2: nat,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_740_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: b > sigma_measure_real,X5: b > a > real,I3: set_b,Y5: b > real > extend8495563244428889912nnreal,N2: b > sigma_7234349610311085201nnreal] :
( ( prob_k7188637637266226338b_real @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k3585524288665042990nnreal @ m @ K3 @ N2
@ ^ [I2: b,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_741_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: a > sigma_measure_real,X5: a > a > real,I3: set_a,Y5: a > real > extend8495563244428889912nnreal,N2: a > sigma_7234349610311085201nnreal] :
( ( prob_k1592419022078095649a_real @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k6689048371381128237nnreal @ m @ K3 @ N2
@ ^ [I2: a,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_742_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: extend8495563244428889912nnreal > sigma_measure_real,X5: extend8495563244428889912nnreal > a > real,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,N2: extend8495563244428889912nnreal > sigma_7234349610311085201nnreal] :
( ( prob_k3587330016639965743l_real @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k5415967778188962875nnreal @ m @ K3 @ N2
@ ^ [I2: extend8495563244428889912nnreal,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_743_k__wise__indep__vars__compose,axiom,
! [K3: nat,M2: real > sigma_measure_real,X5: real > a > real,I3: set_real,Y5: real > real > extend8495563244428889912nnreal,N2: real > sigma_7234349610311085201nnreal] :
( ( prob_k4854350642655679139l_real @ m @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k5555076286090509999nnreal @ m @ K3 @ N2
@ ^ [I2: real,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% k_wise_indep_vars_compose
thf(fact_744_integral__completion,axiom,
! [F: a > real,M: sigma_measure_a] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( bochne378719280626478695a_real @ ( comple3428971583294703880tion_a @ M ) @ F )
= ( bochne378719280626478695a_real @ M @ F ) ) ) ).
% integral_completion
thf(fact_745_integrable__completion,axiom,
! [F: a > real,M: sigma_measure_a] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( bochne2139062162225249880a_real @ ( comple3428971583294703880tion_a @ M ) @ F )
= ( bochne2139062162225249880a_real @ M @ F ) ) ) ).
% integrable_completion
thf(fact_746_borel__measurable__sum,axiom,
! [S2: set_nat,F: nat > a > real,M: sigma_measure_a] :
( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( member_a_real @ ( F @ I4 ) @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) )
=> ( member_a_real
@ ^ [X: a] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_sum
thf(fact_747_borel__measurable__sum,axiom,
! [S2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > a > real,M: sigma_measure_a] :
( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ S2 )
=> ( member_a_real @ ( F @ I4 ) @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) )
=> ( member_a_real
@ ^ [X: a] :
( groups2265062954415509024l_real
@ ^ [I2: extend8495563244428889912nnreal] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_sum
thf(fact_748_borel__measurable__sum,axiom,
! [S2: set_real,F: real > a > real,M: sigma_measure_a] :
( ! [I4: real] :
( ( member_real @ I4 @ S2 )
=> ( member_a_real @ ( F @ I4 ) @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) )
=> ( member_a_real
@ ^ [X: a] :
( groups8097168146408367636l_real
@ ^ [I2: real] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_sum
thf(fact_749_borel__measurable__sum,axiom,
! [S2: set_b,F: b > a > real,M: sigma_measure_a] :
( ! [I4: b] :
( ( member_b @ I4 @ S2 )
=> ( member_a_real @ ( F @ I4 ) @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) )
=> ( member_a_real
@ ^ [X: a] :
( groups8336678772925405937b_real
@ ^ [I2: b] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_sum
thf(fact_750_borel__measurable__sum,axiom,
! [S2: set_a,F: a > a > real,M: sigma_measure_a] :
( ! [I4: a] :
( ( member_a @ I4 @ S2 )
=> ( member_a_real @ ( F @ I4 ) @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) )
=> ( member_a_real
@ ^ [X: a] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ).
% borel_measurable_sum
thf(fact_751_borel__measurable__sum,axiom,
! [S2: set_nat,F: nat > real > extend8495563244428889912nnreal,M: sigma_measure_real] :
( ! [I4: nat] :
( ( member_nat @ I4 @ S2 )
=> ( member2919562650594848410nnreal @ ( F @ I4 ) @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) )
=> ( member2919562650594848410nnreal
@ ^ [X: real] :
( groups4868793261593263428nnreal
@ ^ [I2: nat] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).
% borel_measurable_sum
thf(fact_752_borel__measurable__sum,axiom,
! [S2: set_b,F: b > real > extend8495563244428889912nnreal,M: sigma_measure_real] :
( ! [I4: b] :
( ( member_b @ I4 @ S2 )
=> ( member2919562650594848410nnreal @ ( F @ I4 ) @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) )
=> ( member2919562650594848410nnreal
@ ^ [X: real] :
( groups9167310395270569469nnreal
@ ^ [I2: b] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).
% borel_measurable_sum
thf(fact_753_borel__measurable__sum,axiom,
! [S2: set_a,F: a > real > extend8495563244428889912nnreal,M: sigma_measure_real] :
( ! [I4: a] :
( ( member_a @ I4 @ S2 )
=> ( member2919562650594848410nnreal @ ( F @ I4 ) @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) )
=> ( member2919562650594848410nnreal
@ ^ [X: real] :
( groups3047462441131878908nnreal
@ ^ [I2: a] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).
% borel_measurable_sum
thf(fact_754_borel__measurable__sum,axiom,
! [S2: set_Ex3793607809372303086nnreal,F: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,M: sigma_measure_real] :
( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ S2 )
=> ( member2919562650594848410nnreal @ ( F @ I4 ) @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) )
=> ( member2919562650594848410nnreal
@ ^ [X: real] :
( groups4193414088831287468nnreal
@ ^ [I2: extend8495563244428889912nnreal] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).
% borel_measurable_sum
thf(fact_755_borel__measurable__sum,axiom,
! [S2: set_real,F: real > real > extend8495563244428889912nnreal,M: sigma_measure_real] :
( ! [I4: real] :
( ( member_real @ I4 @ S2 )
=> ( member2919562650594848410nnreal @ ( F @ I4 ) @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) )
=> ( member2919562650594848410nnreal
@ ^ [X: real] :
( groups4232809223866053280nnreal
@ ^ [I2: real] : ( F @ I2 @ X )
@ S2 )
@ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ).
% borel_measurable_sum
thf(fact_756_borel__measurable__diff,axiom,
! [F: a > real,M: sigma_measure_a,G: a > real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_a_real
@ ^ [X: a] : ( minus_minus_real @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_diff
thf(fact_757_borel__measurable__times,axiom,
! [F: a > real,M: sigma_measure_a,G: a > real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
=> ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) )
=> ( member_a_real
@ ^ [X: a] : ( times_times_real @ ( F @ X ) @ ( G @ X ) )
@ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ) ) ).
% borel_measurable_times
thf(fact_758_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B5 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_759_mult__mono__nonpos__nonneg,axiom,
! [A: real,C: real,D2: real,B: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ D2 )
=> ( ( ord_less_eq_real @ D2 @ B )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ C @ D2 ) ) ) ) ) ) ).
% mult_mono_nonpos_nonneg
thf(fact_760_finite__Collect__le__nat,axiom,
! [K3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ N @ K3 ) ) ) ).
% finite_Collect_le_nat
thf(fact_761_mult__mono__nonpos__nonpos,axiom,
! [C: real,A: real,D2: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ D2 @ B )
=> ( ( ord_less_eq_real @ D2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ C @ D2 ) ) ) ) ) ) ).
% mult_mono_nonpos_nonpos
thf(fact_762_measurable__completion,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N2: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N2 ) )
=> ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( comple3506806835435775778n_real @ M ) @ N2 ) ) ) ).
% measurable_completion
thf(fact_763_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_764_borel__measurable__const,axiom,
! [C: real,M: sigma_measure_a] :
( member_a_real
@ ^ [X: a] : C
@ ( sigma_9116425665531756122a_real @ M @ borel_5078946678739801102l_real ) ) ).
% borel_measurable_const
thf(fact_765_borel__measurable__const,axiom,
! [C: extend8495563244428889912nnreal,M: sigma_measure_real] :
( member2919562650594848410nnreal
@ ^ [X: real] : C
@ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ).
% borel_measurable_const
thf(fact_766_mult__mono__nonneg__nonpos,axiom,
! [A: real,C: real,D2: real,B: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ D2 )
=> ( ( ord_less_eq_real @ D2 @ B )
=> ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ D2 @ C ) ) ) ) ) ) ).
% mult_mono_nonneg_nonpos
thf(fact_767_diff__is__0__eq,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_768_diff__is__0__eq_H,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_769_diff__diff__cancel,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_770_is__borel__def,axiom,
( borel_4993665998515044718a_real
= ( ^ [F3: a > real,M4: sigma_measure_a] : ( member_a_real @ F3 @ ( sigma_9116425665531756122a_real @ M4 @ borel_5078946678739801102l_real ) ) ) ) ).
% is_borel_def
thf(fact_771_is__borel__def,axiom,
( borel_3656262399657348386nnreal
= ( ^ [F3: real > extend8495563244428889912nnreal,M4: sigma_measure_real] : ( member2919562650594848410nnreal @ F3 @ ( sigma_9017504469962657078nnreal @ M4 @ borel_6524799422816628122nnreal ) ) ) ) ).
% is_borel_def
thf(fact_772_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: nat > sigma_measure_a,X5: nat > a > a,I3: set_nat,Y5: nat > a > real,N2: nat > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k6325968634923510307_nat_a @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k6744671002923369159t_real @ M @ K3 @ N2
@ ^ [I2: nat,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_773_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: b > sigma_measure_a,X5: b > a > a,I3: set_b,Y5: b > a > real,N2: b > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k6574085460301583240_a_b_a @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k7188637637266226338b_real @ M @ K3 @ N2
@ ^ [I2: b,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_774_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: a > sigma_measure_a,X5: a > a > a,I3: set_a,Y5: a > a > real,N2: a > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k138169005419483465_a_a_a @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k1592419022078095649a_real @ M @ K3 @ N2
@ ^ [I2: a,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_775_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: extend8495563244428889912nnreal > sigma_measure_a,X5: extend8495563244428889912nnreal > a > a,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > a > real,N2: extend8495563244428889912nnreal > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k282204728494123195real_a @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k3587330016639965743l_real @ M @ K3 @ N2
@ ^ [I2: extend8495563244428889912nnreal,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_776_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: real > sigma_measure_a,X5: real > a > a,I3: set_real,Y5: real > a > real,N2: real > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k2222437789551045447real_a @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k4854350642655679139l_real @ M @ K3 @ N2
@ ^ [I2: real,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_777_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: nat > sigma_measure_real,X5: nat > a > real,I3: set_nat,Y5: nat > real > extend8495563244428889912nnreal,N2: nat > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k6744671002923369159t_real @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k3632087671423262931nnreal @ M @ K3 @ N2
@ ^ [I2: nat,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_778_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: b > sigma_measure_real,X5: b > a > real,I3: set_b,Y5: b > real > extend8495563244428889912nnreal,N2: b > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k7188637637266226338b_real @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k3585524288665042990nnreal @ M @ K3 @ N2
@ ^ [I2: b,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_779_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: a > sigma_measure_real,X5: a > a > real,I3: set_a,Y5: a > real > extend8495563244428889912nnreal,N2: a > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k1592419022078095649a_real @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k6689048371381128237nnreal @ M @ K3 @ N2
@ ^ [I2: a,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_780_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: extend8495563244428889912nnreal > sigma_measure_real,X5: extend8495563244428889912nnreal > a > real,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,N2: extend8495563244428889912nnreal > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k3587330016639965743l_real @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k5415967778188962875nnreal @ M @ K3 @ N2
@ ^ [I2: extend8495563244428889912nnreal,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_781_prob__space_Ok__wise__indep__vars__compose,axiom,
! [M: sigma_measure_a,K3: nat,M2: real > sigma_measure_real,X5: real > a > real,I3: set_real,Y5: real > real > extend8495563244428889912nnreal,N2: real > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( prob_k4854350642655679139l_real @ M @ K3 @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( prob_k5555076286090509999nnreal @ M @ K3 @ N2
@ ^ [I2: real,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.k_wise_indep_vars_compose
thf(fact_782_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_783_order__refl,axiom,
! [X2: real] : ( ord_less_eq_real @ X2 @ X2 ) ).
% order_refl
thf(fact_784_order__refl,axiom,
! [X2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ X2 @ X2 ) ).
% order_refl
thf(fact_785_order__refl,axiom,
! [X2: set_set_a] : ( ord_le3724670747650509150_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_786_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_787_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_788_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_789_dual__order_Orefl,axiom,
! [A: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ A @ A ) ).
% dual_order.refl
thf(fact_790_dual__order_Orefl,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_791_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_792_mult__cancel2,axiom,
! [M3: nat,K3: nat,N3: nat] :
( ( ( times_times_nat @ M3 @ K3 )
= ( times_times_nat @ N3 @ K3 ) )
= ( ( M3 = N3 )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_793_mult__cancel1,axiom,
! [K3: nat,M3: nat,N3: nat] :
( ( ( times_times_nat @ K3 @ M3 )
= ( times_times_nat @ K3 @ N3 ) )
= ( ( M3 = N3 )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_794_mult__0__right,axiom,
! [M3: nat] :
( ( times_times_nat @ M3 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_795_mult__is__0,axiom,
! [M3: nat,N3: nat] :
( ( ( times_times_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
| ( N3 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_796_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_797_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_798_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_799_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_800_mult__0,axiom,
! [N3: nat] :
( ( times_times_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% mult_0
thf(fact_801_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_802_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_803_nle__le,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ~ ( ord_le3935885782089961368nnreal @ A @ B ) )
= ( ( ord_le3935885782089961368nnreal @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_804_le__cases3,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_805_le__cases3,axiom,
! [X2: real,Y3: real,Z2: real] :
( ( ( ord_less_eq_real @ X2 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X2 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y3 )
=> ~ ( ord_less_eq_real @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_806_le__cases3,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ~ ( ord_le3935885782089961368nnreal @ Y3 @ Z2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ~ ( ord_le3935885782089961368nnreal @ X2 @ Z2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ X2 @ Z2 )
=> ~ ( ord_le3935885782089961368nnreal @ Z2 @ Y3 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Z2 @ Y3 )
=> ~ ( ord_le3935885782089961368nnreal @ Y3 @ X2 ) )
=> ( ( ( ord_le3935885782089961368nnreal @ Y3 @ Z2 )
=> ~ ( ord_le3935885782089961368nnreal @ Z2 @ X2 ) )
=> ~ ( ( ord_le3935885782089961368nnreal @ Z2 @ X2 )
=> ~ ( ord_le3935885782089961368nnreal @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_807_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_808_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_809_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y4 = Z ) )
= ( ^ [X: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X @ Y )
& ( ord_le3935885782089961368nnreal @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_810_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [X: set_set_a,Y: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X @ Y )
& ( ord_le3724670747650509150_set_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_811_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_812_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_813_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_814_ord__eq__le__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A = B )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_815_ord__eq__le__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( A = B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_816_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_817_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_818_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_819_ord__le__eq__trans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( B = C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_820_ord__le__eq__trans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_821_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_822_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_823_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_824_order__antisym,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_825_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_826_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_827_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_828_order_Otrans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_829_order_Otrans,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ord_le3935885782089961368nnreal @ A @ C ) ) ) ).
