TPTP Problem File: SLH0652^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Universal_Hash_Families/0033_Preliminary_Results/prob_00128_004878__18524758_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1535 ( 450 unt; 256 typ; 0 def)
% Number of atoms : 4091 (1175 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 12249 ( 556 ~; 51 |; 219 &;9300 @)
% ( 0 <=>;2123 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 606 ( 606 >; 0 *; 0 +; 0 <<)
% Number of symbols : 232 ( 229 usr; 35 con; 0-5 aty)
% Number of variables : 3402 ( 163 ^;3148 !; 91 ?;3402 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:45:20.065
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
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thf(ty_n_t__Probability____Mass____Function__Opmf_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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thf(ty_n_t__Probability____Mass____Function__Opmf_Itf__a_J,type,
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thf(ty_n_t__Probability____Mass____Function__Opmf_I_Eo_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_I_Eo_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__a,type,
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% Explicit typings (229)
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thf(sy_c_Complete__Measure_Omain__part_001_Eo,type,
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thf(sy_c_Complete__Measure_Onull__part_001_Eo,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_I_Eo_M_Eo_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_M_Eo_J,type,
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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_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__a_001_Eo,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__events_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__set_001tf__a,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001_Eo,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Independent__Family_Oprob__space_Oindep__sets_001tf__a_001tf__a,type,
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thf(sy_c_Independent__Family_Oprob__space_Otail__events_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
inf_inf_real: real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
sup_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Real__Oreal,type,
sup_sup_real: real > real > real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Real__Oreal,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001_Eo,type,
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thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
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thf(sy_c_Measure__Space_Ofinite__measure_001_Eo,type,
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thf(sy_c_Measure__Space_Ofinite__measure_001tf__a,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
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ord_less_eq_o_real: ( $o > real ) > ( $o > real ) > $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__Probability____Mass____Function__Opmf_I_Eo_J_M_Eo_J,type,
ord_le3567021052573903211mf_o_o: ( probab1498759712122475378_pmf_o > $o ) > ( probab1498759712122475378_pmf_o > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Probability____Mass____Function__Opmf_It__Nat__Onat_J_M_Eo_J,type,
ord_le6862693511789568227_nat_o: ( probab469873468395307276mf_nat > $o ) > ( probab469873468395307276mf_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Probability____Mass____Function__Opmf_Itf__a_J_M_Eo_J,type,
ord_le4654290998161896325mf_a_o: ( probab3364570286911266904_pmf_a > $o ) > ( probab3364570286911266904_pmf_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_Eo,type,
ord_less_eq_o: $o > $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_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_I_Eo_J_J,type,
ord_le4374716579403074808_set_o: set_set_o > set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_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_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nonnegative____Real__Oennreal,type,
top_to1496364449551166952nnreal: extend8495563244428889912nnreal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_I_Eo_J_J,type,
top_top_set_option_o: set_option_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
top_to7721136755696657239od_o_o: set_Product_prod_o_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
top_to7022684507342537725_o_nat: set_Pr2101469702781467981_o_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
top_to8070287629520841379_nat_o: set_Pr3149072824959771635_nat_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
top_top_set_set_o: set_set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
top_top_set_set_a: set_set_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_Eo_M_Eo_J_J,type,
top_to1686961084667892491um_o_o: set_Sum_sum_o_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J_J,type,
top_to6072511757011528009_o_nat: set_Sum_sum_o_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_M_Eo_J_J,type,
top_to7120114879189831663_nat_o: set_Sum_sum_nat_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Probability__Mass__Function_Obernoulli__pmf,type,
probab6844364797682710202li_pmf: real > probab1498759712122475378_pmf_o ).
thf(sy_c_Probability__Mass__Function_Ocond__pmf_001_Eo,type,
probab8494970989125154181_pmf_o: probab1498759712122475378_pmf_o > set_o > probab1498759712122475378_pmf_o ).
thf(sy_c_Probability__Mass__Function_Ocond__pmf_001t__Nat__Onat,type,
probab7431941403989380899mf_nat: probab469873468395307276mf_nat > set_nat > probab469873468395307276mf_nat ).
thf(sy_c_Probability__Mass__Function_Ocond__pmf_001tf__a,type,
probab4270644268839999083_pmf_a: probab3364570286911266904_pmf_a > set_a > probab3364570286911266904_pmf_a ).
thf(sy_c_Probability__Mass__Function_Ogeometric__pmf,type,
probab1729510186672448573ic_pmf: real > probab469873468395307276mf_nat ).
thf(sy_c_Probability__Mass__Function_Opmf_001_Eo,type,
probab7541796623121487107_pmf_o: probab1498759712122475378_pmf_o > $o > real ).
thf(sy_c_Probability__Mass__Function_Opmf_001t__Nat__Onat,type,
probab2040650700041456421mf_nat: probab469873468395307276mf_nat > nat > real ).
thf(sy_c_Probability__Mass__Function_Opmf_001tf__a,type,
probab3485170606694471401_pmf_a: probab3364570286911266904_pmf_a > a > real ).
thf(sy_c_Probability__Mass__Function_Opmf_OAbs__pmf_001_Eo,type,
probab597269709993677180_pmf_o: sigma_measure_o > probab1498759712122475378_pmf_o ).
thf(sy_c_Probability__Mass__Function_Opmf_OAbs__pmf_001t__Nat__Onat,type,
probab5843134691084328684mf_nat: sigma_measure_nat > probab469873468395307276mf_nat ).
thf(sy_c_Probability__Mass__Function_Opmf_OAbs__pmf_001tf__a,type,
probab1189994150051702498_pmf_a: sigma_measure_a > probab3364570286911266904_pmf_a ).
thf(sy_c_Probability__Mass__Function_Opmf_Omeasure__pmf_001_Eo,type,
probab7036721048548158344_pmf_o: probab1498759712122475378_pmf_o > sigma_measure_o ).
thf(sy_c_Probability__Mass__Function_Opmf_Omeasure__pmf_001t__Nat__Onat,type,
probab1352011410425470944mf_nat: probab469873468395307276mf_nat > sigma_measure_nat ).
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__Mass__Function_Opmf_Opred__pmf_001_Eo,type,
probab7340661830756014643_pmf_o: ( $o > $o ) > probab1498759712122475378_pmf_o > $o ).
thf(sy_c_Probability__Mass__Function_Opmf_Opred__pmf_001t__Nat__Onat,type,
probab7504333880715489781mf_nat: ( nat > $o ) > probab469873468395307276mf_nat > $o ).
thf(sy_c_Probability__Mass__Function_Opmf_Opred__pmf_001tf__a,type,
probab7006109922600289305_pmf_a: ( a > $o ) > probab3364570286911266904_pmf_a > $o ).
thf(sy_c_Probability__Mass__Function_Opmf__as__measure_Ocr__pmf_001_Eo,type,
probab2081575740620332389_pmf_o: sigma_measure_o > probab1498759712122475378_pmf_o > $o ).
thf(sy_c_Probability__Mass__Function_Opmf__as__measure_Ocr__pmf_001t__Nat__Onat,type,
probab6917518385137753923mf_nat: sigma_measure_nat > probab469873468395307276mf_nat > $o ).
thf(sy_c_Probability__Mass__Function_Opmf__as__measure_Ocr__pmf_001tf__a,type,
probab451167891469848139_pmf_a: sigma_measure_a > probab3364570286911266904_pmf_a > $o ).
thf(sy_c_Probability__Mass__Function_Opmf__of__set_001_Eo,type,
probab8295283590095636704_set_o: set_o > probab1498759712122475378_pmf_o ).
thf(sy_c_Probability__Mass__Function_Opmf__of__set_001t__Nat__Onat,type,
probab1830274953030043784et_nat: set_nat > probab469873468395307276mf_nat ).
thf(sy_c_Probability__Mass__Function_Opmf__of__set_001tf__a,type,
probab3131728818378861638_set_a: set_a > probab3364570286911266904_pmf_a ).
thf(sy_c_Probability__Mass__Function_Opoisson__pmf,type,
probab4011777617282711093on_pmf: real > probab469873468395307276mf_nat ).
thf(sy_c_Probability__Mass__Function_Oset__pmf_001_Eo,type,
probab7458556812659319003_pmf_o: probab1498759712122475378_pmf_o > set_o ).
thf(sy_c_Probability__Mass__Function_Oset__pmf_001t__Nat__Onat,type,
probab3271515132085200205mf_nat: probab469873468395307276mf_nat > set_nat ).
thf(sy_c_Probability__Mass__Function_Oset__pmf_001tf__a,type,
probab49036049091589825_pmf_a: probab3364570286911266904_pmf_a > set_a ).
thf(sy_c_Probability__Measure_Opair__prob__space_001tf__a_001tf__a,type,
probab9084394580607489186ce_a_a: sigma_measure_a > sigma_measure_a > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001_Eo,type,
probab1190487603588612059pace_o: sigma_measure_o > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001t__Nat__Onat,type,
probab2904919403188438605ce_nat: sigma_measure_nat > $o ).
thf(sy_c_Probability__Measure_Oprob__space_001tf__a,type,
probab7247484486040049089pace_a: sigma_measure_a > $o ).
thf(sy_c_Probability__Measure_Oprob__space__axioms_001tf__a,type,
probab8302655048591552734ioms_a: sigma_measure_a > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OBall_001_Eo,type,
ball_o: set_o > ( $o > $o ) > $o ).
thf(sy_c_Set_OBall_001t__Nat__Onat,type,
ball_nat: set_nat > ( nat > $o ) > $o ).
thf(sy_c_Set_OBall_001tf__a,type,
ball_a: set_a > ( a > $o ) > $o ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001_Eo,type,
pow_o: set_o > set_set_o ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
insert_set_o: set_o > set_set_o > set_set_o ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001_Eo,type,
is_empty_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001_Eo,type,
remove_o: $o > set_o > set_o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Sigma__Algebra_Oalgebra_001_Eo,type,
sigma_algebra_o: set_o > set_set_o > $o ).
thf(sy_c_Sigma__Algebra_Oemeasure_001_Eo,type,
sigma_emeasure_o: sigma_measure_o > set_o > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001t__Nat__Onat,type,
sigma_emeasure_nat: sigma_measure_nat > set_nat > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Oemeasure_001tf__a,type,
sigma_emeasure_a: sigma_measure_a > set_a > extend8495563244428889912nnreal ).
thf(sy_c_Sigma__Algebra_Omeasure_001_Eo,type,
sigma_measure_o2: sigma_measure_o > set_o > real ).
thf(sy_c_Sigma__Algebra_Omeasure_001t__Nat__Onat,type,
sigma_measure_nat2: sigma_measure_nat > set_nat > real ).
thf(sy_c_Sigma__Algebra_Omeasure_001tf__a,type,
sigma_measure_a2: sigma_measure_a > set_a > real ).
thf(sy_c_Sigma__Algebra_Orestrict__space_001_Eo,type,
sigma_8520893325391096540pace_o: sigma_measure_o > set_o > sigma_measure_o ).
thf(sy_c_Sigma__Algebra_Orestrict__space_001t__Nat__Onat,type,
sigma_744083341818469772ce_nat: sigma_measure_nat > set_nat > sigma_measure_nat ).
thf(sy_c_Sigma__Algebra_Orestrict__space_001tf__a,type,
sigma_8692839461743104066pace_a: sigma_measure_a > set_a > sigma_measure_a ).
thf(sy_c_Sigma__Algebra_Oring__of__sets_001_Eo,type,
sigma_ring_of_sets_o: set_o > set_set_o > $o ).
thf(sy_c_Sigma__Algebra_Osets_001_Eo,type,
sigma_sets_o: sigma_measure_o > set_set_o ).
thf(sy_c_Sigma__Algebra_Osets_001t__Nat__Onat,type,
sigma_sets_nat: sigma_measure_nat > set_set_nat ).
thf(sy_c_Sigma__Algebra_Osets_001tf__a,type,
sigma_sets_a: sigma_measure_a > set_set_a ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001_Eo,type,
sigma_3687534776968752773ebra_o: set_o > set_set_o > $o ).
thf(sy_c_Sigma__Algebra_Osigma__algebra_001tf__a,type,
sigma_4968961713055010667ebra_a: set_a > set_set_a > $o ).
thf(sy_c_Sigma__Algebra_Osmallest__ccdi__sets_001_Eo,type,
sigma_7164253587203589747sets_o: set_o > set_set_o > set_set_o ).
thf(sy_c_Sigma__Algebra_Ospace_001_Eo,type,
sigma_space_o: sigma_measure_o > set_o ).
thf(sy_c_Sigma__Algebra_Ospace_001t__Nat__Onat,type,
sigma_space_nat: sigma_measure_nat > set_nat ).
thf(sy_c_Sigma__Algebra_Ospace_001tf__a,type,
sigma_space_a: sigma_measure_a > set_a ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $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_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_v_M,type,
m: sigma_measure_a ).
thf(sy_v_p,type,
p: probab3364570286911266904_pmf_a ).
% Relevant facts (1278)
thf(fact_0_assms,axiom,
( m
= ( probab7257411610070727406_pmf_a @ p ) ) ).
% assms
thf(fact_1_prob__space_Oaxioms_I2_J,axiom,
! [M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( probab8302655048591552734ioms_a @ M ) ) ).
% prob_space.axioms(2)
thf(fact_2_pmf_Opred__cong,axiom,
! [X: probab3364570286911266904_pmf_a,Ya: probab3364570286911266904_pmf_a,P: a > $o,Pa: a > $o] :
( ( X = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( probab49036049091589825_pmf_a @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( probab7006109922600289305_pmf_a @ P @ X )
= ( probab7006109922600289305_pmf_a @ Pa @ Ya ) ) ) ) ).
% pmf.pred_cong
thf(fact_3_pmf_Opred__cong,axiom,
! [X: probab1498759712122475378_pmf_o,Ya: probab1498759712122475378_pmf_o,P: $o > $o,Pa: $o > $o] :
( ( X = Ya )
=> ( ! [Z: $o] :
( ( member_o @ Z @ ( probab7458556812659319003_pmf_o @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( probab7340661830756014643_pmf_o @ P @ X )
= ( probab7340661830756014643_pmf_o @ Pa @ Ya ) ) ) ) ).
% pmf.pred_cong
thf(fact_4_pmf_Opred__cong,axiom,
! [X: probab469873468395307276mf_nat,Ya: probab469873468395307276mf_nat,P: nat > $o,Pa: nat > $o] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( probab3271515132085200205mf_nat @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( probab7504333880715489781mf_nat @ P @ X )
= ( probab7504333880715489781mf_nat @ Pa @ Ya ) ) ) ) ).