% order.trans
thf(fact_830_order_Otrans,axiom,
! [A: set_set_a,B: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ C )
=> ( ord_le3724670747650509150_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_831_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_832_order__trans,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_833_order__trans,axiom,
! [X2: real,Y3: real,Z2: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ( ord_less_eq_real @ Y3 @ Z2 )
=> ( ord_less_eq_real @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_834_order__trans,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal,Z2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ( ord_le3935885782089961368nnreal @ Y3 @ Z2 )
=> ( ord_le3935885782089961368nnreal @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_835_order__trans,axiom,
! [X2: set_set_a,Y3: set_set_a,Z2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( ord_le3724670747650509150_set_a @ Y3 @ Z2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_836_order__trans,axiom,
! [X2: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( ord_less_eq_set_a @ Y3 @ Z2 )
=> ( ord_less_eq_set_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_837_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_838_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_eq_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_839_linorder__wlog,axiom,
! [P: extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o,A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ! [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: extend8495563244428889912nnreal,B3: extend8495563244428889912nnreal] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_840_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A3 )
& ( ord_less_eq_nat @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_841_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A3: real,B5: real] :
( ( ord_less_eq_real @ B5 @ A3 )
& ( ord_less_eq_real @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_842_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y4 = Z ) )
= ( ^ [A3: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B5 @ A3 )
& ( ord_le3935885782089961368nnreal @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_843_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B5 @ A3 )
& ( ord_le3724670747650509150_set_a @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_844_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_845_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_846_dual__order_Oantisym,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_847_dual__order_Oantisym,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_848_dual__order_Oantisym,axiom,
! [B: set_set_a,A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_849_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_850_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_851_dual__order_Otrans,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_852_dual__order_Otrans,axiom,
! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( ( ord_le3935885782089961368nnreal @ C @ B )
=> ( ord_le3935885782089961368nnreal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_853_dual__order_Otrans,axiom,
! [B: set_set_a,A: set_set_a,C: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( ( ord_le3724670747650509150_set_a @ C @ B )
=> ( ord_le3724670747650509150_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_854_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_855_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_856_antisym,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_857_antisym,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_858_antisym,axiom,
! [A: set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B )
=> ( ( ord_le3724670747650509150_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_859_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_860_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A3: nat,B5: nat] :
( ( ord_less_eq_nat @ A3 @ B5 )
& ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_861_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
= ( ^ [A3: real,B5: real] :
( ( ord_less_eq_real @ A3 @ B5 )
& ( ord_less_eq_real @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_862_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: extend8495563244428889912nnreal,Z: extend8495563244428889912nnreal] : ( Y4 = Z ) )
= ( ^ [A3: extend8495563244428889912nnreal,B5: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A3 @ B5 )
& ( ord_le3935885782089961368nnreal @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_863_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_set_a,Z: set_set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_set_a,B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_864_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A3: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A3 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_865_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ 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_866_order__subst1,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ 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_867_order__subst1,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_868_order__subst1,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ 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_869_order__subst1,axiom,
! [A: real,F: real > real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ 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_870_order__subst1,axiom,
! [A: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ 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: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_872_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_873_order__subst1,axiom,
! [A: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_874_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ 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_875_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ 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_876_order__subst2,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ 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_877_order__subst2,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_878_order__subst2,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ 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_879_order__subst2,axiom,
! [A: real,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ 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_880_order__subst2,axiom,
! [A: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_881_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_882_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_less_eq_real @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_883_order__subst2,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ord_le3935885782089961368nnreal @ ( F @ B ) @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_884_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ 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_885_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_886_order__eq__refl,axiom,
! [X2: real,Y3: real] :
( ( X2 = Y3 )
=> ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_887_order__eq__refl,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( X2 = Y3 )
=> ( ord_le3935885782089961368nnreal @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_888_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_889_order__eq__refl,axiom,
! [X2: set_a,Y3: set_a] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_890_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_891_linorder__linear,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
| ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_892_linorder__linear,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
| ( ord_le3935885782089961368nnreal @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_893_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ 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_894_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ 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_895_ord__eq__le__subst,axiom,
! [A: extend8495563244428889912nnreal,F: nat > extend8495563244428889912nnreal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_896_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ 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,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ 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: extend8495563244428889912nnreal,F: real > extend8495563244428889912nnreal,B: real,C: real] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_899_ord__eq__le__subst,axiom,
! [A: nat,F: extend8495563244428889912nnreal > nat,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_900_ord__eq__le__subst,axiom,
! [A: real,F: extend8495563244428889912nnreal > real,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_901_ord__eq__le__subst,axiom,
! [A: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le3935885782089961368nnreal @ B @ C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_902_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ 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_903_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= 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_904_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= 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_905_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_906_ord__le__eq__subst,axiom,
! [A: real,B: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= 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,B: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= 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: real,B: real,F: real > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_909_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > nat,C: nat] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_910_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > real,C: real] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_911_ord__le__eq__subst,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ X3 @ Y2 )
=> ( ord_le3935885782089961368nnreal @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le3935885782089961368nnreal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_912_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= 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_913_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_914_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_915_linorder__le__cases,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ~ ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ 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: real,X2: real] :
( ( ord_less_eq_real @ Y3 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_918_order__antisym__conv,axiom,
! [Y3: extend8495563244428889912nnreal,X2: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ Y3 @ X2 )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_919_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_920_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_921_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_922_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_923_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_924_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_925_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_926_le__trans,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K3 )
=> ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_927_eq__imp__le,axiom,
! [M3: nat,N3: nat] :
( ( M3 = N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% eq_imp_le
thf(fact_928_le__antisym,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M3 )
=> ( M3 = N3 ) ) ) ).
% le_antisym
thf(fact_929_nat__le__linear,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
| ( ord_less_eq_nat @ N3 @ M3 ) ) ).
% nat_le_linear
thf(fact_930_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K3: nat,B: nat] :
( ( P @ K3 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_931_mult__le__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ).
% mult_le_mono2
thf(fact_932_mult__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ).
% mult_le_mono1
thf(fact_933_mult__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_934_le__square,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ).
% le_square
thf(fact_935_le__cube,axiom,
! [M3: nat] : ( ord_less_eq_nat @ M3 @ ( times_times_nat @ M3 @ ( times_times_nat @ M3 @ M3 ) ) ) ).
% le_cube
thf(fact_936_diff__mult__distrib2,axiom,
! [K3: nat,M3: nat,N3: nat] :
( ( times_times_nat @ K3 @ ( minus_minus_nat @ M3 @ N3 ) )
= ( minus_minus_nat @ ( times_times_nat @ K3 @ M3 ) @ ( times_times_nat @ K3 @ N3 ) ) ) ).
% diff_mult_distrib2
thf(fact_937_diff__mult__distrib,axiom,
! [M3: nat,N3: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M3 @ N3 ) @ K3 )
= ( minus_minus_nat @ ( times_times_nat @ M3 @ K3 ) @ ( times_times_nat @ N3 @ K3 ) ) ) ).
% diff_mult_distrib
thf(fact_938_diffs0__imp__equal,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M3 )
= zero_zero_nat )
=> ( M3 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_939_diff__commute,axiom,
! [I: nat,J: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K3 ) @ J ) ) ).
% diff_commute
thf(fact_940_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_941_diff__le__mono2,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_942_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_943_diff__le__self,axiom,
! [M3: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N3 ) @ M3 ) ).
% diff_le_self
thf(fact_944_diff__le__mono,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_945_Nat_Odiff__diff__eq,axiom,
! [K3: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K3 @ M3 )
=> ( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K3 ) @ ( minus_minus_nat @ N3 @ K3 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_946_le__diff__iff,axiom,
! [K3: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K3 @ M3 )
=> ( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K3 ) @ ( minus_minus_nat @ N3 @ K3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_947_eq__diff__iff,axiom,
! [K3: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K3 @ M3 )
=> ( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ( ( minus_minus_nat @ M3 @ K3 )
= ( minus_minus_nat @ N3 @ K3 ) )
= ( M3 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_948_indep__var__compose,axiom,
! [M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > real,N1: sigma_measure_real,Y22: a > real,N22: sigma_measure_real] :
( ( indepe2440653194691626188ar_a_a @ m @ M1 @ X1 @ M22 @ X22 )
=> ( ( member_a_real @ Y1 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
=> ( ( member_a_real @ Y22 @ ( sigma_9116425665531756122a_real @ M22 @ N22 ) )
=> ( indepe8958435565499147358a_real @ m @ N1 @ ( comp_a_real_a @ Y1 @ X1 ) @ N22 @ ( comp_a_real_a @ Y22 @ X22 ) ) ) ) ) ).
% indep_var_compose
thf(fact_949_indep__var__compose,axiom,
! [M1: sigma_measure_real,X1: a > real,M22: sigma_measure_real,X22: a > real,Y1: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y22: real > extend8495563244428889912nnreal,N22: sigma_7234349610311085201nnreal] :
( ( indepe8958435565499147358a_real @ m @ M1 @ X1 @ M22 @ X22 )
=> ( ( member2919562650594848410nnreal @ Y1 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
=> ( ( member2919562650594848410nnreal @ Y22 @ ( sigma_9017504469962657078nnreal @ M22 @ N22 ) )
=> ( indepe3534117692041274858nnreal @ m @ N1 @ ( comp_r7806941060661185781real_a @ Y1 @ X1 ) @ N22 @ ( comp_r7806941060661185781real_a @ Y22 @ X22 ) ) ) ) ) ).
% indep_var_compose
thf(fact_950_pred__subset__eq,axiom,
! [R2: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R2 )
@ ^ [X: nat] : ( member_nat @ X @ S2 ) )
= ( ord_less_eq_set_nat @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_951_pred__subset__eq,axiom,
! [R2: set_re5328672808648366137nnreal,S2: set_re5328672808648366137nnreal] :
( ( ord_le8425755247534135084real_o
@ ^ [X: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X @ R2 )
@ ^ [X: real > extend8495563244428889912nnreal] : ( member2919562650594848410nnreal @ X @ S2 ) )
= ( ord_le2462468573666744473nnreal @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_952_pred__subset__eq,axiom,
! [R2: set_a_real,S2: set_a_real] :
( ( ord_less_eq_a_real_o
@ ^ [X: a > real] : ( member_a_real @ X @ R2 )
@ ^ [X: a > real] : ( member_a_real @ X @ S2 ) )
= ( ord_le3334967407727675675a_real @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_953_pred__subset__eq,axiom,
! [R2: set_b,S2: set_b] :
( ( ord_less_eq_b_o
@ ^ [X: b] : ( member_b @ X @ R2 )
@ ^ [X: b] : ( member_b @ X @ S2 ) )
= ( ord_less_eq_set_b @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_954_pred__subset__eq,axiom,
! [R2: set_Ex3793607809372303086nnreal,S2: set_Ex3793607809372303086nnreal] :
( ( ord_le7025323315894483639real_o
@ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ R2 )
@ ^ [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ S2 ) )
= ( ord_le6787938422905777998nnreal @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_955_pred__subset__eq,axiom,
! [R2: set_real,S2: set_real] :
( ( ord_less_eq_real_o
@ ^ [X: real] : ( member_real @ X @ R2 )
@ ^ [X: real] : ( member_real @ X @ S2 ) )
= ( ord_less_eq_set_real @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_956_pred__subset__eq,axiom,
! [R2: set_set_a,S2: set_set_a] :
( ( ord_less_eq_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ R2 )
@ ^ [X: set_a] : ( member_set_a @ X @ S2 ) )
= ( ord_le3724670747650509150_set_a @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_957_pred__subset__eq,axiom,
! [R2: set_a,S2: set_a] :
( ( ord_less_eq_a_o
@ ^ [X: a] : ( member_a @ X @ R2 )
@ ^ [X: a] : ( member_a @ X @ S2 ) )
= ( ord_less_eq_set_a @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_958_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_959_prob__space__distr,axiom,
! [F: a > real,M2: sigma_measure_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ M2 ) )
=> ( probab535871623910865577e_real @ ( measure_distr_a_real @ m @ M2 @ F ) ) ) ).
% prob_space_distr
thf(fact_960_prob__space__distr,axiom,
! [F: a > a,M2: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ M2 ) )
=> ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ m @ M2 @ F ) ) ) ).
% prob_space_distr
thf(fact_961_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M5: nat] :
? [N: nat] :
( ( ord_less_eq_nat @ M5 @ N )
& ( member_nat @ N @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_962_distr__completion,axiom,
! [X5: real > extend8495563244428889912nnreal,M: sigma_measure_real,N2: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ X5 @ ( sigma_9017504469962657078nnreal @ M @ N2 ) )
=> ( ( measur8829990298702910942nnreal @ ( comple3506806835435775778n_real @ M ) @ N2 @ X5 )
= ( measur8829990298702910942nnreal @ M @ N2 @ X5 ) ) ) ).
% distr_completion
thf(fact_963_distr__completion,axiom,
! [X5: a > real,M: sigma_measure_a,N2: sigma_measure_real] :
( ( member_a_real @ X5 @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ( measure_distr_a_real @ ( comple3428971583294703880tion_a @ M ) @ N2 @ X5 )
= ( measure_distr_a_real @ M @ N2 @ X5 ) ) ) ).
% distr_completion
thf(fact_964_prob__space_Oprob__space__distr,axiom,
! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,M2: sigma_7234349610311085201nnreal] :
( ( probab535871623910865577e_real @ M )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ M2 ) )
=> ( probab6612481188548237749nnreal @ ( measur8829990298702910942nnreal @ M @ M2 @ F ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_965_prob__space_Oprob__space__distr,axiom,
! [M: sigma_measure_a,F: a > real,M2: sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M2 ) )
=> ( probab535871623910865577e_real @ ( measure_distr_a_real @ M @ M2 @ F ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_966_prob__space_Oprob__space__distr,axiom,
! [M: sigma_measure_a,F: a > a,M2: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ M2 ) )
=> ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ M @ M2 @ F ) ) ) ) ).
% prob_space.prob_space_distr
thf(fact_967_prob__space__distrD,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N2: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N2 ) )
=> ( ( probab6612481188548237749nnreal @ ( measur8829990298702910942nnreal @ M @ N2 @ F ) )
=> ( probab535871623910865577e_real @ M ) ) ) ).
% prob_space_distrD
thf(fact_968_prob__space__distrD,axiom,
! [F: a > real,M: sigma_measure_a,N2: sigma_measure_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ( probab535871623910865577e_real @ ( measure_distr_a_real @ M @ N2 @ F ) )
=> ( probab7247484486040049089pace_a @ M ) ) ) ).
% prob_space_distrD
thf(fact_969_prob__space__distrD,axiom,
! [F: a > a,M: sigma_measure_a,N2: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ( probab7247484486040049089pace_a @ ( measure_distr_a_a @ M @ N2 @ F ) )
=> ( probab7247484486040049089pace_a @ M ) ) ) ).
% prob_space_distrD
thf(fact_970_integrable__distr,axiom,
! [T2: a > real,M: sigma_measure_a,M2: sigma_measure_real,F: real > real] :
( ( member_a_real @ T2 @ ( sigma_9116425665531756122a_real @ M @ M2 ) )
=> ( ( bochne3340023020068487468l_real @ ( measure_distr_a_real @ M @ M2 @ T2 ) @ F )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : ( F @ ( T2 @ X ) ) ) ) ) ).
% integrable_distr
thf(fact_971_integrable__distr,axiom,
! [T2: a > a,M: sigma_measure_a,M2: sigma_measure_a,F: a > real] :
( ( member_a_a @ T2 @ ( sigma_measurable_a_a @ M @ M2 ) )
=> ( ( bochne2139062162225249880a_real @ ( measure_distr_a_a @ M @ M2 @ T2 ) @ F )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : ( F @ ( T2 @ X ) ) ) ) ) ).
% integrable_distr
thf(fact_972_measure__pmf_Oprob__space__distr,axiom,
! [F: a > real,M: probab3364570286911266904_pmf_a,M2: sigma_measure_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7257411610070727406_pmf_a @ M ) @ M2 ) )
=> ( probab535871623910865577e_real @ ( measure_distr_a_real @ ( probab7257411610070727406_pmf_a @ M ) @ M2 @ F ) ) ) ).