% pmf.pred_cong
thf(fact_5_pmf_Opred__mono__strong,axiom,
! [P: a > $o,X: probab3364570286911266904_pmf_a,Pa: a > $o] :
( ( probab7006109922600289305_pmf_a @ P @ X )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( probab49036049091589825_pmf_a @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( probab7006109922600289305_pmf_a @ Pa @ X ) ) ) ).
% pmf.pred_mono_strong
thf(fact_6_pmf_Opred__mono__strong,axiom,
! [P: $o > $o,X: probab1498759712122475378_pmf_o,Pa: $o > $o] :
( ( probab7340661830756014643_pmf_o @ P @ X )
=> ( ! [Z: $o] :
( ( member_o @ Z @ ( probab7458556812659319003_pmf_o @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( probab7340661830756014643_pmf_o @ Pa @ X ) ) ) ).
% pmf.pred_mono_strong
thf(fact_7_pmf_Opred__mono__strong,axiom,
! [P: nat > $o,X: probab469873468395307276mf_nat,Pa: nat > $o] :
( ( probab7504333880715489781mf_nat @ P @ X )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( probab3271515132085200205mf_nat @ X ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( probab7504333880715489781mf_nat @ Pa @ X ) ) ) ).
% pmf.pred_mono_strong
thf(fact_8_prob__space_Oindep__sets__cong,axiom,
! [M: sigma_measure_a,I: set_nat,J: set_nat,F: nat > set_set_a,G: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( I = J )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe6267730027088848354_a_nat @ M @ F @ I )
= ( indepe6267730027088848354_a_nat @ M @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_9_prob__space_Oindep__sets__cong,axiom,
! [M: sigma_measure_a,I: set_o,J: set_o,F: $o > set_set_a,G: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( I = J )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe7780107833195774214ts_a_o @ M @ F @ I )
= ( indepe7780107833195774214ts_a_o @ M @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_10_prob__space_Oindep__sets__cong,axiom,
! [M: sigma_measure_a,I: set_a,J: set_a,F: a > set_set_a,G: a > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( I = J )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ( F @ I2 )
= ( G @ I2 ) ) )
=> ( ( indepe8927441866673418604ts_a_a @ M @ F @ I )
= ( indepe8927441866673418604ts_a_a @ M @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_cong
thf(fact_11_measure__pmf__inject,axiom,
! [X: probab3364570286911266904_pmf_a,Y: probab3364570286911266904_pmf_a] :
( ( ( probab7257411610070727406_pmf_a @ X )
= ( probab7257411610070727406_pmf_a @ Y ) )
= ( X = Y ) ) ).
% measure_pmf_inject
thf(fact_12_prob__space__measure__pmf,axiom,
! [P2: probab3364570286911266904_pmf_a] : ( probab7247484486040049089pace_a @ ( probab7257411610070727406_pmf_a @ P2 ) ) ).
% prob_space_measure_pmf
thf(fact_13_measure__pmf_Oprob__space__axioms,axiom,
! [M: probab3364570286911266904_pmf_a] : ( probab7247484486040049089pace_a @ ( probab7257411610070727406_pmf_a @ M ) ) ).
% measure_pmf.prob_space_axioms
thf(fact_14_pair__prob__space__def,axiom,
( probab9084394580607489186ce_a_a
= ( ^ [M1: sigma_measure_a,M2: sigma_measure_a] :
( ( binary9071631243165066622te_a_a @ M1 @ M2 )
& ( probab7247484486040049089pace_a @ M1 )
& ( probab7247484486040049089pace_a @ M2 ) ) ) ) ).
% pair_prob_space_def
thf(fact_15_pair__prob__space_Ointro,axiom,
! [M12: sigma_measure_a,M22: sigma_measure_a] :
( ( binary9071631243165066622te_a_a @ M12 @ M22 )
=> ( ( probab7247484486040049089pace_a @ M12 )
=> ( ( probab7247484486040049089pace_a @ M22 )
=> ( probab9084394580607489186ce_a_a @ M12 @ M22 ) ) ) ) ).
% pair_prob_space.intro
thf(fact_16_prob__space__def,axiom,
( probab7247484486040049089pace_a
= ( ^ [M3: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M3 )
& ( probab8302655048591552734ioms_a @ M3 ) ) ) ) ).
% prob_space_def
thf(fact_17_prob__space_Ointro,axiom,
! [M: sigma_measure_a] :
( ( measur930452917991658466sure_a @ M )
=> ( ( probab8302655048591552734ioms_a @ M )
=> ( probab7247484486040049089pace_a @ M ) ) ) ).
% prob_space.intro
thf(fact_18_prob__space_Oindep__sets__mono__sets,axiom,
! [M: sigma_measure_a,F: nat > set_set_a,I: set_nat,G: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe6267730027088848354_a_nat @ M @ F @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6267730027088848354_a_nat @ M @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_19_prob__space_Oindep__sets__mono__sets,axiom,
! [M: sigma_measure_a,F: $o > set_set_a,I: set_o,G: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe7780107833195774214ts_a_o @ M @ F @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7780107833195774214ts_a_o @ M @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_20_prob__space_Oindep__sets__mono__sets,axiom,
! [M: sigma_measure_a,F: a > set_set_a,I: set_a,G: a > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe8927441866673418604ts_a_a @ M @ F @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ I )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe8927441866673418604ts_a_a @ M @ G @ I ) ) ) ) ).
% prob_space.indep_sets_mono_sets
thf(fact_21_pmf_Opred__set,axiom,
( probab7504333880715489781mf_nat
= ( ^ [P3: nat > $o,X2: probab469873468395307276mf_nat] :
! [Y2: nat] :
( ( member_nat @ Y2 @ ( probab3271515132085200205mf_nat @ X2 ) )
=> ( P3 @ Y2 ) ) ) ) ).
% pmf.pred_set
thf(fact_22_pmf_Opred__set,axiom,
( probab7340661830756014643_pmf_o
= ( ^ [P3: $o > $o,X2: probab1498759712122475378_pmf_o] :
! [Y2: $o] :
( ( member_o @ Y2 @ ( probab7458556812659319003_pmf_o @ X2 ) )
=> ( P3 @ Y2 ) ) ) ) ).
% pmf.pred_set
thf(fact_23_pmf_Opred__set,axiom,
( probab7006109922600289305_pmf_a
= ( ^ [P3: a > $o,X2: probab3364570286911266904_pmf_a] :
! [Y2: a] :
( ( member_a @ Y2 @ ( probab49036049091589825_pmf_a @ X2 ) )
=> ( P3 @ Y2 ) ) ) ) ).
% pmf.pred_set
thf(fact_24_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_25_measure__pmf_Ofinite__measure__axioms,axiom,
! [M: probab3364570286911266904_pmf_a] : ( measur930452917991658466sure_a @ ( probab7257411610070727406_pmf_a @ M ) ) ).
% measure_pmf.finite_measure_axioms
thf(fact_26_prob__space_Ofinite__measure,axiom,
! [M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( measur930452917991658466sure_a @ M ) ) ).