% measure_pmf.prob_space_distr
thf(fact_973_measure__pmf_Oprob__space__distr,axiom,
! [F: real > extend8495563244428889912nnreal,M: probab5323077797692357992f_real,M2: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( probab3584279362189474492f_real @ M ) @ M2 ) )
=> ( probab6612481188548237749nnreal @ ( measur8829990298702910942nnreal @ ( probab3584279362189474492f_real @ M ) @ M2 @ F ) ) ) ).
% measure_pmf.prob_space_distr
thf(fact_974_integrable__distr__eq,axiom,
! [G: real > extend8495563244428889912nnreal,M: sigma_measure_real,N2: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ N2 ) )
=> ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ N2 @ borel_5078946678739801102l_real ) )
=> ( ( bochne9025062821074728248l_real @ ( measur8829990298702910942nnreal @ M @ N2 @ G ) @ F )
= ( bochne3340023020068487468l_real @ M
@ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).
% integrable_distr_eq
thf(fact_975_integrable__distr__eq,axiom,
! [G: a > real,M: sigma_measure_a,N2: sigma_measure_real,F: real > real] :
( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N2 @ borel_5078946678739801102l_real ) )
=> ( ( bochne3340023020068487468l_real @ ( measure_distr_a_real @ M @ N2 @ G ) @ F )
= ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).
% integrable_distr_eq
thf(fact_976_integrable__distr__eq,axiom,
! [G: a > a,M: sigma_measure_a,N2: sigma_measure_a,F: a > real] :
( ( member_a_a @ G @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ N2 @ borel_5078946678739801102l_real ) )
=> ( ( bochne2139062162225249880a_real @ ( measure_distr_a_a @ M @ N2 @ G ) @ F )
= ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).
% integrable_distr_eq
thf(fact_977_integral__distr,axiom,
! [G: real > extend8495563244428889912nnreal,M: sigma_measure_real,N2: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ N2 ) )
=> ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ N2 @ borel_5078946678739801102l_real ) )
=> ( ( bochne2458729288719820649l_real @ ( measur8829990298702910942nnreal @ M @ N2 @ G ) @ F )
= ( bochne3715101410578510557l_real @ M
@ ^ [X: real] : ( F @ ( G @ X ) ) ) ) ) ) ).
% integral_distr
thf(fact_978_integral__distr,axiom,
! [G: a > real,M: sigma_measure_a,N2: sigma_measure_real,F: real > real] :
( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N2 @ borel_5078946678739801102l_real ) )
=> ( ( bochne3715101410578510557l_real @ ( measure_distr_a_real @ M @ N2 @ G ) @ F )
= ( bochne378719280626478695a_real @ M
@ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).
% integral_distr
thf(fact_979_integral__distr,axiom,
! [G: a > a,M: sigma_measure_a,N2: sigma_measure_a,F: a > real] :
( ( member_a_a @ G @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ N2 @ borel_5078946678739801102l_real ) )
=> ( ( bochne378719280626478695a_real @ ( measure_distr_a_a @ M @ N2 @ G ) @ F )
= ( bochne378719280626478695a_real @ M
@ ^ [X: a] : ( F @ ( G @ X ) ) ) ) ) ) ).
% integral_distr
thf(fact_980_prob__space_Oindep__var__compose,axiom,
! [M: sigma_measure_a,M1: sigma_measure_a,X1: a > a,M22: sigma_measure_a,X22: a > a,Y1: a > real,N1: sigma_measure_real,Y22: a > real,N22: sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe2440653194691626188ar_a_a @ M @ M1 @ X1 @ M22 @ X22 )
=> ( ( member_a_real @ Y1 @ ( sigma_9116425665531756122a_real @ M1 @ N1 ) )
=> ( ( member_a_real @ Y22 @ ( sigma_9116425665531756122a_real @ M22 @ N22 ) )
=> ( indepe8958435565499147358a_real @ M @ N1 @ ( comp_a_real_a @ Y1 @ X1 ) @ N22 @ ( comp_a_real_a @ Y22 @ X22 ) ) ) ) ) ) ).
% prob_space.indep_var_compose
thf(fact_981_prob__space_Oindep__var__compose,axiom,
! [M: sigma_measure_a,M1: sigma_measure_real,X1: a > real,M22: sigma_measure_real,X22: a > real,Y1: real > extend8495563244428889912nnreal,N1: sigma_7234349610311085201nnreal,Y22: real > extend8495563244428889912nnreal,N22: sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe8958435565499147358a_real @ M @ M1 @ X1 @ M22 @ X22 )
=> ( ( member2919562650594848410nnreal @ Y1 @ ( sigma_9017504469962657078nnreal @ M1 @ N1 ) )
=> ( ( member2919562650594848410nnreal @ Y22 @ ( sigma_9017504469962657078nnreal @ M22 @ N22 ) )
=> ( indepe3534117692041274858nnreal @ M @ N1 @ ( comp_r7806941060661185781real_a @ Y1 @ X1 ) @ N22 @ ( comp_r7806941060661185781real_a @ Y22 @ X22 ) ) ) ) ) ) ).
% prob_space.indep_var_compose
thf(fact_982_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_983_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M5: nat] :
! [X: nat] :
( ( member_nat @ X @ N5 )
=> ( ord_less_eq_nat @ X @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_984_distr__distr,axiom,
! [G: real > extend8495563244428889912nnreal,N2: sigma_measure_real,L2: sigma_7234349610311085201nnreal,F: a > real,M: sigma_measure_a] :
( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ N2 @ L2 ) )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ( measur8829990298702910942nnreal @ ( measure_distr_a_real @ M @ N2 @ F ) @ L2 @ G )
= ( measur4839436603801885502nnreal @ M @ L2 @ ( comp_r7806941060661185781real_a @ G @ F ) ) ) ) ) ).
% distr_distr
thf(fact_985_indep__vars__compose,axiom,
! [M2: nat > sigma_measure_a,X5: nat > a > a,I3: set_nat,Y5: nat > a > real,N2: nat > sigma_measure_real] :
( ( indepe3245197900929106294_nat_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3903564294488106548t_real @ m @ N2
@ ^ [I2: nat] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_986_indep__vars__compose,axiom,
! [M2: b > sigma_measure_a,X5: b > a > a,I3: set_b,Y5: b > a > real,N2: b > sigma_measure_real] :
( ( indepe7639357355105118965_a_b_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8265442546547513973b_real @ m @ N2
@ ^ [I2: b] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_987_indep__vars__compose,axiom,
! [M2: a > sigma_measure_a,X5: a > a > a,I3: set_a,Y5: a > a > real,N2: a > sigma_measure_real] :
( ( indepe1203440900223019190_a_a_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe2669223931359383284a_real @ m @ N2
@ ^ [I2: a] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_988_indep__vars__compose,axiom,
! [M2: extend8495563244428889912nnreal > sigma_measure_a,X5: extend8495563244428889912nnreal > a > a,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > a > real,N2: extend8495563244428889912nnreal > sigma_measure_real] :
( ( indepe432285032990533646real_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3928765670445028764l_real @ m @ N2
@ ^ [I2: extend8495563244428889912nnreal] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_989_indep__vars__compose,axiom,
! [M2: real > sigma_measure_a,X5: real > a > a,I3: set_real,Y5: real > a > real,N2: real > sigma_measure_real] :
( ( indepe3299242698832333082real_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe7095277112020832144l_real @ m @ N2
@ ^ [I2: real] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_990_indep__vars__compose,axiom,
! [M2: nat > sigma_measure_real,X5: nat > a > real,I3: set_nat,Y5: nat > real > extend8495563244428889912nnreal,N2: nat > sigma_7234349610311085201nnreal] :
( ( indepe3903564294488106548t_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8764986580812350144nnreal @ m @ N2
@ ^ [I2: nat] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_991_indep__vars__compose,axiom,
! [M2: b > sigma_measure_real,X5: b > a > real,I3: set_b,Y5: b > real > extend8495563244428889912nnreal,N2: b > sigma_7234349610311085201nnreal] :
( ( indepe8265442546547513973b_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3735604593161453441nnreal @ m @ N2
@ ^ [I2: b] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_992_indep__vars__compose,axiom,
! [M2: a > sigma_measure_real,X5: a > a > real,I3: set_a,Y5: a > real > extend8495563244428889912nnreal,N2: a > sigma_7234349610311085201nnreal] :
( ( indepe2669223931359383284a_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe6839128675877538688nnreal @ m @ N2
@ ^ [I2: a] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_993_indep__vars__compose,axiom,
! [M2: extend8495563244428889912nnreal > sigma_measure_real,X5: extend8495563244428889912nnreal > a > real,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,N2: extend8495563244428889912nnreal > sigma_7234349610311085201nnreal] :
( ( indepe3928765670445028764l_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8330679734821070376nnreal @ m @ N2
@ ^ [I2: extend8495563244428889912nnreal] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_994_indep__vars__compose,axiom,
! [M2: real > sigma_measure_real,X5: real > a > real,I3: set_real,Y5: real > real > extend8495563244428889912nnreal,N2: real > sigma_7234349610311085201nnreal] :
( ( indepe7095277112020832144l_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe5896511939895573020nnreal @ m @ N2
@ ^ [I2: real] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ).
% indep_vars_compose
thf(fact_995_finite__measure__distr,axiom,
! [F: a > real,M2: sigma_measure_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ M2 ) )
=> ( measur3606880022600206024e_real @ ( measure_distr_a_real @ m @ M2 @ F ) ) ) ).
% finite_measure_distr
thf(fact_996_finite__measure__distr,axiom,
! [F: a > a,M2: sigma_measure_a] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ M2 ) )
=> ( measur930452917991658466sure_a @ ( measure_distr_a_a @ m @ M2 @ F ) ) ) ).
% finite_measure_distr
thf(fact_997_increasingD,axiom,
! [M: set_set_set_a,F: set_set_a > nat,X2: set_set_a,Y3: set_set_a] :
( ( measur1244951900059291067_a_nat @ M @ F )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( member_set_set_a @ X2 @ M )
=> ( ( member_set_set_a @ Y3 @ M )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_998_increasingD,axiom,
! [M: set_set_set_a,F: set_set_a > real,X2: set_set_a,Y3: set_set_a] :
( ( measur2331856671108623127a_real @ M @ F )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( member_set_set_a @ X2 @ M )
=> ( ( member_set_set_a @ Y3 @ M )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_999_increasingD,axiom,
! [M: set_set_set_a,F: set_set_a > extend8495563244428889912nnreal,X2: set_set_a,Y3: set_set_a] :
( ( measur1771626496591458595nnreal @ M @ F )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( member_set_set_a @ X2 @ M )
=> ( ( member_set_set_a @ Y3 @ M )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_1000_increasingD,axiom,
! [M: set_set_set_a,F: set_set_a > set_set_a,X2: set_set_a,Y3: set_set_a] :
( ( measur2197171192767378579_set_a @ M @ F )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( member_set_set_a @ X2 @ M )
=> ( ( member_set_set_a @ Y3 @ M )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_1001_increasingD,axiom,
! [M: set_set_set_a,F: set_set_a > set_a,X2: set_set_a,Y3: set_set_a] :
( ( measur5181028491126448947_set_a @ M @ F )
=> ( ( ord_le3724670747650509150_set_a @ X2 @ Y3 )
=> ( ( member_set_set_a @ X2 @ M )
=> ( ( member_set_set_a @ Y3 @ M )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_1002_increasingD,axiom,
! [M: set_set_a,F: set_a > nat,X2: set_a,Y3: set_a] :
( ( measur8151441426001876059_a_nat @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_1003_increasingD,axiom,
! [M: set_set_a,F: set_a > real,X2: set_a,Y3: set_a] :
( ( measur1776380161843274167a_real @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_1004_increasingD,axiom,
! [M: set_set_a,F: set_a > extend8495563244428889912nnreal,X2: set_a,Y3: set_a] :
( ( measur5393715408109795267nnreal @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_le3935885782089961368nnreal @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_1005_increasingD,axiom,
! [M: set_set_a,F: set_a > set_set_a,X2: set_a,Y3: set_a] :
( ( measur8202069185322079731_set_a @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_le3724670747650509150_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_1006_increasingD,axiom,
! [M: set_set_a,F: set_a > set_a,X2: set_a,Y3: set_a] :
( ( measur7842569353079325843_set_a @ M @ F )
=> ( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ( member_set_a @ X2 @ M )
=> ( ( member_set_a @ Y3 @ M )
=> ( ord_less_eq_set_a @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% increasingD
thf(fact_1007_increasing__def,axiom,
( measur1244951900059291067_a_nat
= ( ^ [M4: set_set_set_a,Mu: set_set_a > nat] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M4 )
=> ! [Y: set_set_a] :
( ( member_set_set_a @ Y @ M4 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ord_less_eq_nat @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1008_increasing__def,axiom,
( measur2331856671108623127a_real
= ( ^ [M4: set_set_set_a,Mu: set_set_a > real] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M4 )
=> ! [Y: set_set_a] :
( ( member_set_set_a @ Y @ M4 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ord_less_eq_real @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1009_increasing__def,axiom,
( measur1771626496591458595nnreal
= ( ^ [M4: set_set_set_a,Mu: set_set_a > extend8495563244428889912nnreal] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M4 )
=> ! [Y: set_set_a] :
( ( member_set_set_a @ Y @ M4 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ord_le3935885782089961368nnreal @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1010_increasing__def,axiom,
( measur2197171192767378579_set_a
= ( ^ [M4: set_set_set_a,Mu: set_set_a > set_set_a] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M4 )
=> ! [Y: set_set_a] :
( ( member_set_set_a @ Y @ M4 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ord_le3724670747650509150_set_a @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1011_increasing__def,axiom,
( measur5181028491126448947_set_a
= ( ^ [M4: set_set_set_a,Mu: set_set_a > set_a] :
! [X: set_set_a] :
( ( member_set_set_a @ X @ M4 )
=> ! [Y: set_set_a] :
( ( member_set_set_a @ Y @ M4 )
=> ( ( ord_le3724670747650509150_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1012_increasing__def,axiom,
( measur8151441426001876059_a_nat
= ( ^ [M4: set_set_a,Mu: set_a > nat] :
! [X: set_a] :
( ( member_set_a @ X @ M4 )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ M4 )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_nat @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1013_increasing__def,axiom,
( measur1776380161843274167a_real
= ( ^ [M4: set_set_a,Mu: set_a > real] :
! [X: set_a] :
( ( member_set_a @ X @ M4 )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ M4 )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_real @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1014_increasing__def,axiom,
( measur5393715408109795267nnreal
= ( ^ [M4: set_set_a,Mu: set_a > extend8495563244428889912nnreal] :
! [X: set_a] :
( ( member_set_a @ X @ M4 )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ M4 )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_le3935885782089961368nnreal @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1015_increasing__def,axiom,
( measur8202069185322079731_set_a
= ( ^ [M4: set_set_a,Mu: set_a > set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ M4 )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ M4 )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_le3724670747650509150_set_a @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1016_increasing__def,axiom,
( measur7842569353079325843_set_a
= ( ^ [M4: set_set_a,Mu: set_a > set_a] :
! [X: set_a] :
( ( member_set_a @ X @ M4 )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ M4 )
=> ( ( ord_less_eq_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ ( Mu @ X ) @ ( Mu @ Y ) ) ) ) ) ) ) ).
% increasing_def
thf(fact_1017_nat__mult__eq__cancel__disj,axiom,
! [K3: nat,M3: nat,N3: nat] :
( ( ( times_times_nat @ K3 @ M3 )
= ( times_times_nat @ K3 @ N3 ) )
= ( ( K3 = zero_zero_nat )
| ( M3 = N3 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1018_measurable__comp,axiom,
! [F: a > a,M: sigma_measure_a,N2: sigma_measure_a,G: a > real,L2: sigma_measure_real] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ N2 @ L2 ) )
=> ( member_a_real @ ( comp_a_real_a @ G @ F ) @ ( sigma_9116425665531756122a_real @ M @ L2 ) ) ) ) ).
% measurable_comp
thf(fact_1019_measurable__comp,axiom,
! [F: real > real,M: sigma_measure_real,N2: sigma_measure_real,G: real > extend8495563244428889912nnreal,L2: sigma_7234349610311085201nnreal] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N2 ) )
=> ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ N2 @ L2 ) )
=> ( member2919562650594848410nnreal @ ( comp_r6279034453215524981l_real @ G @ F ) @ ( sigma_9017504469962657078nnreal @ M @ L2 ) ) ) ) ).
% measurable_comp
thf(fact_1020_measurable__comp,axiom,
! [F: a > real,M: sigma_measure_a,N2: sigma_measure_real,G: real > real,L2: sigma_measure_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N2 @ L2 ) )
=> ( member_a_real @ ( comp_real_real_a @ G @ F ) @ ( sigma_9116425665531756122a_real @ M @ L2 ) ) ) ) ).
% measurable_comp
thf(fact_1021_measurable__comp,axiom,
! [F: a > real,M: sigma_measure_a,N2: sigma_measure_real,G: real > extend8495563244428889912nnreal,L2: sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ N2 @ L2 ) )
=> ( member298456594901751504nnreal @ ( comp_r7806941060661185781real_a @ G @ F ) @ ( sigma_214952329563889126nnreal @ M @ L2 ) ) ) ) ).
% measurable_comp
thf(fact_1022_measurable__comp,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N2: sigma_7234349610311085201nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L2: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N2 ) )
=> ( ( member8329810500450651686nnreal @ G @ ( sigma_7926153774531450434nnreal @ N2 @ L2 ) )
=> ( member2919562650594848410nnreal @ ( comp_E4178961840025359489l_real @ G @ F ) @ ( sigma_9017504469962657078nnreal @ M @ L2 ) ) ) ) ).
% measurable_comp
thf(fact_1023_finite__measure__axioms,axiom,
measur930452917991658466sure_a @ m ).
% finite_measure_axioms
thf(fact_1024_indep__vars__compose2,axiom,
! [M2: nat > sigma_measure_a,X5: nat > a > a,I3: set_nat,Y5: nat > a > real,N2: nat > sigma_measure_real] :
( ( indepe3245197900929106294_nat_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3903564294488106548t_real @ m @ N2
@ ^ [I2: nat,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1025_indep__vars__compose2,axiom,
! [M2: b > sigma_measure_a,X5: b > a > a,I3: set_b,Y5: b > a > real,N2: b > sigma_measure_real] :
( ( indepe7639357355105118965_a_b_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8265442546547513973b_real @ m @ N2
@ ^ [I2: b,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1026_indep__vars__compose2,axiom,
! [M2: a > sigma_measure_a,X5: a > a > a,I3: set_a,Y5: a > a > real,N2: a > sigma_measure_real] :
( ( indepe1203440900223019190_a_a_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe2669223931359383284a_real @ m @ N2
@ ^ [I2: a,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1027_indep__vars__compose2,axiom,
! [M2: extend8495563244428889912nnreal > sigma_measure_a,X5: extend8495563244428889912nnreal > a > a,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > a > real,N2: extend8495563244428889912nnreal > sigma_measure_real] :
( ( indepe432285032990533646real_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3928765670445028764l_real @ m @ N2
@ ^ [I2: extend8495563244428889912nnreal,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1028_indep__vars__compose2,axiom,
! [M2: real > sigma_measure_a,X5: real > a > a,I3: set_real,Y5: real > a > real,N2: real > sigma_measure_real] :
( ( indepe3299242698832333082real_a @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe7095277112020832144l_real @ m @ N2
@ ^ [I2: real,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1029_indep__vars__compose2,axiom,
! [M2: nat > sigma_measure_real,X5: nat > a > real,I3: set_nat,Y5: nat > real > extend8495563244428889912nnreal,N2: nat > sigma_7234349610311085201nnreal] :
( ( indepe3903564294488106548t_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8764986580812350144nnreal @ m @ N2
@ ^ [I2: nat,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1030_indep__vars__compose2,axiom,
! [M2: b > sigma_measure_real,X5: b > a > real,I3: set_b,Y5: b > real > extend8495563244428889912nnreal,N2: b > sigma_7234349610311085201nnreal] :
( ( indepe8265442546547513973b_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3735604593161453441nnreal @ m @ N2
@ ^ [I2: b,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1031_indep__vars__compose2,axiom,
! [M2: a > sigma_measure_real,X5: a > a > real,I3: set_a,Y5: a > real > extend8495563244428889912nnreal,N2: a > sigma_7234349610311085201nnreal] :
( ( indepe2669223931359383284a_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe6839128675877538688nnreal @ m @ N2
@ ^ [I2: a,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1032_indep__vars__compose2,axiom,
! [M2: extend8495563244428889912nnreal > sigma_measure_real,X5: extend8495563244428889912nnreal > a > real,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,N2: extend8495563244428889912nnreal > sigma_7234349610311085201nnreal] :
( ( indepe3928765670445028764l_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8330679734821070376nnreal @ m @ N2
@ ^ [I2: extend8495563244428889912nnreal,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1033_indep__vars__compose2,axiom,
! [M2: real > sigma_measure_real,X5: real > a > real,I3: set_real,Y5: real > real > extend8495563244428889912nnreal,N2: real > sigma_7234349610311085201nnreal] :
( ( indepe7095277112020832144l_real @ m @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe5896511939895573020nnreal @ m @ N2
@ ^ [I2: real,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ).
% indep_vars_compose2
thf(fact_1034_indep__vars__distr,axiom,
! [F: a > a,N2: sigma_measure_a,I3: set_nat,X6: nat > a > real,M2: nat > sigma_measure_real] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ N2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe3903564294488106548t_real @ m @ M2
@ ^ [I2: nat] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe3903564294488106548t_real @ ( measure_distr_a_a @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1035_indep__vars__distr,axiom,
! [F: a > a,N2: sigma_measure_a,I3: set_b,X6: b > a > real,M2: b > sigma_measure_real] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ N2 ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe8265442546547513973b_real @ m @ M2
@ ^ [I2: b] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe8265442546547513973b_real @ ( measure_distr_a_a @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1036_indep__vars__distr,axiom,
! [F: a > a,N2: sigma_measure_a,I3: set_a,X6: a > a > real,M2: a > sigma_measure_real] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ N2 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe2669223931359383284a_real @ m @ M2
@ ^ [I2: a] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe2669223931359383284a_real @ ( measure_distr_a_a @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1037_indep__vars__distr,axiom,
! [F: a > a,N2: sigma_measure_a,I3: set_Ex3793607809372303086nnreal,X6: extend8495563244428889912nnreal > a > real,M2: extend8495563244428889912nnreal > sigma_measure_real] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ N2 ) )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe3928765670445028764l_real @ m @ M2
@ ^ [I2: extend8495563244428889912nnreal] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe3928765670445028764l_real @ ( measure_distr_a_a @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1038_indep__vars__distr,axiom,
! [F: a > a,N2: sigma_measure_a,I3: set_real,X6: real > a > real,M2: real > sigma_measure_real] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ m @ N2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe7095277112020832144l_real @ m @ M2
@ ^ [I2: real] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe7095277112020832144l_real @ ( measure_distr_a_a @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1039_indep__vars__distr,axiom,
! [F: a > real,N2: sigma_measure_real,I3: set_nat,X6: nat > real > extend8495563244428889912nnreal,M2: nat > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ N2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe8764986580812350144nnreal @ m @ M2
@ ^ [I2: nat] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe7397540011872310654nnreal @ ( measure_distr_a_real @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1040_indep__vars__distr,axiom,
! [F: a > real,N2: sigma_measure_real,I3: set_b,X6: b > real > extend8495563244428889912nnreal,M2: b > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ N2 ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe3735604593161453441nnreal @ m @ M2
@ ^ [I2: b] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe6789649827632570435nnreal @ ( measure_distr_a_real @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1041_indep__vars__distr,axiom,
! [F: a > real,N2: sigma_measure_real,I3: set_a,X6: a > real > extend8495563244428889912nnreal,M2: a > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ N2 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe6839128675877538688nnreal @ m @ M2
@ ^ [I2: a] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe669801873493879874nnreal @ ( measure_distr_a_real @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1042_indep__vars__distr,axiom,
! [F: a > real,N2: sigma_measure_real,I3: set_Ex3793607809372303086nnreal,X6: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,M2: extend8495563244428889912nnreal > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ N2 ) )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe8330679734821070376nnreal @ m @ M2
@ ^ [I2: extend8495563244428889912nnreal] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe2251227809676183782nnreal @ ( measure_distr_a_real @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1043_indep__vars__distr,axiom,
! [F: a > real,N2: sigma_measure_real,I3: set_real,X6: real > real > extend8495563244428889912nnreal,M2: real > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ m @ N2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe5896511939895573020nnreal @ m @ M2
@ ^ [I2: real] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe6989957652004097498nnreal @ ( measure_distr_a_real @ m @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% indep_vars_distr
thf(fact_1044_measure__pmf_Ofinite__measure__axioms,axiom,
! [M: probab3364570286911266904_pmf_a] : ( measur930452917991658466sure_a @ ( probab7257411610070727406_pmf_a @ M ) ) ).
% measure_pmf.finite_measure_axioms
thf(fact_1045_prob__space_Ofinite__measure,axiom,
! [M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( measur930452917991658466sure_a @ M ) ) ).
% prob_space.finite_measure
thf(fact_1046_subprob__space_Oaxioms_I1_J,axiom,
! [M: sigma_measure_a] :
( ( giry_subprob_space_a @ M )
=> ( measur930452917991658466sure_a @ M ) ) ).
% subprob_space.axioms(1)
thf(fact_1047_finite__measure_Ointegrable__const,axiom,
! [M: sigma_measure_a,A: real] :
( ( measur930452917991658466sure_a @ M )
=> ( bochne2139062162225249880a_real @ M
@ ^ [X: a] : A ) ) ).
% finite_measure.integrable_const
thf(fact_1048_finite__measure_Ofinite__measure__distr,axiom,
! [M: sigma_measure_real,F: real > extend8495563244428889912nnreal,M2: sigma_7234349610311085201nnreal] :
( ( measur3606880022600206024e_real @ M )
=> ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ M2 ) )
=> ( measur8478876643349974356nnreal @ ( measur8829990298702910942nnreal @ M @ M2 @ F ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_1049_finite__measure_Ofinite__measure__distr,axiom,
! [M: sigma_measure_a,F: a > real,M2: sigma_measure_real] :
( ( measur930452917991658466sure_a @ M )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ M2 ) )
=> ( measur3606880022600206024e_real @ ( measure_distr_a_real @ M @ M2 @ F ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_1050_finite__measure_Ofinite__measure__distr,axiom,
! [M: sigma_measure_a,F: a > a,M2: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ M2 ) )
=> ( measur930452917991658466sure_a @ ( measure_distr_a_a @ M @ M2 @ F ) ) ) ) ).
% finite_measure.finite_measure_distr
thf(fact_1051_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: nat > sigma_measure_a,X5: nat > a > a,I3: set_nat,Y5: nat > a > real,N2: nat > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe3245197900929106294_nat_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3903564294488106548t_real @ M @ N2
@ ^ [I2: nat,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1052_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: b > sigma_measure_a,X5: b > a > a,I3: set_b,Y5: b > a > real,N2: b > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe7639357355105118965_a_b_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8265442546547513973b_real @ M @ N2
@ ^ [I2: b,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1053_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: a > sigma_measure_a,X5: a > a > a,I3: set_a,Y5: a > a > real,N2: a > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe1203440900223019190_a_a_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe2669223931359383284a_real @ M @ N2
@ ^ [I2: a,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1054_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: extend8495563244428889912nnreal > sigma_measure_a,X5: extend8495563244428889912nnreal > a > a,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > a > real,N2: extend8495563244428889912nnreal > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe432285032990533646real_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3928765670445028764l_real @ M @ N2
@ ^ [I2: extend8495563244428889912nnreal,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1055_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: real > sigma_measure_a,X5: real > a > a,I3: set_real,Y5: real > a > real,N2: real > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe3299242698832333082real_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe7095277112020832144l_real @ M @ N2
@ ^ [I2: real,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1056_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: nat > sigma_measure_real,X5: nat > a > real,I3: set_nat,Y5: nat > real > extend8495563244428889912nnreal,N2: nat > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe3903564294488106548t_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8764986580812350144nnreal @ M @ N2
@ ^ [I2: nat,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1057_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: b > sigma_measure_real,X5: b > a > real,I3: set_b,Y5: b > real > extend8495563244428889912nnreal,N2: b > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe8265442546547513973b_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3735604593161453441nnreal @ M @ N2
@ ^ [I2: b,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1058_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: a > sigma_measure_real,X5: a > a > real,I3: set_a,Y5: a > real > extend8495563244428889912nnreal,N2: a > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe2669223931359383284a_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe6839128675877538688nnreal @ M @ N2
@ ^ [I2: a,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1059_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: extend8495563244428889912nnreal > sigma_measure_real,X5: extend8495563244428889912nnreal > a > real,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,N2: extend8495563244428889912nnreal > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe3928765670445028764l_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8330679734821070376nnreal @ M @ N2
@ ^ [I2: extend8495563244428889912nnreal,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1060_prob__space_Oindep__vars__compose2,axiom,
! [M: sigma_measure_a,M2: real > sigma_measure_real,X5: real > a > real,I3: set_real,Y5: real > real > extend8495563244428889912nnreal,N2: real > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe7095277112020832144l_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe5896511939895573020nnreal @ M @ N2
@ ^ [I2: real,X: a] : ( Y5 @ I2 @ ( X5 @ I2 @ X ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose2
thf(fact_1061_measure__pmf_Ofinite__measure__distr,axiom,
! [F: a > real,M: probab3364570286911266904_pmf_a,M2: sigma_measure_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7257411610070727406_pmf_a @ M ) @ M2 ) )
=> ( measur3606880022600206024e_real @ ( measure_distr_a_real @ ( probab7257411610070727406_pmf_a @ M ) @ M2 @ F ) ) ) ).
% measure_pmf.finite_measure_distr
thf(fact_1062_measure__pmf_Ofinite__measure__distr,axiom,
! [F: real > extend8495563244428889912nnreal,M: probab5323077797692357992f_real,M2: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ ( probab3584279362189474492f_real @ M ) @ M2 ) )
=> ( measur8478876643349974356nnreal @ ( measur8829990298702910942nnreal @ ( probab3584279362189474492f_real @ M ) @ M2 @ F ) ) ) ).