% prob_space.finite_measure
thf(fact_27_prob__space_Oindep__sets__mono,axiom,
! [M: sigma_measure_a,F: nat > set_set_a,I: set_nat,J: set_nat,G: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe6267730027088848354_a_nat @ M @ F @ I )
=> ( ( ord_less_eq_set_nat @ J @ I )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe6267730027088848354_a_nat @ M @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_28_prob__space_Oindep__sets__mono,axiom,
! [M: sigma_measure_a,F: $o > set_set_a,I: set_o,J: set_o,G: $o > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe7780107833195774214ts_a_o @ M @ F @ I )
=> ( ( ord_less_eq_set_o @ J @ I )
=> ( ! [I2: $o] :
( ( member_o @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe7780107833195774214ts_a_o @ M @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_29_prob__space_Oindep__sets__mono,axiom,
! [M: sigma_measure_a,F: a > set_set_a,I: set_a,J: set_a,G: a > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe8927441866673418604ts_a_a @ M @ F @ I )
=> ( ( ord_less_eq_set_a @ J @ I )
=> ( ! [I2: a] :
( ( member_a @ I2 @ J )
=> ( ord_le3724670747650509150_set_a @ ( G @ I2 ) @ ( F @ I2 ) ) )
=> ( indepe8927441866673418604ts_a_a @ M @ G @ J ) ) ) ) ) ).
% prob_space.indep_sets_mono
thf(fact_30_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_31_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_32_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_33_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_34_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_35_dual__order_Orefl,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_36_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_37_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_38_ball__reg,axiom,
! [R: set_nat,P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ R )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_39_ball__reg,axiom,
! [R: set_o,P: $o > $o,Q: $o > $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ R )
=> ( P @ X3 ) )
=> ! [X4: $o] :
( ( member_o @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_40_ball__reg,axiom,
! [R: set_a,P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ R )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ R )
=> ( P @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ R )
=> ( Q @ X4 ) ) ) ) ).
% ball_reg
thf(fact_41_Ball__def,axiom,
( ball_nat
= ( ^ [A3: set_nat,P3: nat > $o] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( P3 @ X2 ) ) ) ) ).
% Ball_def
thf(fact_42_Ball__def,axiom,
( ball_o
= ( ^ [A3: set_o,P3: $o > $o] :
! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( P3 @ X2 ) ) ) ) ).
% Ball_def
thf(fact_43_Ball__def,axiom,
( ball_a
= ( ^ [A3: set_a,P3: a > $o] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( P3 @ X2 ) ) ) ) ).
% Ball_def
thf(fact_44_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X2: $o] : ( member_o @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_50_pmf_Opred__mono,axiom,
! [P: nat > $o,Pa: nat > $o] :
( ( ord_less_eq_nat_o @ P @ Pa )
=> ( ord_le6862693511789568227_nat_o @ ( probab7504333880715489781mf_nat @ P ) @ ( probab7504333880715489781mf_nat @ Pa ) ) ) ).
% pmf.pred_mono
thf(fact_51_pmf_Opred__mono,axiom,
! [P: $o > $o,Pa: $o > $o] :
( ( ord_less_eq_o_o @ P @ Pa )
=> ( ord_le3567021052573903211mf_o_o @ ( probab7340661830756014643_pmf_o @ P ) @ ( probab7340661830756014643_pmf_o @ Pa ) ) ) ).
% pmf.pred_mono
thf(fact_52_pmf_Opred__mono,axiom,
! [P: a > $o,Pa: a > $o] :
( ( ord_less_eq_a_o @ P @ Pa )
=> ( ord_le4654290998161896325mf_a_o @ ( probab7006109922600289305_pmf_a @ P ) @ ( probab7006109922600289305_pmf_a @ Pa ) ) ) ).
% pmf.pred_mono
thf(fact_53_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_54_nle__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A2 @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_55_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_56_le__cases3,axiom,
! [X: real,Y: real,Z2: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_57_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_58_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_59_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_60_ord__eq__le__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_61_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_62_ord__le__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_63_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_64_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_65_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_66_order_Otrans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A2 @ C ) ) ) ).
% order.trans
thf(fact_67_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_68_order__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_69_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: 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 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_70_linorder__wlog,axiom,
! [P: real > real > $o,A2: real,B2: 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 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_71_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_72_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_73_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_74_dual__order_Oantisym,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_75_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_76_dual__order_Otrans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_77_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_78_antisym,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_79_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_80_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z3: real] : ( Y3 = Z3 ) )
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_81_order__subst1,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_82_order__subst1,axiom,
! [A2: nat,F2: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_83_order__subst1,axiom,
! [A2: real,F2: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_84_order__subst1,axiom,
! [A2: real,F2: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_85_order__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_86_order__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_87_order__subst2,axiom,
! [A2: real,B2: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst2,axiom,
! [A2: real,B2: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F2 @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_90_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_91_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_92_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_93_ord__eq__le__subst,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_94_ord__eq__le__subst,axiom,
! [A2: real,F2: nat > real,B2: nat,C: nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_95_ord__eq__le__subst,axiom,
! [A2: nat,F2: real > nat,B2: real,C: real] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_96_ord__eq__le__subst,axiom,
! [A2: real,F2: real > real,B2: real,C: real] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_97_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_98_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_99_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_100_ord__le__eq__subst,axiom,
! [A2: real,B2: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_101_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_102_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_103_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_104_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_105_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_106_in__mono,axiom,
! [A: set_o,B: set_o,X: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X @ A )
=> ( member_o @ X @ B ) ) ) ).
% in_mono
thf(fact_107_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B ) ) ) ).
% in_mono
thf(fact_108_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_109_subsetD,axiom,
! [A: set_o,B: set_o,C: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% subsetD
thf(fact_110_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% subsetD
thf(fact_111_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_112_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B5: set_o] :
! [X2: $o] :
( ( member_o @ X2 @ A3 )
=> ( member_o @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_113_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_114_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_115_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B5: set_o] :
! [T: $o] :
( ( member_o @ T @ A3 )
=> ( member_o @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_116_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_117_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_118_GreatestI2__order,axiom,
! [P: real > $o,X: real,Q: real > $o] :
( ( P @ X )
=> ( ! [Y4: real] :
( ( P @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X ) )
=> ( ! [X3: real] :
( ( P @ X3 )
=> ( ! [Y5: real] :
( ( P @ Y5 )
=> ( ord_less_eq_real @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_119_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_120_Greatest__equality,axiom,
! [P: real > $o,X: real] :
( ( P @ X )
=> ( ! [Y4: real] :
( ( P @ Y4 )
=> ( ord_less_eq_real @ Y4 @ X ) )
=> ( ( order_Greatest_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_121_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_122_verit__comp__simplify1_I2_J,axiom,
! [A2: real] : ( ord_less_eq_real @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_123_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_124_verit__la__disequality,axiom,
! [A2: real,B2: real] :
( ( A2 = B2 )
| ~ ( ord_less_eq_real @ A2 @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_125_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X5: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_126_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_real
= ( ^ [X5: $o > real,Y6: $o > real] :
( ( ord_less_eq_real @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_real @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_127_le__left__mono,axiom,
! [X: nat,Y: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ A2 )
=> ( ord_less_eq_nat @ X @ A2 ) ) ) ).
% le_left_mono
thf(fact_128_le__left__mono,axiom,
! [X: real,Y: real,A2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ A2 )
=> ( ord_less_eq_real @ X @ A2 ) ) ) ).
% le_left_mono
thf(fact_129_finite__measure__restrict__space,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( measur930452917991658466sure_a @ M )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( measur930452917991658466sure_a @ ( sigma_8692839461743104066pace_a @ M @ A ) ) ) ) ).
% finite_measure_restrict_space
thf(fact_130_mono__restrict__space,axiom,
! [M: sigma_measure_a,N: sigma_measure_a,X6: set_a] :
( ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ N ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ X6 ) ) @ ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ N @ X6 ) ) ) ) ).