% measure_pmf.finite_measure_distr
thf(fact_1063_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: nat > sigma_measure_a,X5: nat > a > a,I3: set_nat,Y5: nat > a > real,N2: nat > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe3245197900929106294_nat_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3903564294488106548t_real @ M @ N2
@ ^ [I2: nat] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1064_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: b > sigma_measure_a,X5: b > a > a,I3: set_b,Y5: b > a > real,N2: b > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe7639357355105118965_a_b_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8265442546547513973b_real @ M @ N2
@ ^ [I2: b] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1065_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: a > sigma_measure_a,X5: a > a > a,I3: set_a,Y5: a > a > real,N2: a > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe1203440900223019190_a_a_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe2669223931359383284a_real @ M @ N2
@ ^ [I2: a] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1066_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: extend8495563244428889912nnreal > sigma_measure_a,X5: extend8495563244428889912nnreal > a > a,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > a > real,N2: extend8495563244428889912nnreal > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe432285032990533646real_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3928765670445028764l_real @ M @ N2
@ ^ [I2: extend8495563244428889912nnreal] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1067_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: real > sigma_measure_a,X5: real > a > a,I3: set_real,Y5: real > a > real,N2: real > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe3299242698832333082real_a @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member_a_real @ ( Y5 @ I4 ) @ ( sigma_9116425665531756122a_real @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe7095277112020832144l_real @ M @ N2
@ ^ [I2: real] : ( comp_a_real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1068_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: nat > sigma_measure_real,X5: nat > a > real,I3: set_nat,Y5: nat > real > extend8495563244428889912nnreal,N2: nat > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe3903564294488106548t_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8764986580812350144nnreal @ M @ N2
@ ^ [I2: nat] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1069_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: b > sigma_measure_real,X5: b > a > real,I3: set_b,Y5: b > real > extend8495563244428889912nnreal,N2: b > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe8265442546547513973b_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe3735604593161453441nnreal @ M @ N2
@ ^ [I2: b] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1070_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: a > sigma_measure_real,X5: a > a > real,I3: set_a,Y5: a > real > extend8495563244428889912nnreal,N2: a > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe2669223931359383284a_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe6839128675877538688nnreal @ M @ N2
@ ^ [I2: a] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1071_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: extend8495563244428889912nnreal > sigma_measure_real,X5: extend8495563244428889912nnreal > a > real,I3: set_Ex3793607809372303086nnreal,Y5: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,N2: extend8495563244428889912nnreal > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe3928765670445028764l_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe8330679734821070376nnreal @ M @ N2
@ ^ [I2: extend8495563244428889912nnreal] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1072_prob__space_Oindep__vars__compose,axiom,
! [M: sigma_measure_a,M2: real > sigma_measure_real,X5: real > a > real,I3: set_real,Y5: real > real > extend8495563244428889912nnreal,N2: real > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe7095277112020832144l_real @ M @ M2 @ X5 @ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( Y5 @ I4 ) @ ( sigma_9017504469962657078nnreal @ ( M2 @ I4 ) @ ( N2 @ I4 ) ) ) )
=> ( indepe5896511939895573020nnreal @ M @ N2
@ ^ [I2: real] : ( comp_r7806941060661185781real_a @ ( Y5 @ I2 ) @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ).
% prob_space.indep_vars_compose
thf(fact_1073_measurable__compose__rev,axiom,
! [F: real > real,L2: sigma_measure_real,N2: sigma_measure_real,G: a > real,M: sigma_measure_a] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ L2 @ N2 ) )
=> ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ L2 ) )
=> ( member_a_real
@ ^ [X: a] : ( F @ ( G @ X ) )
@ ( sigma_9116425665531756122a_real @ M @ N2 ) ) ) ) ).
% measurable_compose_rev
thf(fact_1074_measurable__compose__rev,axiom,
! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L2: sigma_7234349610311085201nnreal,N2: sigma_7234349610311085201nnreal,G: real > extend8495563244428889912nnreal,M: sigma_measure_real] :
( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ L2 @ N2 ) )
=> ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ L2 ) )
=> ( member2919562650594848410nnreal
@ ^ [X: real] : ( F @ ( G @ X ) )
@ ( sigma_9017504469962657078nnreal @ M @ N2 ) ) ) ) ).
% measurable_compose_rev
thf(fact_1075_measurable__compose__rev,axiom,
! [F: a > real,L2: sigma_measure_a,N2: sigma_measure_real,G: a > a,M: sigma_measure_a] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ L2 @ N2 ) )
=> ( ( member_a_a @ G @ ( sigma_measurable_a_a @ M @ L2 ) )
=> ( member_a_real
@ ^ [X: a] : ( F @ ( G @ X ) )
@ ( sigma_9116425665531756122a_real @ M @ N2 ) ) ) ) ).
% measurable_compose_rev
thf(fact_1076_measurable__compose__rev,axiom,
! [F: real > extend8495563244428889912nnreal,L2: sigma_measure_real,N2: sigma_7234349610311085201nnreal,G: real > real,M: sigma_measure_real] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ L2 @ N2 ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ L2 ) )
=> ( member2919562650594848410nnreal
@ ^ [X: real] : ( F @ ( G @ X ) )
@ ( sigma_9017504469962657078nnreal @ M @ N2 ) ) ) ) ).
% measurable_compose_rev
thf(fact_1077_measurable__compose__rev,axiom,
! [F: real > extend8495563244428889912nnreal,L2: sigma_measure_real,N2: sigma_7234349610311085201nnreal,G: a > real,M: sigma_measure_a] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ L2 @ N2 ) )
=> ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ M @ L2 ) )
=> ( member298456594901751504nnreal
@ ^ [X: a] : ( F @ ( G @ X ) )
@ ( sigma_214952329563889126nnreal @ M @ N2 ) ) ) ) ).
% measurable_compose_rev
thf(fact_1078_measurable__compose,axiom,
! [F: a > a,M: sigma_measure_a,N2: sigma_measure_a,G: a > real,L2: sigma_measure_real] :
( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ( member_a_real @ G @ ( sigma_9116425665531756122a_real @ N2 @ L2 ) )
=> ( member_a_real
@ ^ [X: a] : ( G @ ( F @ X ) )
@ ( sigma_9116425665531756122a_real @ M @ L2 ) ) ) ) ).
% measurable_compose
thf(fact_1079_measurable__compose,axiom,
! [F: real > real,M: sigma_measure_real,N2: sigma_measure_real,G: real > extend8495563244428889912nnreal,L2: sigma_7234349610311085201nnreal] :
( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N2 ) )
=> ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ N2 @ L2 ) )
=> ( member2919562650594848410nnreal
@ ^ [X: real] : ( G @ ( F @ X ) )
@ ( sigma_9017504469962657078nnreal @ M @ L2 ) ) ) ) ).
% measurable_compose
thf(fact_1080_measurable__compose,axiom,
! [F: a > real,M: sigma_measure_a,N2: sigma_measure_real,G: real > real,L2: sigma_measure_real] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N2 @ L2 ) )
=> ( member_a_real
@ ^ [X: a] : ( G @ ( F @ X ) )
@ ( sigma_9116425665531756122a_real @ M @ L2 ) ) ) ) ).
% measurable_compose
thf(fact_1081_measurable__compose,axiom,
! [F: a > real,M: sigma_measure_a,N2: sigma_measure_real,G: real > extend8495563244428889912nnreal,L2: sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ N2 @ L2 ) )
=> ( member298456594901751504nnreal
@ ^ [X: a] : ( G @ ( F @ X ) )
@ ( sigma_214952329563889126nnreal @ M @ L2 ) ) ) ) ).
% measurable_compose
thf(fact_1082_measurable__compose,axiom,
! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N2: sigma_7234349610311085201nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,L2: sigma_7234349610311085201nnreal] :
( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N2 ) )
=> ( ( member8329810500450651686nnreal @ G @ ( sigma_7926153774531450434nnreal @ N2 @ L2 ) )
=> ( member2919562650594848410nnreal
@ ^ [X: real] : ( G @ ( F @ X ) )
@ ( sigma_9017504469962657078nnreal @ M @ L2 ) ) ) ) ).
% measurable_compose
thf(fact_1083_measure__pmf_Oindep__vars__distr,axiom,
! [F: a > real,M: probab3364570286911266904_pmf_a,N2: sigma_measure_real,I3: set_nat,X6: nat > real > extend8495563244428889912nnreal,M2: nat > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe8764986580812350144nnreal @ ( probab7257411610070727406_pmf_a @ M ) @ M2
@ ^ [I2: nat] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe7397540011872310654nnreal @ ( measure_distr_a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% measure_pmf.indep_vars_distr
thf(fact_1084_measure__pmf_Oindep__vars__distr,axiom,
! [F: a > real,M: probab3364570286911266904_pmf_a,N2: sigma_measure_real,I3: set_re5328672808648366137nnreal,X6: ( real > extend8495563244428889912nnreal ) > real > extend8495563244428889912nnreal,M2: ( real > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 ) )
=> ( ! [I4: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe1904861324014696691nnreal @ ( probab7257411610070727406_pmf_a @ M ) @ M2
@ ^ [I2: real > extend8495563244428889912nnreal] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe8147369424511330225nnreal @ ( measure_distr_a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% measure_pmf.indep_vars_distr
thf(fact_1085_measure__pmf_Oindep__vars__distr,axiom,
! [F: a > real,M: probab3364570286911266904_pmf_a,N2: sigma_measure_real,I3: set_a_real,X6: ( a > real ) > real > extend8495563244428889912nnreal,M2: ( a > real ) > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 ) )
=> ( ! [I4: a > real] :
( ( member_a_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe308904231703567133nnreal @ ( probab7257411610070727406_pmf_a @ M ) @ M2
@ ^ [I2: a > real] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe6561983776359739359nnreal @ ( measure_distr_a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% measure_pmf.indep_vars_distr
thf(fact_1086_measure__pmf_Oindep__vars__distr,axiom,
! [F: a > real,M: probab3364570286911266904_pmf_a,N2: sigma_measure_real,I3: set_b,X6: b > real > extend8495563244428889912nnreal,M2: b > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe3735604593161453441nnreal @ ( probab7257411610070727406_pmf_a @ M ) @ M2
@ ^ [I2: b] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe6789649827632570435nnreal @ ( measure_distr_a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% measure_pmf.indep_vars_distr
thf(fact_1087_measure__pmf_Oindep__vars__distr,axiom,
! [F: a > real,M: probab3364570286911266904_pmf_a,N2: sigma_measure_real,I3: set_a,X6: a > real > extend8495563244428889912nnreal,M2: a > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe6839128675877538688nnreal @ ( probab7257411610070727406_pmf_a @ M ) @ M2
@ ^ [I2: a] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe669801873493879874nnreal @ ( measure_distr_a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% measure_pmf.indep_vars_distr
thf(fact_1088_measure__pmf_Oindep__vars__distr,axiom,
! [F: a > real,M: probab3364570286911266904_pmf_a,N2: sigma_measure_real,I3: set_Ex3793607809372303086nnreal,X6: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,M2: extend8495563244428889912nnreal > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 ) )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe8330679734821070376nnreal @ ( probab7257411610070727406_pmf_a @ M ) @ M2
@ ^ [I2: extend8495563244428889912nnreal] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe2251227809676183782nnreal @ ( measure_distr_a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% measure_pmf.indep_vars_distr
thf(fact_1089_measure__pmf_Oindep__vars__distr,axiom,
! [F: a > real,M: probab3364570286911266904_pmf_a,N2: sigma_measure_real,I3: set_real,X6: real > real > extend8495563244428889912nnreal,M2: real > sigma_7234349610311085201nnreal] :
( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe5896511939895573020nnreal @ ( probab7257411610070727406_pmf_a @ M ) @ M2
@ ^ [I2: real] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe6989957652004097498nnreal @ ( measure_distr_a_real @ ( probab7257411610070727406_pmf_a @ M ) @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ).
% measure_pmf.indep_vars_distr
thf(fact_1090_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > a,N2: sigma_measure_a,I3: set_nat,X6: nat > a > real,M2: nat > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe3903564294488106548t_real @ M @ M2
@ ^ [I2: nat] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe3903564294488106548t_real @ ( measure_distr_a_a @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1091_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > a,N2: sigma_measure_a,I3: set_b,X6: b > a > real,M2: b > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe8265442546547513973b_real @ M @ M2
@ ^ [I2: b] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe8265442546547513973b_real @ ( measure_distr_a_a @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1092_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > a,N2: sigma_measure_a,I3: set_a,X6: a > a > real,M2: a > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe2669223931359383284a_real @ M @ M2
@ ^ [I2: a] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe2669223931359383284a_real @ ( measure_distr_a_a @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1093_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > a,N2: sigma_measure_a,I3: set_Ex3793607809372303086nnreal,X6: extend8495563244428889912nnreal > a > real,M2: extend8495563244428889912nnreal > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe3928765670445028764l_real @ M @ M2
@ ^ [I2: extend8495563244428889912nnreal] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe3928765670445028764l_real @ ( measure_distr_a_a @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1094_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > a,N2: sigma_measure_a,I3: set_real,X6: real > a > real,M2: real > sigma_measure_real] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_a @ F @ ( sigma_measurable_a_a @ M @ N2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member_a_real @ ( X6 @ I4 ) @ ( sigma_9116425665531756122a_real @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe7095277112020832144l_real @ M @ M2
@ ^ [I2: real] : ( comp_a_real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe7095277112020832144l_real @ ( measure_distr_a_a @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1095_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > real,N2: sigma_measure_real,I3: set_nat,X6: nat > real > extend8495563244428889912nnreal,M2: nat > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe8764986580812350144nnreal @ M @ M2
@ ^ [I2: nat] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe7397540011872310654nnreal @ ( measure_distr_a_real @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1096_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > real,N2: sigma_measure_real,I3: set_b,X6: b > real > extend8495563244428889912nnreal,M2: b > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe3735604593161453441nnreal @ M @ M2
@ ^ [I2: b] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe6789649827632570435nnreal @ ( measure_distr_a_real @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1097_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > real,N2: sigma_measure_real,I3: set_a,X6: a > real > extend8495563244428889912nnreal,M2: a > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe6839128675877538688nnreal @ M @ M2
@ ^ [I2: a] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe669801873493879874nnreal @ ( measure_distr_a_real @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1098_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > real,N2: sigma_measure_real,I3: set_Ex3793607809372303086nnreal,X6: extend8495563244428889912nnreal > real > extend8495563244428889912nnreal,M2: extend8495563244428889912nnreal > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe8330679734821070376nnreal @ M @ M2
@ ^ [I2: extend8495563244428889912nnreal] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe2251227809676183782nnreal @ ( measure_distr_a_real @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1099_prob__space_Oindep__vars__distr,axiom,
! [M: sigma_measure_a,F: a > real,N2: sigma_measure_real,I3: set_real,X6: real > real > extend8495563244428889912nnreal,M2: real > sigma_7234349610311085201nnreal] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_a_real @ F @ ( sigma_9116425665531756122a_real @ M @ N2 ) )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( member2919562650594848410nnreal @ ( X6 @ I4 ) @ ( sigma_9017504469962657078nnreal @ N2 @ ( M2 @ I4 ) ) ) )
=> ( ( indepe5896511939895573020nnreal @ M @ M2
@ ^ [I2: real] : ( comp_r7806941060661185781real_a @ ( X6 @ I2 ) @ F )
@ I3 )
=> ( indepe6989957652004097498nnreal @ ( measure_distr_a_real @ M @ N2 @ F ) @ M2 @ X6 @ I3 ) ) ) ) ) ).
% prob_space.indep_vars_distr
thf(fact_1100_indep__vars__lebesgue__integral,axiom,
! [I3: set_re5328672808648366137nnreal,X5: ( real > extend8495563244428889912nnreal ) > a > real] :
( ( finite7684081742213514138nnreal @ I3 )
=> ( ( indepe8810216666108218087l_real @ m
@ ^ [Uu: real > extend8495563244428889912nnreal] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] :
( groups8287757765882971734l_real
@ ^ [I2: real > extend8495563244428889912nnreal] : ( X5 @ I2 @ Omega )
@ I3 ) )
= ( groups8287757765882971734l_real
@ ^ [I2: real > extend8495563244428889912nnreal] : ( bochne378719280626478695a_real @ m @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ) ).
% indep_vars_lebesgue_integral
thf(fact_1101_indep__vars__lebesgue__integral,axiom,
! [I3: set_a_real,X5: ( a > real ) > a > real] :
( ( finite_finite_a_real @ I3 )
=> ( ( indepe3165004708401367057l_real @ m
@ ^ [Uu: a > real] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: a > real] :
( ( member_a_real @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] :
( groups6588801770659435106l_real
@ ^ [I2: a > real] : ( X5 @ I2 @ Omega )
@ I3 ) )
= ( groups6588801770659435106l_real
@ ^ [I2: a > real] : ( bochne378719280626478695a_real @ m @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ) ).
% indep_vars_lebesgue_integral
thf(fact_1102_indep__vars__lebesgue__integral,axiom,
! [I3: set_Ex3793607809372303086nnreal,X5: extend8495563244428889912nnreal > a > real] :
( ( finite3782138982310603983nnreal @ I3 )
=> ( ( indepe3928765670445028764l_real @ m
@ ^ [Uu: extend8495563244428889912nnreal] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] :
( groups8574036886795004683l_real
@ ^ [I2: extend8495563244428889912nnreal] : ( X5 @ I2 @ Omega )
@ I3 ) )
= ( groups8574036886795004683l_real
@ ^ [I2: extend8495563244428889912nnreal] : ( bochne378719280626478695a_real @ m @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ) ).
% indep_vars_lebesgue_integral
thf(fact_1103_indep__vars__lebesgue__integral,axiom,
! [I3: set_real,X5: real > a > real] :
( ( finite_finite_real @ I3 )
=> ( ( indepe7095277112020832144l_real @ m
@ ^ [Uu: real] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] :
( groups1681761925125756287l_real
@ ^ [I2: real] : ( X5 @ I2 @ Omega )
@ I3 ) )
= ( groups1681761925125756287l_real
@ ^ [I2: real] : ( bochne378719280626478695a_real @ m @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ) ).
% indep_vars_lebesgue_integral
thf(fact_1104_indep__vars__lebesgue__integral,axiom,
! [I3: set_b,X5: b > a > real] :
( ( finite_finite_b @ I3 )
=> ( ( indepe8265442546547513973b_real @ m
@ ^ [Uu: b] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] :
( groups7542908600250600134b_real
@ ^ [I2: b] : ( X5 @ I2 @ Omega )
@ I3 ) )
= ( groups7542908600250600134b_real
@ ^ [I2: b] : ( bochne378719280626478695a_real @ m @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ) ).
% indep_vars_lebesgue_integral
thf(fact_1105_indep__vars__lebesgue__integral,axiom,
! [I3: set_nat,X5: nat > a > real] :
( ( finite_finite_nat @ I3 )
=> ( ( indepe3903564294488106548t_real @ m
@ ^ [Uu: nat] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] :
( groups129246275422532515t_real
@ ^ [I2: nat] : ( X5 @ I2 @ Omega )
@ I3 ) )
= ( groups129246275422532515t_real
@ ^ [I2: nat] : ( bochne378719280626478695a_real @ m @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ) ).
% indep_vars_lebesgue_integral
thf(fact_1106_indep__vars__lebesgue__integral,axiom,
! [I3: set_a,X5: a > a > real] :
( ( finite_finite_a @ I3 )
=> ( ( indepe2669223931359383284a_real @ m
@ ^ [Uu: a] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] :
( groups1946689985062469445a_real
@ ^ [I2: a] : ( X5 @ I2 @ Omega )
@ I3 ) )
= ( groups1946689985062469445a_real
@ ^ [I2: a] : ( bochne378719280626478695a_real @ m @ ( X5 @ I2 ) )
@ I3 ) ) ) ) ) ).
% indep_vars_lebesgue_integral
thf(fact_1107_indep__vars__sum,axiom,
! [I3: set_re5328672808648366137nnreal,I: real > extend8495563244428889912nnreal,X5: ( real > extend8495563244428889912nnreal ) > a > real] :
( ( finite7684081742213514138nnreal @ I3 )
=> ( ~ ( member2919562650594848410nnreal @ I @ I3 )
=> ( ( indepe8810216666108218087l_real @ m
@ ^ [Uu: real > extend8495563244428889912nnreal] : borel_5078946678739801102l_real
@ X5
@ ( insert152533262698245683nnreal @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups8532390058693651307l_real
@ ^ [I2: real > extend8495563244428889912nnreal] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_sum
thf(fact_1108_indep__vars__sum,axiom,
! [I3: set_a_real,I: a > real,X5: ( a > real ) > a > real] :
( ( finite_finite_a_real @ I3 )
=> ( ~ ( member_a_real @ I @ I3 )
=> ( ( indepe3165004708401367057l_real @ m
@ ^ [Uu: a > real] : borel_5078946678739801102l_real
@ X5
@ ( insert_a_real @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups6125259628802515085l_real
@ ^ [I2: a > real] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_sum
thf(fact_1109_indep__vars__sum,axiom,
! [I3: set_Ex3793607809372303086nnreal,I: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal > a > real] :
( ( finite3782138982310603983nnreal @ I3 )
=> ( ~ ( member7908768830364227535nnreal @ I @ I3 )
=> ( ( indepe3928765670445028764l_real @ m
@ ^ [Uu: extend8495563244428889912nnreal] : borel_5078946678739801102l_real
@ X5
@ ( insert7407984058720857448nnreal @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups2265062954415509024l_real
@ ^ [I2: extend8495563244428889912nnreal] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_sum
thf(fact_1110_indep__vars__sum,axiom,
! [I3: set_real,I: real,X5: real > a > real] :
( ( finite_finite_real @ I3 )
=> ( ~ ( member_real @ I @ I3 )
=> ( ( indepe7095277112020832144l_real @ m
@ ^ [Uu: real] : borel_5078946678739801102l_real
@ X5
@ ( insert_real @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups8097168146408367636l_real
@ ^ [I2: real] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_sum
thf(fact_1111_indep__vars__sum,axiom,
! [I3: set_nat,I: nat,X5: nat > a > real] :
( ( finite_finite_nat @ I3 )
=> ( ~ ( member_nat @ I @ I3 )
=> ( ( indepe3903564294488106548t_real @ m
@ ^ [Uu: nat] : borel_5078946678739801102l_real
@ X5
@ ( insert_nat @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups6591440286371151544t_real
@ ^ [I2: nat] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_sum
thf(fact_1112_indep__vars__sum,axiom,
! [I3: set_b,I: b,X5: b > a > real] :
( ( finite_finite_b @ I3 )
=> ( ~ ( member_b @ I @ I3 )
=> ( ( indepe8265442546547513973b_real @ m
@ ^ [Uu: b] : borel_5078946678739801102l_real
@ X5
@ ( insert_b @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups8336678772925405937b_real
@ ^ [I2: b] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_sum
thf(fact_1113_indep__vars__sum,axiom,
! [I3: set_a,I: a,X5: a > a > real] :
( ( finite_finite_a @ I3 )
=> ( ~ ( member_a @ I @ I3 )
=> ( ( indepe2669223931359383284a_real @ m
@ ^ [Uu: a] : borel_5078946678739801102l_real
@ X5
@ ( insert_a @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups2740460157737275248a_real
@ ^ [I2: a] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_sum
thf(fact_1114_indep__vars__integrable,axiom,
! [I3: set_re5328672808648366137nnreal,X5: ( real > extend8495563244428889912nnreal ) > a > real] :
( ( finite7684081742213514138nnreal @ I3 )
=> ( ( indepe8810216666108218087l_real @ m
@ ^ [Uu: real > extend8495563244428889912nnreal] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: real > extend8495563244428889912nnreal] :
( ( member2919562650594848410nnreal @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ m
@ ^ [Omega: a] :
( groups8287757765882971734l_real
@ ^ [I2: real > extend8495563244428889912nnreal] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_integrable
thf(fact_1115_indep__vars__integrable,axiom,
! [I3: set_a_real,X5: ( a > real ) > a > real] :
( ( finite_finite_a_real @ I3 )
=> ( ( indepe3165004708401367057l_real @ m
@ ^ [Uu: a > real] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: a > real] :
( ( member_a_real @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ m
@ ^ [Omega: a] :
( groups6588801770659435106l_real
@ ^ [I2: a > real] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_integrable
thf(fact_1116_indep__vars__integrable,axiom,
! [I3: set_Ex3793607809372303086nnreal,X5: extend8495563244428889912nnreal > a > real] :
( ( finite3782138982310603983nnreal @ I3 )
=> ( ( indepe3928765670445028764l_real @ m
@ ^ [Uu: extend8495563244428889912nnreal] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ m
@ ^ [Omega: a] :
( groups8574036886795004683l_real
@ ^ [I2: extend8495563244428889912nnreal] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_integrable
thf(fact_1117_indep__vars__integrable,axiom,
! [I3: set_real,X5: real > a > real] :
( ( finite_finite_real @ I3 )
=> ( ( indepe7095277112020832144l_real @ m
@ ^ [Uu: real] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: real] :
( ( member_real @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ m
@ ^ [Omega: a] :
( groups1681761925125756287l_real
@ ^ [I2: real] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_integrable
thf(fact_1118_indep__vars__integrable,axiom,
! [I3: set_b,X5: b > a > real] :
( ( finite_finite_b @ I3 )
=> ( ( indepe8265442546547513973b_real @ m
@ ^ [Uu: b] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: b] :
( ( member_b @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ m
@ ^ [Omega: a] :
( groups7542908600250600134b_real
@ ^ [I2: b] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_integrable
thf(fact_1119_indep__vars__integrable,axiom,
! [I3: set_nat,X5: nat > a > real] :
( ( finite_finite_nat @ I3 )
=> ( ( indepe3903564294488106548t_real @ m
@ ^ [Uu: nat] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: nat] :
( ( member_nat @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ m
@ ^ [Omega: a] :
( groups129246275422532515t_real
@ ^ [I2: nat] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_integrable
thf(fact_1120_indep__vars__integrable,axiom,
! [I3: set_a,X5: a > a > real] :
( ( finite_finite_a @ I3 )
=> ( ( indepe2669223931359383284a_real @ m
@ ^ [Uu: a] : borel_5078946678739801102l_real
@ X5
@ I3 )
=> ( ! [I4: a] :
( ( member_a @ I4 @ I3 )
=> ( bochne2139062162225249880a_real @ m @ ( X5 @ I4 ) ) )
=> ( bochne2139062162225249880a_real @ m
@ ^ [Omega: a] :
( groups1946689985062469445a_real
@ ^ [I2: a] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_integrable
thf(fact_1121_finite__indexed__bound,axiom,
! [A2: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > nat > $o] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M6: nat] :
! [X4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X4 @ A2 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1122_finite__indexed__bound,axiom,
! [A2: set_real,P: real > nat > $o] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M6: nat] :
! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1123_finite__indexed__bound,axiom,
! [A2: set_b,P: b > nat > $o] :
( ( finite_finite_b @ A2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M6: nat] :
! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1124_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M6: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1125_finite__indexed__bound,axiom,
! [A2: set_a,P: a > nat > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M6: nat] :
! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1126_finite__indexed__bound,axiom,
! [A2: set_Ex3793607809372303086nnreal,P: extend8495563244428889912nnreal > real > $o] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ! [X3: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X3 @ A2 )
=> ? [X_12: real] : ( P @ X3 @ X_12 ) )
=> ? [M6: real] :
! [X4: extend8495563244428889912nnreal] :
( ( member7908768830364227535nnreal @ X4 @ A2 )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1127_finite__indexed__bound,axiom,
! [A2: set_real,P: real > real > $o] :
( ( finite_finite_real @ A2 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ? [X_12: real] : ( P @ X3 @ X_12 ) )
=> ? [M6: real] :
! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1128_finite__indexed__bound,axiom,
! [A2: set_b,P: b > real > $o] :
( ( finite_finite_b @ A2 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A2 )
=> ? [X_12: real] : ( P @ X3 @ X_12 ) )
=> ? [M6: real] :
! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1129_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > real > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [X_12: real] : ( P @ X3 @ X_12 ) )
=> ? [M6: real] :
! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1130_finite__indexed__bound,axiom,
! [A2: set_a,P: a > real > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [X_12: real] : ( P @ X3 @ X_12 ) )
=> ? [M6: real] :
! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M6 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_1131_insert__absorb2,axiom,
! [X2: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A2 ) )
= ( insert_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_1132_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1133_insert__iff,axiom,
! [A: real > extend8495563244428889912nnreal,B: real > extend8495563244428889912nnreal,A2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ A @ ( insert152533262698245683nnreal @ B @ A2 ) )
= ( ( A = B )
| ( member2919562650594848410nnreal @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1134_insert__iff,axiom,
! [A: a > real,B: a > real,A2: set_a_real] :
( ( member_a_real @ A @ ( insert_a_real @ B @ A2 ) )
= ( ( A = B )
| ( member_a_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1135_insert__iff,axiom,
! [A: b,B: b,A2: set_b] :
( ( member_b @ A @ ( insert_b @ B @ A2 ) )
= ( ( A = B )
| ( member_b @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1136_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1137_insert__iff,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ A2 ) )
= ( ( A = B )
| ( member7908768830364227535nnreal @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1138_insert__iff,axiom,
! [A: real,B: real,A2: set_real] :
( ( member_real @ A @ ( insert_real @ B @ A2 ) )
= ( ( A = B )
| ( member_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1139_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1140_insertCI,axiom,
! [A: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal,B: real > extend8495563244428889912nnreal] :
( ( ~ ( member2919562650594848410nnreal @ A @ B2 )
=> ( A = B ) )
=> ( member2919562650594848410nnreal @ A @ ( insert152533262698245683nnreal @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1141_insertCI,axiom,
! [A: a > real,B2: set_a_real,B: a > real] :
( ( ~ ( member_a_real @ A @ B2 )
=> ( A = B ) )
=> ( member_a_real @ A @ ( insert_a_real @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1142_insertCI,axiom,
! [A: b,B2: set_b,B: b] :
( ( ~ ( member_b @ A @ B2 )
=> ( A = B ) )
=> ( member_b @ A @ ( insert_b @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1143_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1144_insertCI,axiom,
! [A: extend8495563244428889912nnreal,B2: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
( ( ~ ( member7908768830364227535nnreal @ A @ B2 )
=> ( A = B ) )
=> ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1145_insertCI,axiom,
! [A: real,B2: set_real,B: real] :
( ( ~ ( member_real @ A @ B2 )
=> ( A = B ) )
=> ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1146_insert__subset,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1147_insert__subset,axiom,
! [X2: real > extend8495563244428889912nnreal,A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ( ord_le2462468573666744473nnreal @ ( insert152533262698245683nnreal @ X2 @ A2 ) @ B2 )
= ( ( member2919562650594848410nnreal @ X2 @ B2 )
& ( ord_le2462468573666744473nnreal @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1148_insert__subset,axiom,
! [X2: a > real,A2: set_a_real,B2: set_a_real] :
( ( ord_le3334967407727675675a_real @ ( insert_a_real @ X2 @ A2 ) @ B2 )
= ( ( member_a_real @ X2 @ B2 )
& ( ord_le3334967407727675675a_real @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1149_insert__subset,axiom,
! [X2: b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( insert_b @ X2 @ A2 ) @ B2 )
= ( ( member_b @ X2 @ B2 )
& ( ord_less_eq_set_b @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1150_insert__subset,axiom,
! [X2: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ( ord_le6787938422905777998nnreal @ ( insert7407984058720857448nnreal @ X2 @ A2 ) @ B2 )
= ( ( member7908768830364227535nnreal @ X2 @ B2 )
& ( ord_le6787938422905777998nnreal @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1151_insert__subset,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ( ord_less_eq_set_real @ ( insert_real @ X2 @ A2 ) @ B2 )
= ( ( member_real @ X2 @ B2 )
& ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1152_insert__subset,axiom,
! [X2: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X2 @ A2 ) @ B2 )
= ( ( member_set_a @ X2 @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1153_insert__subset,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( ( member_a @ X2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_1154_finite__insert,axiom,
! [A: b,A2: set_b] :
( ( finite_finite_b @ ( insert_b @ A @ A2 ) )
= ( finite_finite_b @ A2 ) ) ).