% mono_restrict_space
thf(fact_131_sets__restrict__space__subset,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) )
=> ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ ( comple3428971583294703880tion_a @ M ) @ S ) ) @ ( sigma_sets_a @ ( comple3428971583294703880tion_a @ M ) ) ) ) ).
% sets_restrict_space_subset
thf(fact_132_finite__product__sigma__finite__axioms__def,axiom,
finite8532872553530597074ms_nat = finite_finite_nat ).
% finite_product_sigma_finite_axioms_def
thf(fact_133_finite__product__sigma__finite__axioms_Ointro,axiom,
! [I: set_nat] :
( ( finite_finite_nat @ I )
=> ( finite8532872553530597074ms_nat @ I ) ) ).
% finite_product_sigma_finite_axioms.intro
thf(fact_134_finite__Pow__iff,axiom,
! [A: set_nat] :
( ( finite1152437895449049373et_nat @ ( pow_nat @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_Pow_iff
thf(fact_135_finite__has__maximal2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( ord_less_eq_o @ A2 @ X3 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_136_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_137_finite__has__maximal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ord_less_eq_real @ A2 @ X3 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_138_finite__has__minimal2,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ( ord_less_eq_o @ X3 @ A2 )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_139_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_140_finite__has__minimal2,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ( ord_less_eq_real @ X3 @ A2 )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_141_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_142_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_143_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_144_sets__restrict__space__cong,axiom,
! [M: sigma_measure_a,N: sigma_measure_a,Omega: set_a] :
( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N ) )
=> ( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ Omega ) )
= ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ N @ Omega ) ) ) ) ).
% sets_restrict_space_cong
thf(fact_145_restrict__space__sets__cong,axiom,
! [A: set_a,B: set_a,M: sigma_measure_a,N: sigma_measure_a] :
( ( A = B )
=> ( ( ( sigma_sets_a @ M )
= ( sigma_sets_a @ N ) )
=> ( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ A ) )
= ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ N @ B ) ) ) ) ) ).
% restrict_space_sets_cong
thf(fact_146_prob__space_Otail__events__sets,axiom,
! [M: sigma_measure_a,A: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ! [I2: nat] : ( ord_le3724670747650509150_set_a @ ( A @ I2 ) @ ( sigma_sets_a @ M ) )
=> ( ord_le3724670747650509150_set_a @ ( indepe7538416700049374166_a_nat @ M @ A ) @ ( sigma_sets_a @ M ) ) ) ) ).
% prob_space.tail_events_sets
thf(fact_147_finite__indexed__bound,axiom,
! [A: set_o,P: $o > nat > $o] :
( ( finite_finite_o @ A )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M4: nat] :
! [X4: $o] :
( ( member_o @ X4 @ A )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M4 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_148_finite__indexed__bound,axiom,
! [A: set_a,P: a > nat > $o] :
( ( finite_finite_a @ A )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M4: nat] :
! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M4 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_149_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ M4 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_150_finite__indexed__bound,axiom,
! [A: set_o,P: $o > real > $o] :
( ( finite_finite_o @ A )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ? [X_1: real] : ( P @ X3 @ X_1 ) )
=> ? [M4: real] :
! [X4: $o] :
( ( member_o @ X4 @ A )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M4 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_151_finite__indexed__bound,axiom,
! [A: set_a,P: a > real > $o] :
( ( finite_finite_a @ A )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ? [X_1: real] : ( P @ X3 @ X_1 ) )
=> ? [M4: real] :
! [X4: a] :
( ( member_a @ X4 @ A )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M4 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_152_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > real > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [X_1: real] : ( P @ X3 @ X_1 ) )
=> ? [M4: real] :
! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [K: real] :
( ( ord_less_eq_real @ K @ M4 )
& ( P @ X4 @ K ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_153_prob__space_Oindep__setD__ev2,axiom,
! [M: sigma_measure_a,A: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe2041756565122539606_set_a @ M @ A @ B )
=> ( ord_le3724670747650509150_set_a @ B @ ( sigma_sets_a @ M ) ) ) ) ).
% prob_space.indep_setD_ev2
thf(fact_154_prob__space_Oindep__setD__ev1,axiom,
! [M: sigma_measure_a,A: set_set_a,B: set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe2041756565122539606_set_a @ M @ A @ B )
=> ( ord_le3724670747650509150_set_a @ A @ ( sigma_sets_a @ M ) ) ) ) ).
% prob_space.indep_setD_ev1
thf(fact_155_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M: sigma_measure_a,F: nat > set_set_a,I: set_nat] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe6267730027088848354_a_nat @ M @ F @ I )
= ( ! [J2: set_nat] :
( ( ord_less_eq_set_nat @ J2 @ I )
=> ( ( J2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ J2 )
=> ( indepe6267730027088848354_a_nat @ M @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_156_prob__space_Oindep__sets__finite__index__sets,axiom,
! [M: sigma_measure_a,F: $o > set_set_a,I: set_o] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe7780107833195774214ts_a_o @ M @ F @ I )
= ( ! [J2: set_o] :
( ( ord_less_eq_set_o @ J2 @ I )
=> ( ( J2 != bot_bot_set_o )
=> ( ( finite_finite_o @ J2 )
=> ( indepe7780107833195774214ts_a_o @ M @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_sets_finite_index_sets
thf(fact_157_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_158_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_159_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_160_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_161_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_162_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_163_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_164_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_165_empty__subsetI,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% empty_subsetI
thf(fact_166_subset__empty,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_167_sets_Oempty__sets,axiom,
! [M: sigma_measure_o] : ( member_set_o @ bot_bot_set_o @ ( sigma_sets_o @ M ) ) ).
% sets.empty_sets
thf(fact_168_Set_Oball__empty,axiom,
! [P: $o > $o,X4: $o] :
( ( member_o @ X4 @ bot_bot_set_o )
=> ( P @ X4 ) ) ).
% Set.ball_empty
thf(fact_169_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_170_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_171_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_172_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_173_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_174_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_175_equals0I,axiom,
! [A: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_176_equals0I,axiom,
! [A: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_177_equals0I,axiom,
! [A: set_o] :
( ! [Y4: $o] :
~ ( member_o @ Y4 @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_178_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_179_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_180_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_181_bot_Oextremum__uniqueI,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
=> ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_182_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_183_bot_Oextremum__unique,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_184_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_185_bot_Oextremum,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% bot.extremum
thf(fact_186_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_187_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_188_infinite__imp__nonempty,axiom,
! [S: set_o] :
( ~ ( finite_finite_o @ S )
=> ( S != bot_bot_set_o ) ) ).
% infinite_imp_nonempty
thf(fact_189_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_190_finite_OemptyI,axiom,
finite_finite_o @ bot_bot_set_o ).
% finite.emptyI
thf(fact_191_Pow__bottom,axiom,
! [B: set_o] : ( member_set_o @ bot_bot_set_o @ ( pow_o @ B ) ) ).
% Pow_bottom
thf(fact_192_set__pmf__not__empty,axiom,
! [M: probab3364570286911266904_pmf_a] :
( ( probab49036049091589825_pmf_a @ M )
!= bot_bot_set_a ) ).
% set_pmf_not_empty
thf(fact_193_set__pmf__not__empty,axiom,
! [M: probab1498759712122475378_pmf_o] :
( ( probab7458556812659319003_pmf_o @ M )
!= bot_bot_set_o ) ).
% set_pmf_not_empty
thf(fact_194_set__pmf__not__empty,axiom,
! [M: probab469873468395307276mf_nat] :
( ( probab3271515132085200205mf_nat @ M )
!= bot_bot_set_nat ) ).