% finite_insert
thf(fact_1155_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_1156_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_1157_insert__Diff1,axiom,
! [X2: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1158_insert__Diff1,axiom,
! [X2: real > extend8495563244428889912nnreal,B2: set_re5328672808648366137nnreal,A2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ X2 @ B2 )
=> ( ( minus_3708639258518406418nnreal @ ( insert152533262698245683nnreal @ X2 @ A2 ) @ B2 )
= ( minus_3708639258518406418nnreal @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1159_insert__Diff1,axiom,
! [X2: a > real,B2: set_a_real,A2: set_a_real] :
( ( member_a_real @ X2 @ B2 )
=> ( ( minus_4124197362600706274a_real @ ( insert_a_real @ X2 @ A2 ) @ B2 )
= ( minus_4124197362600706274a_real @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1160_insert__Diff1,axiom,
! [X2: b,B2: set_b,A2: set_b] :
( ( member_b @ X2 @ B2 )
=> ( ( minus_minus_set_b @ ( insert_b @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1161_insert__Diff1,axiom,
! [X2: extend8495563244428889912nnreal,B2: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ X2 @ B2 )
=> ( ( minus_104578273773384135nnreal @ ( insert7407984058720857448nnreal @ X2 @ A2 ) @ B2 )
= ( minus_104578273773384135nnreal @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1162_insert__Diff1,axiom,
! [X2: real,B2: set_real,A2: set_real] :
( ( member_real @ X2 @ B2 )
=> ( ( minus_minus_set_real @ ( insert_real @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1163_insert__Diff1,axiom,
! [X2: a,B2: set_a,A2: set_a] :
( ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1164_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1165_Diff__insert0,axiom,
! [X2: real > extend8495563244428889912nnreal,A2: set_re5328672808648366137nnreal,B2: set_re5328672808648366137nnreal] :
( ~ ( member2919562650594848410nnreal @ X2 @ A2 )
=> ( ( minus_3708639258518406418nnreal @ A2 @ ( insert152533262698245683nnreal @ X2 @ B2 ) )
= ( minus_3708639258518406418nnreal @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1166_Diff__insert0,axiom,
! [X2: a > real,A2: set_a_real,B2: set_a_real] :
( ~ ( member_a_real @ X2 @ A2 )
=> ( ( minus_4124197362600706274a_real @ A2 @ ( insert_a_real @ X2 @ B2 ) )
= ( minus_4124197362600706274a_real @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1167_Diff__insert0,axiom,
! [X2: b,A2: set_b,B2: set_b] :
( ~ ( member_b @ X2 @ A2 )
=> ( ( minus_minus_set_b @ A2 @ ( insert_b @ X2 @ B2 ) )
= ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1168_Diff__insert0,axiom,
! [X2: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal,B2: set_Ex3793607809372303086nnreal] :
( ~ ( member7908768830364227535nnreal @ X2 @ A2 )
=> ( ( minus_104578273773384135nnreal @ A2 @ ( insert7407984058720857448nnreal @ X2 @ B2 ) )
= ( minus_104578273773384135nnreal @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1169_Diff__insert0,axiom,
! [X2: real,A2: set_real,B2: set_real] :
( ~ ( member_real @ X2 @ A2 )
=> ( ( minus_minus_set_real @ A2 @ ( insert_real @ X2 @ B2 ) )
= ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1170_Diff__insert0,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1171_indep__vars__prod,axiom,
! [I3: set_re5328672808648366137nnreal,I: real > extend8495563244428889912nnreal,X5: ( real > extend8495563244428889912nnreal ) > a > real] :
( ( finite7684081742213514138nnreal @ I3 )
=> ( ~ ( member2919562650594848410nnreal @ I @ I3 )
=> ( ( indepe8810216666108218087l_real @ m
@ ^ [Uu: real > extend8495563244428889912nnreal] : borel_5078946678739801102l_real
@ X5
@ ( insert152533262698245683nnreal @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups8287757765882971734l_real
@ ^ [I2: real > extend8495563244428889912nnreal] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_prod
thf(fact_1172_indep__vars__prod,axiom,
! [I3: set_a_real,I: a > real,X5: ( a > real ) > a > real] :
( ( finite_finite_a_real @ I3 )
=> ( ~ ( member_a_real @ I @ I3 )
=> ( ( indepe3165004708401367057l_real @ m
@ ^ [Uu: a > real] : borel_5078946678739801102l_real
@ X5
@ ( insert_a_real @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups6588801770659435106l_real
@ ^ [I2: a > real] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_prod
thf(fact_1173_indep__vars__prod,axiom,
! [I3: set_Ex3793607809372303086nnreal,I: extend8495563244428889912nnreal,X5: extend8495563244428889912nnreal > a > real] :
( ( finite3782138982310603983nnreal @ I3 )
=> ( ~ ( member7908768830364227535nnreal @ I @ I3 )
=> ( ( indepe3928765670445028764l_real @ m
@ ^ [Uu: extend8495563244428889912nnreal] : borel_5078946678739801102l_real
@ X5
@ ( insert7407984058720857448nnreal @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups8574036886795004683l_real
@ ^ [I2: extend8495563244428889912nnreal] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_prod
thf(fact_1174_indep__vars__prod,axiom,
! [I3: set_real,I: real,X5: real > a > real] :
( ( finite_finite_real @ I3 )
=> ( ~ ( member_real @ I @ I3 )
=> ( ( indepe7095277112020832144l_real @ m
@ ^ [Uu: real] : borel_5078946678739801102l_real
@ X5
@ ( insert_real @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups1681761925125756287l_real
@ ^ [I2: real] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_prod
thf(fact_1175_indep__vars__prod,axiom,
! [I3: set_b,I: b,X5: b > a > real] :
( ( finite_finite_b @ I3 )
=> ( ~ ( member_b @ I @ I3 )
=> ( ( indepe8265442546547513973b_real @ m
@ ^ [Uu: b] : borel_5078946678739801102l_real
@ X5
@ ( insert_b @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups7542908600250600134b_real
@ ^ [I2: b] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_prod
thf(fact_1176_indep__vars__prod,axiom,
! [I3: set_nat,I: nat,X5: nat > a > real] :
( ( finite_finite_nat @ I3 )
=> ( ~ ( member_nat @ I @ I3 )
=> ( ( indepe3903564294488106548t_real @ m
@ ^ [Uu: nat] : borel_5078946678739801102l_real
@ X5
@ ( insert_nat @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups129246275422532515t_real
@ ^ [I2: nat] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_prod
thf(fact_1177_indep__vars__prod,axiom,
! [I3: set_a,I: a,X5: a > a > real] :
( ( finite_finite_a @ I3 )
=> ( ~ ( member_a @ I @ I3 )
=> ( ( indepe2669223931359383284a_real @ m
@ ^ [Uu: a] : borel_5078946678739801102l_real
@ X5
@ ( insert_a @ I @ I3 ) )
=> ( indepe8958435565499147358a_real @ m @ borel_5078946678739801102l_real @ ( X5 @ I ) @ borel_5078946678739801102l_real
@ ^ [Omega: a] :
( groups1946689985062469445a_real
@ ^ [I2: a] : ( X5 @ I2 @ Omega )
@ I3 ) ) ) ) ) ).
% indep_vars_prod
thf(fact_1178_prod__zero__iff,axiom,
! [A2: set_b,F: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( ( groups7542908600250600134b_real @ F @ A2 )
= zero_zero_real )
= ( ? [X: b] :
( ( member_b @ X @ A2 )
& ( ( F @ X )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_1179_prod__zero__iff,axiom,
! [A2: set_nat,F: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups129246275422532515t_real @ F @ A2 )
= zero_zero_real )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ( F @ X )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_1180_prod__zero__iff,axiom,
! [A2: set_a,F: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( ( groups1946689985062469445a_real @ F @ A2 )
= zero_zero_real )
= ( ? [X: a] :
( ( member_a @ X @ A2 )
& ( ( F @ X )
= zero_zero_real ) ) ) ) ) ).
% prod_zero_iff
thf(fact_1181_prod__zero__iff,axiom,
! [A2: set_b,F: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ( ( groups8336835798719075562_b_nat @ F @ A2 )
= zero_zero_nat )
= ( ? [X: b] :
( ( member_b @ X @ A2 )
& ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_1182_prod__zero__iff,axiom,
! [A2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( groups708209901874060359at_nat @ F @ A2 )
= zero_zero_nat )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_1183_prod__zero__iff,axiom,
! [A2: set_a,F: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( ( groups7101391469762681065_a_nat @ F @ A2 )
= zero_zero_nat )
= ( ? [X: a] :
( ( member_a @ X @ A2 )
& ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_1184_card_Oinfinite,axiom,
! [A2: set_b] :
( ~ ( finite_finite_b @ A2 )
=> ( ( finite_card_b @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1185_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1186_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_1187_finite__Diff__insert,axiom,
! [A2: set_b,A: b,B2: set_b] :
( ( finite_finite_b @ ( minus_minus_set_b @ A2 @ ( insert_b @ A @ B2 ) ) )
= ( finite_finite_b @ ( minus_minus_set_b @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1188_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1189_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1190_prod_Oinsert,axiom,
! [A2: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ~ ( member7908768830364227535nnreal @ X2 @ A2 )
=> ( ( groups8574036886795004683l_real @ G @ ( insert7407984058720857448nnreal @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups8574036886795004683l_real @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1191_prod_Oinsert,axiom,
! [A2: set_real,X2: real,G: real > real] :
( ( finite_finite_real @ A2 )
=> ( ~ ( member_real @ X2 @ A2 )
=> ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1192_prod_Oinsert,axiom,
! [A2: set_b,X2: b,G: b > real] :
( ( finite_finite_b @ A2 )
=> ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups7542908600250600134b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups7542908600250600134b_real @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1193_prod_Oinsert,axiom,
! [A2: set_nat,X2: nat,G: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X2 @ A2 )
=> ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1194_prod_Oinsert,axiom,
! [A2: set_a,X2: a,G: a > real] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups1946689985062469445a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups1946689985062469445a_real @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1195_prod_Oinsert,axiom,
! [A2: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ~ ( member7908768830364227535nnreal @ X2 @ A2 )
=> ( ( groups3213843972782180271al_nat @ G @ ( insert7407984058720857448nnreal @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups3213843972782180271al_nat @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1196_prod_Oinsert,axiom,
! [A2: set_real,X2: real,G: real > nat] :
( ( finite_finite_real @ A2 )
=> ( ~ ( member_real @ X2 @ A2 )
=> ( ( groups4696554848551431203al_nat @ G @ ( insert_real @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1197_prod_Oinsert,axiom,
! [A2: set_b,X2: b,G: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups8336835798719075562_b_nat @ G @ ( insert_b @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups8336835798719075562_b_nat @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1198_prod_Oinsert,axiom,
! [A2: set_nat,X2: nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X2 @ A2 )
=> ( ( groups708209901874060359at_nat @ G @ ( insert_nat @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups708209901874060359at_nat @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1199_prod_Oinsert,axiom,
! [A2: set_a,X2: a,G: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups7101391469762681065_a_nat @ G @ ( insert_a @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups7101391469762681065_a_nat @ G @ A2 ) ) ) ) ) ).
% prod.insert
thf(fact_1200_card__insert__le,axiom,
! [A2: set_a,X2: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_1201_prod_Oinsert__if,axiom,
! [A2: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( ( member7908768830364227535nnreal @ X2 @ A2 )
=> ( ( groups8574036886795004683l_real @ G @ ( insert7407984058720857448nnreal @ X2 @ A2 ) )
= ( groups8574036886795004683l_real @ G @ A2 ) ) )
& ( ~ ( member7908768830364227535nnreal @ X2 @ A2 )
=> ( ( groups8574036886795004683l_real @ G @ ( insert7407984058720857448nnreal @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups8574036886795004683l_real @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1202_prod_Oinsert__if,axiom,
! [A2: set_real,X2: real,G: real > real] :
( ( finite_finite_real @ A2 )
=> ( ( ( member_real @ X2 @ A2 )
=> ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X2 @ A2 ) )
= ( groups1681761925125756287l_real @ G @ A2 ) ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1203_prod_Oinsert__if,axiom,
! [A2: set_b,X2: b,G: b > real] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ X2 @ A2 )
=> ( ( groups7542908600250600134b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( groups7542908600250600134b_real @ G @ A2 ) ) )
& ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups7542908600250600134b_real @ G @ ( insert_b @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups7542908600250600134b_real @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1204_prod_Oinsert__if,axiom,
! [A2: set_nat,X2: nat,G: nat > real] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X2 @ A2 ) )
= ( groups129246275422532515t_real @ G @ A2 ) ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1205_prod_Oinsert__if,axiom,
! [A2: set_a,X2: a,G: a > real] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X2 @ A2 )
=> ( ( groups1946689985062469445a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( groups1946689985062469445a_real @ G @ A2 ) ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups1946689985062469445a_real @ G @ ( insert_a @ X2 @ A2 ) )
= ( times_times_real @ ( G @ X2 ) @ ( groups1946689985062469445a_real @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1206_prod_Oinsert__if,axiom,
! [A2: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > nat] :
( ( finite3782138982310603983nnreal @ A2 )
=> ( ( ( member7908768830364227535nnreal @ X2 @ A2 )
=> ( ( groups3213843972782180271al_nat @ G @ ( insert7407984058720857448nnreal @ X2 @ A2 ) )
= ( groups3213843972782180271al_nat @ G @ A2 ) ) )
& ( ~ ( member7908768830364227535nnreal @ X2 @ A2 )
=> ( ( groups3213843972782180271al_nat @ G @ ( insert7407984058720857448nnreal @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups3213843972782180271al_nat @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1207_prod_Oinsert__if,axiom,
! [A2: set_real,X2: real,G: real > nat] :
( ( finite_finite_real @ A2 )
=> ( ( ( member_real @ X2 @ A2 )
=> ( ( groups4696554848551431203al_nat @ G @ ( insert_real @ X2 @ A2 ) )
= ( groups4696554848551431203al_nat @ G @ A2 ) ) )
& ( ~ ( member_real @ X2 @ A2 )
=> ( ( groups4696554848551431203al_nat @ G @ ( insert_real @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1208_prod_Oinsert__if,axiom,
! [A2: set_b,X2: b,G: b > nat] :
( ( finite_finite_b @ A2 )
=> ( ( ( member_b @ X2 @ A2 )
=> ( ( groups8336835798719075562_b_nat @ G @ ( insert_b @ X2 @ A2 ) )
= ( groups8336835798719075562_b_nat @ G @ A2 ) ) )
& ( ~ ( member_b @ X2 @ A2 )
=> ( ( groups8336835798719075562_b_nat @ G @ ( insert_b @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups8336835798719075562_b_nat @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1209_prod_Oinsert__if,axiom,
! [A2: set_nat,X2: nat,G: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ( groups708209901874060359at_nat @ G @ ( insert_nat @ X2 @ A2 ) )
= ( groups708209901874060359at_nat @ G @ A2 ) ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( groups708209901874060359at_nat @ G @ ( insert_nat @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups708209901874060359at_nat @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1210_prod_Oinsert__if,axiom,
! [A2: set_a,X2: a,G: a > nat] :
( ( finite_finite_a @ A2 )
=> ( ( ( member_a @ X2 @ A2 )
=> ( ( groups7101391469762681065_a_nat @ G @ ( insert_a @ X2 @ A2 ) )
= ( groups7101391469762681065_a_nat @ G @ A2 ) ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ( groups7101391469762681065_a_nat @ G @ ( insert_a @ X2 @ A2 ) )
= ( times_times_nat @ ( G @ X2 ) @ ( groups7101391469762681065_a_nat @ G @ A2 ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_1211_mk__disjoint__insert,axiom,
! [A: real > extend8495563244428889912nnreal,A2: set_re5328672808648366137nnreal] :
( ( member2919562650594848410nnreal @ A @ A2 )
=> ? [B6: set_re5328672808648366137nnreal] :
( ( A2
= ( insert152533262698245683nnreal @ A @ B6 ) )
& ~ ( member2919562650594848410nnreal @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1212_mk__disjoint__insert,axiom,
! [A: a > real,A2: set_a_real] :
( ( member_a_real @ A @ A2 )
=> ? [B6: set_a_real] :
( ( A2
= ( insert_a_real @ A @ B6 ) )
& ~ ( member_a_real @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1213_mk__disjoint__insert,axiom,
! [A: b,A2: set_b] :
( ( member_b @ A @ A2 )
=> ? [B6: set_b] :
( ( A2
= ( insert_b @ A @ B6 ) )
& ~ ( member_b @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1214_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B6: set_a] :
( ( A2
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1215_mk__disjoint__insert,axiom,
! [A: extend8495563244428889912nnreal,A2: set_Ex3793607809372303086nnreal] :
( ( member7908768830364227535nnreal @ A @ A2 )
=> ? [B6: set_Ex3793607809372303086nnreal] :
( ( A2
= ( insert7407984058720857448nnreal @ A @ B6 ) )
& ~ ( member7908768830364227535nnreal @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1216_mk__disjoint__insert,axiom,
! [A: real,A2: set_real] :
( ( member_real @ A @ A2 )
=> ? [B6: set_real] :
( ( A2
= ( insert_real @ A @ B6 ) )
& ~ ( member_real @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_1217_ennreal__minus__zero,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ A @ zero_z7100319975126383169nnreal )
= A ) ).
% ennreal_minus_zero
thf(fact_1218_zero__minus__ennreal,axiom,
! [A: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ zero_z7100319975126383169nnreal @ A )
= zero_z7100319975126383169nnreal ) ).
% zero_minus_ennreal
thf(fact_1219_ennreal__mult__left__cong,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( A != zero_z7100319975126383169nnreal )
=> ( B = C ) )
=> ( ( times_1893300245718287421nnreal @ A @ B )
= ( times_1893300245718287421nnreal @ A @ C ) ) ) ).
% ennreal_mult_left_cong
thf(fact_1220_ennreal__mult__right__cong,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ( A != zero_z7100319975126383169nnreal )
=> ( B = C ) )
=> ( ( times_1893300245718287421nnreal @ B @ A )
= ( times_1893300245718287421nnreal @ C @ A ) ) ) ).
% ennreal_mult_right_cong
thf(fact_1221_ennreal__minus__eq__0,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ( minus_8429688780609304081nnreal @ A @ B )
= zero_z7100319975126383169nnreal )
=> ( ord_le3935885782089961368nnreal @ A @ B ) ) ).
% ennreal_minus_eq_0
thf(fact_1222_diff__diff__commute__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ C )
= ( minus_8429688780609304081nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).
% diff_diff_commute_ennreal
thf(fact_1223_ennreal__minus__mono,axiom,
! [A: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ C )
=> ( ( ord_le3935885782089961368nnreal @ D2 @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ C @ D2 ) ) ) ) ).
% ennreal_minus_mono
thf(fact_1224_ennreal__mono__minus,axiom,
! [C: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ C @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ ( minus_8429688780609304081nnreal @ A @ C ) ) ) ).
% ennreal_mono_minus
thf(fact_1225_diff__le__self__ennreal,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ B ) @ A ) ).
% diff_le_self_ennreal
thf(fact_1226_ennreal__diff__le__mono__left,axiom,
! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal] :
( ( ord_le3935885782089961368nnreal @ A @ B )
=> ( ord_le3935885782089961368nnreal @ ( minus_8429688780609304081nnreal @ A @ C ) @ B ) ) ).
% ennreal_diff_le_mono_left
thf(fact_1227_real__scaleR__def,axiom,
real_V1485227260804924795R_real = times_times_real ).
% real_scaleR_def
thf(fact_1228_subprob__not__empty,axiom,
( ( sigma_space_a @ m )
!= bot_bot_set_a ) ).
% subprob_not_empty
thf(fact_1229_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1230_bot__ennreal,axiom,
bot_bo841427958541957580nnreal = zero_z7100319975126383169nnreal ).
% bot_ennreal
thf(fact_1231_ennreal__inj,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( extend7643940197134561352nnreal @ A )
= ( extend7643940197134561352nnreal @ B ) )
= ( A = B ) ) ) ) ).
% ennreal_inj
thf(fact_1232_ennreal__0,axiom,
( ( extend7643940197134561352nnreal @ zero_zero_real )
= zero_z7100319975126383169nnreal ) ).
% ennreal_0
thf(fact_1233_ennreal__eq__zero__iff,axiom,
! [X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X2 )
=> ( ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal )
= ( X2 = zero_zero_real ) ) ) ).
% ennreal_eq_zero_iff
thf(fact_1234_ennreal__le__iff,axiom,
! [Y3: real,X2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y3 )
=> ( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ ( extend7643940197134561352nnreal @ Y3 ) )
= ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% ennreal_le_iff
thf(fact_1235_tail__events__sets,axiom,
! [A2: nat > set_set_a] :
( ! [I4: nat] : ( ord_le3724670747650509150_set_a @ ( A2 @ I4 ) @ ( sigma_sets_a @ m ) )
=> ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ m @ A2 ) @ ( sigma_sets_a @ m ) ) ) ).
% tail_events_sets
thf(fact_1236_ennreal__leI,axiom,
! [X2: real,Y3: real] :
( ( ord_less_eq_real @ X2 @ Y3 )
=> ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ ( extend7643940197134561352nnreal @ Y3 ) ) ) ).
% ennreal_leI
thf(fact_1237_measurable__ennreal,axiom,
member2919562650594848410nnreal @ extend7643940197134561352nnreal @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).
% measurable_ennreal
thf(fact_1238_ennreal__neg,axiom,
! [X2: real] :
( ( ord_less_eq_real @ X2 @ zero_zero_real )
=> ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal ) ) ).
% ennreal_neg
thf(fact_1239_ennreal__eq__0__iff,axiom,
! [X2: real] :
( ( ( extend7643940197134561352nnreal @ X2 )
= zero_z7100319975126383169nnreal )
= ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% ennreal_eq_0_iff
thf(fact_1240_le__ennreal__iff,axiom,
! [R: real,X2: extend8495563244428889912nnreal] :
( ( ord_less_eq_real @ zero_zero_real @ R )
=> ( ( ord_le3935885782089961368nnreal @ X2 @ ( extend7643940197134561352nnreal @ R ) )
= ( ? [Q2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Q2 )
& ( X2
= ( extend7643940197134561352nnreal @ Q2 ) )
& ( ord_less_eq_real @ Q2 @ R ) ) ) ) ) ).
% le_ennreal_iff
thf(fact_1241_ennreal__le__iff2,axiom,
! [X2: real,Y3: real] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ ( extend7643940197134561352nnreal @ Y3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
& ( ord_less_eq_real @ X2 @ Y3 ) )
| ( ( ord_less_eq_real @ X2 @ zero_zero_real )
& ( ord_less_eq_real @ Y3 @ zero_zero_real ) ) ) ) ).
% ennreal_le_iff2
thf(fact_1242_mult__right__ennreal__cancel,axiom,
! [A: extend8495563244428889912nnreal,C: real,B: extend8495563244428889912nnreal] :
( ( ( times_1893300245718287421nnreal @ A @ ( extend7643940197134561352nnreal @ C ) )
= ( times_1893300245718287421nnreal @ B @ ( extend7643940197134561352nnreal @ C ) ) )
= ( ( A = B )
| ( ord_less_eq_real @ C @ zero_zero_real ) ) ) ).
% mult_right_ennreal_cancel
thf(fact_1243_ennreal__mult_H_H,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
= ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ).
% ennreal_mult''
thf(fact_1244_ennreal__mult_H,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
= ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ).
% ennreal_mult'
thf(fact_1245_ennreal__mult,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( extend7643940197134561352nnreal @ ( times_times_real @ A @ B ) )
= ( times_1893300245718287421nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) ) ) ) ) ).
% ennreal_mult
thf(fact_1246_ennreal__minus,axiom,
! [Q3: real,R: real] :
( ( ord_less_eq_real @ zero_zero_real @ Q3 )
=> ( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ R ) @ ( extend7643940197134561352nnreal @ Q3 ) )
= ( extend7643940197134561352nnreal @ ( minus_minus_real @ R @ Q3 ) ) ) ) ).
% ennreal_minus
thf(fact_1247_ennreal__minus__if,axiom,
! [A: real,B: real] :
( ( minus_8429688780609304081nnreal @ ( extend7643940197134561352nnreal @ A ) @ ( extend7643940197134561352nnreal @ B ) )
= ( extend7643940197134561352nnreal @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ B ) @ ( if_real @ ( ord_less_eq_real @ B @ A ) @ ( minus_minus_real @ A @ B ) @ zero_zero_real ) @ A ) ) ) ).
% ennreal_minus_if
thf(fact_1248_indep__setD__ev2,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ B2 @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev2
thf(fact_1249_indep__setD__ev1,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) ) ) ).
% indep_setD_ev1
thf(fact_1250_measure__increasing,axiom,
measur1776380161843274167a_real @ ( sigma_sets_a @ m ) @ ( sigma_measure_a2 @ m ) ).
% measure_increasing
thf(fact_1251_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_1252_finite__measure__mono,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ m ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B2 ) ) ) ) ).
% finite_measure_mono
thf(fact_1253_finite__measure__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ m ) )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ B2 ) ) ) ) ) ) ).
% finite_measure_Diff
thf(fact_1254_measure__eq__compl,axiom,
! [S: set_a,T: set_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S ) )
= ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ T ) ) )
=> ( ( sigma_measure_a2 @ m @ S )
= ( sigma_measure_a2 @ m @ T ) ) ) ) ) ).
% measure_eq_compl
thf(fact_1255_finite__measure__eq__sum__singleton,axiom,
! [S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ S2 )
=> ( member_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( sigma_sets_a @ m ) ) )
=> ( ( sigma_measure_a2 @ m @ S2 )
= ( groups2740460157737275248a_real
@ ^ [X: a] : ( sigma_measure_a2 @ m @ ( insert_a @ X @ bot_bot_set_a ) )
@ S2 ) ) ) ) ).
% finite_measure_eq_sum_singleton
thf(fact_1256_finite__measure__compl,axiom,
! [S2: set_a] :
( ( member_set_a @ S2 @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ ( sigma_space_a @ m ) @ S2 ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) @ ( sigma_measure_a2 @ m @ S2 ) ) ) ) ).
% finite_measure_compl
thf(fact_1257_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_1258_indep__setI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ ( sigma_sets_a @ m ) )
=> ( ! [A4: set_a,B3: set_a] :
( ( member_set_a @ A4 @ A2 )
=> ( ( member_set_a @ B3 @ B2 )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A4 @ B3 ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A4 ) @ ( sigma_measure_a2 @ m @ B3 ) ) ) ) )
=> ( indepe2041756565122539606_set_a @ m @ A2 @ B2 ) ) ) ) ).
% indep_setI
thf(fact_1259_indep__sets2__eq,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B2 )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( sigma_sets_a @ m ) )
& ( ord_le3724670747650509150_set_a @ B2 @ ( sigma_sets_a @ m ) )
& ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ B2 )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ X @ Y ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ X ) @ ( sigma_measure_a2 @ m @ Y ) ) ) ) ) ) ) ).
% indep_sets2_eq
thf(fact_1260_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_1261_prob__space,axiom,
( ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) )
= one_one_real ) ).
% prob_space
thf(fact_1262_measure__space__inter,axiom,
! [S: set_a,T: set_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ T )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ S @ T ) )
= ( sigma_measure_a2 @ m @ S ) ) ) ) ) ).
% measure_space_inter
thf(fact_1263_indep__setD,axiom,
! [A2: set_set_a,B2: set_set_a,A: set_a,B: set_a] :
( ( indepe2041756565122539606_set_a @ m @ A2 @ B2 )
=> ( ( member_set_a @ A @ A2 )
=> ( ( member_set_a @ B @ B2 )
=> ( ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A @ B ) )
= ( times_times_real @ ( sigma_measure_a2 @ m @ A ) @ ( sigma_measure_a2 @ m @ B ) ) ) ) ) ) ).
% indep_setD
thf(fact_1264_finite__measure__Diff_H,axiom,
! [A2: set_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ m ) )
=> ( ( sigma_measure_a2 @ m @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ m @ A2 ) @ ( sigma_measure_a2 @ m @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% finite_measure_Diff'
thf(fact_1265_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_1266_measure__exclude,axiom,
! [A2: set_a,B2: set_a] :
( ( member_set_a @ A2 @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ B2 @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ A2 )
= ( sigma_measure_a2 @ m @ ( sigma_space_a @ m ) ) )
=> ( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ m @ B2 )
= zero_zero_real ) ) ) ) ) ).
% measure_exclude
thf(fact_1267_subprob__measure__le__1,axiom,
! [X5: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ X5 ) @ one_one_real ) ).
% subprob_measure_le_1
thf(fact_1268_prob__le__1,axiom,
! [A2: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ m @ A2 ) @ one_one_real ) ).
% prob_le_1
thf(fact_1269_obtain__positive__integrable__function,axiom,
~ ! [F4: a > real] :
( ( member_a_real @ F4 @ ( sigma_9116425665531756122a_real @ m @ borel_5078946678739801102l_real ) )
=> ( ! [X4: a] : ( ord_less_real @ zero_zero_real @ ( F4 @ X4 ) )
=> ( ! [X4: a] : ( ord_less_eq_real @ ( F4 @ X4 ) @ one_one_real )
=> ~ ( bochne2139062162225249880a_real @ m @ F4 ) ) ) ) ).
% obtain_positive_integrable_function
thf(fact_1270_countable__support,axiom,
( counta4098120917673242425able_a
@ ( collect_a
@ ^ [X: a] :
( ( sigma_measure_a2 @ m @ ( insert_a @ X @ bot_bot_set_a ) )
!= zero_zero_real ) ) ) ).
% countable_support
thf(fact_1271_measure__zero__union,axiom,
! [S: set_a,T: set_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ m ) )
=> ( ( member_set_a @ T @ ( sigma_sets_a @ m ) )
=> ( ( ( sigma_measure_a2 @ m @ T )
= zero_zero_real )
=> ( ( sigma_measure_a2 @ m @ ( sup_sup_set_a @ S @ T ) )
= ( sigma_measure_a2 @ m @ S ) ) ) ) ) ).
% measure_zero_union
thf(fact_1272_ennreal__le__1,axiom,
! [X2: real] :
( ( ord_le3935885782089961368nnreal @ ( extend7643940197134561352nnreal @ X2 ) @ one_on2969667320475766781nnreal )
= ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% ennreal_le_1
thf(fact_1273_ennreal__ge__1,axiom,
! [X2: real] :
( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( extend7643940197134561352nnreal @ X2 ) )
= ( ord_less_eq_real @ one_one_real @ X2 ) ) ).
% ennreal_ge_1
% Helper facts (7)
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_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 ) ).
thf(help_If_3_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Extended____Nonnegative____Real__Oennreal_T,axiom,
! [X2: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
( ( if_Ext9135588136721118450nnreal @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( bochne378719280626478695a_real @ m
@ ^ [Omega: a] :
( groups8336678772925405937b_real
@ ^ [I2: b] :
( groups8336678772925405937b_real
@ ^ [J2: b] : ( times_times_real @ ( minus_minus_real @ ( f @ I2 @ Omega ) @ ( bochne378719280626478695a_real @ m @ ( f @ I2 ) ) ) @ ( minus_minus_real @ ( f @ J2 @ Omega ) @ ( bochne378719280626478695a_real @ m @ ( f @ J2 ) ) ) )
@ i )
@ i ) )
= ( groups8336678772925405937b_real
@ ^ [I2: b] :
( groups8336678772925405937b_real
@ ^ [J2: b] : ( probab3938396695707481060a_real @ m @ ( f @ I2 ) @ ( f @ J2 ) )
@ i )
@ i ) ) ).
%------------------------------------------------------------------------------