% set_pmf_not_empty
thf(fact_195_empty__in__Fpow,axiom,
! [A: set_o] : ( member_set_o @ bot_bot_set_o @ ( finite_Fpow_o @ A ) ) ).
% empty_in_Fpow
thf(fact_196_finite__has__maximal,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_197_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_198_finite__has__maximal,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_199_finite__has__minimal,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ? [X3: $o] :
( ( member_o @ X3 @ A )
& ! [Xa: $o] :
( ( member_o @ Xa @ A )
=> ( ( ord_less_eq_o @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_200_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_201_finite__has__minimal,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ A )
& ! [Xa: real] :
( ( member_real @ Xa @ A )
=> ( ( ord_less_eq_real @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_202_prob__space_Oindep__events__finite__index__events,axiom,
! [M: sigma_measure_a,F: nat > set_a,I: set_nat] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe1551197314001032186_a_nat @ M @ F @ I )
= ( ! [J2: set_nat] :
( ( ord_less_eq_set_nat @ J2 @ I )
=> ( ( J2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ J2 )
=> ( indepe1551197314001032186_a_nat @ M @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_events_finite_index_events
thf(fact_203_prob__space_Oindep__events__finite__index__events,axiom,
! [M: sigma_measure_a,F: $o > set_a,I: set_o] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( indepe3695496658712714478ts_a_o @ M @ F @ I )
= ( ! [J2: set_o] :
( ( ord_less_eq_set_o @ J2 @ I )
=> ( ( J2 != bot_bot_set_o )
=> ( ( finite_finite_o @ J2 )
=> ( indepe3695496658712714478ts_a_o @ M @ F @ J2 ) ) ) ) ) ) ) ).
% prob_space.indep_events_finite_index_events
thf(fact_204_finite__transitivity__chain,axiom,
! [A: set_a,R: a > a > $o] :
( ( finite_finite_a @ A )
=> ( ! [X3: a] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: a,Y4: a,Z: a] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z )
=> ( R @ X3 @ Z ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ? [Y5: a] :
( ( member_a @ Y5 @ A )
& ( R @ X3 @ Y5 ) ) )
=> ( A = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_205_finite__transitivity__chain,axiom,
! [A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z: nat] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z )
=> ( R @ X3 @ Z ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Y5: nat] :
( ( member_nat @ Y5 @ A )
& ( R @ X3 @ Y5 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_206_finite__transitivity__chain,axiom,
! [A: set_o,R: $o > $o > $o] :
( ( finite_finite_o @ A )
=> ( ! [X3: $o] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: $o,Y4: $o,Z: $o] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z )
=> ( R @ X3 @ Z ) ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ? [Y5: $o] :
( ( member_o @ Y5 @ A )
& ( R @ X3 @ Y5 ) ) )
=> ( A = bot_bot_set_o ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_207_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_208_subset__emptyI,axiom,
! [A: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A )
=> ( ord_less_eq_set_a @ A @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_209_subset__emptyI,axiom,
! [A: set_o] :
( ! [X3: $o] :
~ ( member_o @ X3 @ A )
=> ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_210_prob__space__uniform__count__measure,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( probab2904919403188438605ce_nat @ ( nonneg7031465154080143958re_nat @ A ) ) ) ) ).
% prob_space_uniform_count_measure
thf(fact_211_prob__space__uniform__count__measure,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( probab1190487603588612059pace_o @ ( nonneg5198678888045619090sure_o @ A ) ) ) ) ).
% prob_space_uniform_count_measure
thf(fact_212_prob__space__uniform__count__measure,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ( probab7247484486040049089pace_a @ ( nonneg7367794086797660664sure_a @ A ) ) ) ) ).
% prob_space_uniform_count_measure
thf(fact_213_set__pmf__of__set,axiom,
! [S: set_a] :
( ( S != bot_bot_set_a )
=> ( ( finite_finite_a @ S )
=> ( ( probab49036049091589825_pmf_a @ ( probab3131728818378861638_set_a @ S ) )
= S ) ) ) ).
% set_pmf_of_set
thf(fact_214_set__pmf__of__set,axiom,
! [S: set_o] :
( ( S != bot_bot_set_o )
=> ( ( finite_finite_o @ S )
=> ( ( probab7458556812659319003_pmf_o @ ( probab8295283590095636704_set_o @ S ) )
= S ) ) ) ).
% set_pmf_of_set
thf(fact_215_set__pmf__of__set,axiom,
! [S: set_nat] :
( ( S != bot_bot_set_nat )
=> ( ( finite_finite_nat @ S )
=> ( ( probab3271515132085200205mf_nat @ ( probab1830274953030043784et_nat @ S ) )
= S ) ) ) ).
% set_pmf_of_set
thf(fact_216_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A3: set_o] : ( A3 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_217_finite__Union,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [M5: set_nat] :
( ( member_set_nat @ M5 @ A )
=> ( finite_finite_nat @ M5 ) )
=> ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% finite_Union
thf(fact_218_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_219_finite__UnionD,axiom,
! [A: set_set_nat] :
( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A ) )
=> ( finite1152437895449049373et_nat @ A ) ) ).
% finite_UnionD
thf(fact_220_finite__subset__Union,axiom,
! [A: set_nat,B6: set_set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ B6 ) )
=> ~ ! [F3: set_set_nat] :
( ( finite1152437895449049373et_nat @ F3 )
=> ( ( ord_le6893508408891458716et_nat @ F3 @ B6 )
=> ~ ( ord_less_eq_set_nat @ A @ ( comple7399068483239264473et_nat @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_221_le__cSup__finite,axiom,
! [X6: set_o,X: $o] :
( ( finite_finite_o @ X6 )
=> ( ( member_o @ X @ X6 )
=> ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_222_le__cSup__finite,axiom,
! [X6: set_nat,X: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X @ X6 )
=> ( ord_less_eq_nat @ X @ ( complete_Sup_Sup_nat @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_223_le__cSup__finite,axiom,
! [X6: set_real,X: real] :
( ( finite_finite_real @ X6 )
=> ( ( member_real @ X @ X6 )
=> ( ord_less_eq_real @ X @ ( comple1385675409528146559p_real @ X6 ) ) ) ) ).
% le_cSup_finite
thf(fact_224_cSup__eq__non__empty,axiom,
! [X6: set_o,A2: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ( ord_less_eq_o @ X3 @ A2 ) )
=> ( ! [Y4: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ X6 )
=> ( ord_less_eq_o @ X4 @ Y4 ) )
=> ( ord_less_eq_o @ A2 @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_225_cSup__eq__non__empty,axiom,
! [X6: set_nat,A2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ X3 @ A2 ) )
=> ( ! [Y4: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ X6 )
=> ( ord_less_eq_nat @ X4 @ Y4 ) )
=> ( ord_less_eq_nat @ A2 @ Y4 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_226_cSup__eq__non__empty,axiom,
! [X6: set_real,A2: real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ X3 @ A2 ) )
=> ( ! [Y4: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ X4 @ Y4 ) )
=> ( ord_less_eq_real @ A2 @ Y4 ) )
=> ( ( comple1385675409528146559p_real @ X6 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_227_cSup__least,axiom,
! [X6: set_o,Z2: $o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ( ord_less_eq_o @ X3 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_228_cSup__least,axiom,
! [X6: set_nat,Z2: nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_229_cSup__least,axiom,
! [X6: set_real,Z2: real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ ( comple1385675409528146559p_real @ X6 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_230_cSup__eq,axiom,
! [X6: set_real,A2: real] :
( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ X3 @ A2 ) )
=> ( ! [Y4: real] :
( ! [X4: real] :
( ( member_real @ X4 @ X6 )
=> ( ord_less_eq_real @ X4 @ Y4 ) )
=> ( ord_less_eq_real @ A2 @ Y4 ) )
=> ( ( comple1385675409528146559p_real @ X6 )
= A2 ) ) ) ).
% cSup_eq
thf(fact_231_cSup__eq__maximum,axiom,
! [Z2: $o,X6: set_o] :
( ( member_o @ Z2 @ X6 )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ( ord_less_eq_o @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup_o @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_232_cSup__eq__maximum,axiom,
! [Z2: nat,X6: set_nat] :
( ( member_nat @ Z2 @ X6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_233_cSup__eq__maximum,axiom,
! [Z2: real,X6: set_real] :
( ( member_real @ Z2 @ X6 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ( comple1385675409528146559p_real @ X6 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_234_Sup__upper2,axiom,
! [U: $o,A: set_o,V: $o] :
( ( member_o @ U @ A )
=> ( ( ord_less_eq_o @ V @ U )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_235_Sup__upper,axiom,
! [X: $o,A: set_o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_236_Sup__least,axiom,
! [A: set_o,Z2: $o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_237_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ? [X4: $o] :
( ( member_o @ X4 @ B )
& ( ord_less_eq_o @ A4 @ X4 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_238_Sup__eqI,axiom,
! [A: set_o,X: $o] :
( ! [Y4: $o] :
( ( member_o @ Y4 @ A )
=> ( ord_less_eq_o @ Y4 @ X ) )
=> ( ! [Y4: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A )
=> ( ord_less_eq_o @ Z4 @ Y4 ) )
=> ( ord_less_eq_o @ X @ Y4 ) )
=> ( ( complete_Sup_Sup_o @ A )
= X ) ) ) ).
% Sup_eqI
thf(fact_239_less__eq__Sup,axiom,
! [A: set_o,U: $o] :
( ! [V2: $o] :
( ( member_o @ V2 @ A )
=> ( ord_less_eq_o @ U @ V2 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ U @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_240_emeasure__pmf__of__set__space,axiom,
! [S: set_nat] :
( ( S != bot_bot_set_nat )
=> ( ( finite_finite_nat @ S )
=> ( ( sigma_emeasure_nat @ ( probab1352011410425470944mf_nat @ ( probab1830274953030043784et_nat @ S ) ) @ S )
= one_on2969667320475766781nnreal ) ) ) ).
% emeasure_pmf_of_set_space
thf(fact_241_emeasure__pmf__of__set__space,axiom,
! [S: set_o] :
( ( S != bot_bot_set_o )
=> ( ( finite_finite_o @ S )
=> ( ( sigma_emeasure_o @ ( probab7036721048548158344_pmf_o @ ( probab8295283590095636704_set_o @ S ) ) @ S )
= one_on2969667320475766781nnreal ) ) ) ).
% emeasure_pmf_of_set_space
thf(fact_242_emeasure__pmf__of__set__space,axiom,
! [S: set_a] :
( ( S != bot_bot_set_a )
=> ( ( finite_finite_a @ S )
=> ( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ ( probab3131728818378861638_set_a @ S ) ) @ S )
= one_on2969667320475766781nnreal ) ) ) ).
% emeasure_pmf_of_set_space
thf(fact_243_finite__unsigned__Hahn__decomposition,axiom,
! [M: sigma_measure_o,N: sigma_measure_o] :
( ( measur2447921437955784316sure_o @ M )
=> ( ( measur2447921437955784316sure_o @ N )
=> ( ( ( sigma_sets_o @ N )
= ( sigma_sets_o @ M ) )
=> ? [X3: set_o] :
( ( member_set_o @ X3 @ ( sigma_sets_o @ M ) )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ ( sigma_sets_o @ M ) )
=> ( ( ord_less_eq_set_o @ Xa @ X3 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ N @ Xa ) @ ( sigma_emeasure_o @ M @ Xa ) ) ) )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ ( sigma_sets_o @ M ) )
=> ( ( ( inf_inf_set_o @ Xa @ X3 )
= bot_bot_set_o )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ M @ Xa ) @ ( sigma_emeasure_o @ N @ Xa ) ) ) ) ) ) ) ) ).
% finite_unsigned_Hahn_decomposition
thf(fact_244_measure__eqI__finite,axiom,
! [M: sigma_measure_a,A: set_a,N: sigma_measure_a] :
( ( ( sigma_sets_a @ M )
= ( pow_a @ A ) )
=> ( ( ( sigma_sets_a @ N )
= ( pow_a @ A ) )
=> ( ( finite_finite_a @ A )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A )
=> ( ( sigma_emeasure_a @ M @ ( insert_a @ A4 @ bot_bot_set_a ) )
= ( sigma_emeasure_a @ N @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) )
=> ( M = N ) ) ) ) ) ).
% measure_eqI_finite
thf(fact_245_measure__eqI__finite,axiom,
! [M: sigma_measure_nat,A: set_nat,N: sigma_measure_nat] :
( ( ( sigma_sets_nat @ M )
= ( pow_nat @ A ) )
=> ( ( ( sigma_sets_nat @ N )
= ( pow_nat @ A ) )
=> ( ( finite_finite_nat @ A )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( sigma_emeasure_nat @ M @ ( insert_nat @ A4 @ bot_bot_set_nat ) )
= ( sigma_emeasure_nat @ N @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) )
=> ( M = N ) ) ) ) ) ).
% measure_eqI_finite
thf(fact_246_measure__eqI__finite,axiom,
! [M: sigma_measure_o,A: set_o,N: sigma_measure_o] :
( ( ( sigma_sets_o @ M )
= ( pow_o @ A ) )
=> ( ( ( sigma_sets_o @ N )
= ( pow_o @ A ) )
=> ( ( finite_finite_o @ A )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ( sigma_emeasure_o @ M @ ( insert_o @ A4 @ bot_bot_set_o ) )
= ( sigma_emeasure_o @ N @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) )
=> ( M = N ) ) ) ) ) ).
% measure_eqI_finite
thf(fact_247_prob__space__restrict__space,axiom,
! [S: set_a,M: sigma_measure_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ M ) )
=> ( ( ( sigma_emeasure_a @ M @ S )
= one_on2969667320475766781nnreal )
=> ( probab7247484486040049089pace_a @ ( sigma_8692839461743104066pace_a @ M @ S ) ) ) ) ).
% prob_space_restrict_space
thf(fact_248_Pow__singleton__iff,axiom,
! [X6: set_o,Y7: set_o] :
( ( ( pow_o @ X6 )
= ( insert_set_o @ Y7 @ bot_bot_set_set_o ) )
= ( ( X6 = bot_bot_set_o )
& ( Y7 = bot_bot_set_o ) ) ) ).
% Pow_singleton_iff
thf(fact_249_insert__absorb2,axiom,
! [X: $o,A: set_o] :
( ( insert_o @ X @ ( insert_o @ X @ A ) )
= ( insert_o @ X @ A ) ) ).
% insert_absorb2
thf(fact_250_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_251_insert__iff,axiom,
! [A2: $o,B2: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_o @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_252_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_253_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_254_insertCI,axiom,
! [A2: $o,B: set_o,B2: $o] :
( ( ~ ( member_o @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% insertCI
thf(fact_255_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertCI
thf(fact_256_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_257_Int__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
& ( member_o @ C @ B ) ) ) ).
% Int_iff
thf(fact_258_Int__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B ) ) ) ).
% Int_iff
thf(fact_259_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_260_IntI,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( ( member_o @ C @ B )
=> ( member_o @ C @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% IntI
thf(fact_261_IntI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% IntI
thf(fact_262_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_263_Un__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
| ( member_o @ C @ B ) ) ) ).
% Un_iff
thf(fact_264_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
| ( member_a @ C @ B ) ) ) ).
% Un_iff
thf(fact_265_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_266_UnCI,axiom,
! [C: $o,B: set_o,A: set_o] :
( ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ A ) )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnCI
thf(fact_267_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ A ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_268_insert__subset,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_269_insert__subset,axiom,
! [X: $o,A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X @ A ) @ B )
= ( ( member_o @ X @ B )
& ( ord_less_eq_set_o @ A @ B ) ) ) ).
% insert_subset
thf(fact_270_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B )
= ( ( member_a @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_271_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_272_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_273_singletonI,axiom,
! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_274_finite__insert,axiom,
! [A2: $o,A: set_o] :
( ( finite_finite_o @ ( insert_o @ A2 @ A ) )
= ( finite_finite_o @ A ) ) ).
% finite_insert
thf(fact_275_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_276_finite__Int,axiom,
! [F: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F @ G ) ) ) ).
% finite_Int
thf(fact_277_space__bot,axiom,
( ( sigma_space_o @ bot_bo5758314138661044393sure_o )
= bot_bot_set_o ) ).
% space_bot
thf(fact_278_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_279_Int__insert__right__if1,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_280_Int__insert__right__if1,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_281_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_282_Int__insert__right__if0,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( inf_inf_set_o @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_283_Int__insert__right__if0,axiom,
! [A2: a,A: set_a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_284_insert__inter__insert,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) ).
% insert_inter_insert
thf(fact_285_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_286_Int__insert__left__if1,axiom,
! [A2: $o,C2: set_o,B: set_o] :
( ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_287_Int__insert__left__if1,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_288_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_289_Int__insert__left__if0,axiom,
! [A2: $o,C2: set_o,B: set_o] :
( ~ ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( inf_inf_set_o @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_290_Int__insert__left__if0,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_291_Un__empty,axiom,
! [A: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A @ B )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_292_finite__Un,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
= ( ( finite_finite_nat @ F )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_293_Un__insert__right,axiom,
! [A: set_o,A2: $o,B: set_o] :
( ( sup_sup_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( sup_sup_set_o @ A @ B ) ) ) ).
% Un_insert_right
thf(fact_294_Un__insert__left,axiom,
! [A2: $o,B: set_o,C2: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( insert_o @ A2 @ ( sup_sup_set_o @ B @ C2 ) ) ) ).
% Un_insert_left
thf(fact_295_singleton__insert__inj__eq_H,axiom,
! [A2: $o,A: set_o,B2: $o] :
( ( ( insert_o @ A2 @ A )
= ( insert_o @ B2 @ bot_bot_set_o ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_296_singleton__insert__inj__eq,axiom,
! [B2: $o,A2: $o,A: set_o] :
( ( ( insert_o @ B2 @ bot_bot_set_o )
= ( insert_o @ A2 @ A ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_297_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_298_disjoint__insert_I2_J,axiom,
! [A: set_a,B2: a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A @ ( insert_a @ B2 @ B ) ) )
= ( ~ ( member_a @ B2 @ A )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_299_disjoint__insert_I2_J,axiom,
! [A: set_o,B2: $o,B: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ A @ ( insert_o @ B2 @ B ) ) )
= ( ~ ( member_o @ B2 @ A )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_300_disjoint__insert_I1_J,axiom,
! [B: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ B @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_301_disjoint__insert_I1_J,axiom,
! [B: set_a,A2: a,A: set_a] :
( ( ( inf_inf_set_a @ B @ ( insert_a @ A2 @ A ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ B @ A )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_302_disjoint__insert_I1_J,axiom,
! [B: set_o,A2: $o,A: set_o] :
( ( ( inf_inf_set_o @ B @ ( insert_o @ A2 @ A ) )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B )
& ( ( inf_inf_set_o @ B @ A )
= bot_bot_set_o ) ) ) ).
% disjoint_insert(1)
thf(fact_303_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B ) )
= ( ~ ( member_nat @ A2 @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_304_insert__disjoint_I2_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B ) )
= ( ~ ( member_a @ A2 @ B )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_305_insert__disjoint_I2_J,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( bot_bot_set_o
= ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B ) )
= ( ~ ( member_o @ A2 @ B )
& ( bot_bot_set_o
= ( inf_inf_set_o @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_306_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B )
& ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_307_insert__disjoint_I1_J,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A2 @ A ) @ B )
= bot_bot_set_a )
= ( ~ ( member_a @ A2 @ B )
& ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_308_insert__disjoint_I1_J,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A ) @ B )
= bot_bot_set_o )
= ( ~ ( member_o @ A2 @ B )
& ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ) ).
% insert_disjoint(1)
thf(fact_309_space__restrict__space2,axiom,
! [Omega: set_a,M: sigma_measure_a] :
( ( member_set_a @ Omega @ ( sigma_sets_a @ M ) )
=> ( ( sigma_space_a @ ( sigma_8692839461743104066pace_a @ M @ Omega ) )
= Omega ) ) ).
% space_restrict_space2
thf(fact_310_sets__bot,axiom,
( ( sigma_sets_o @ bot_bo5758314138661044393sure_o )
= ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).
% sets_bot
thf(fact_311_Pow__empty,axiom,
( ( pow_o @ bot_bot_set_o )
= ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).
% Pow_empty
thf(fact_312_insert__is__Un,axiom,
( insert_o
= ( ^ [A5: $o] : ( sup_sup_set_o @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_313_Un__singleton__iff,axiom,
! [A: set_o,B: set_o,X: $o] :
( ( ( sup_sup_set_o @ A @ B )
= ( insert_o @ X @ bot_bot_set_o ) )
= ( ( ( A = bot_bot_set_o )
& ( B
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_314_singleton__Un__iff,axiom,
! [X: $o,A: set_o,B: set_o] :
( ( ( insert_o @ X @ bot_bot_set_o )
= ( sup_sup_set_o @ A @ B ) )
= ( ( ( A = bot_bot_set_o )
& ( B
= ( insert_o @ X @ bot_bot_set_o ) ) )
| ( ( A
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A
= ( insert_o @ X @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_315_space__restrict__space,axiom,
! [M: sigma_measure_a,Omega: set_a] :
( ( sigma_space_a @ ( sigma_8692839461743104066pace_a @ M @ Omega ) )
= ( inf_inf_set_a @ Omega @ ( sigma_space_a @ M ) ) ) ).
% space_restrict_space
thf(fact_316_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B7: set_nat] :
( ( A
= ( insert_nat @ A2 @ B7 ) )
& ~ ( member_nat @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_317_mk__disjoint__insert,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ? [B7: set_o] :
( ( A
= ( insert_o @ A2 @ B7 ) )
& ~ ( member_o @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_318_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B7: set_a] :
( ( A
= ( insert_a @ A2 @ B7 ) )
& ~ ( member_a @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_319_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_320_Int__insert__right,axiom,
! [A2: $o,A: set_o,B: set_o] :
( ( ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( insert_o @ A2 @ ( inf_inf_set_o @ A @ B ) ) ) )
& ( ~ ( member_o @ A2 @ A )
=> ( ( inf_inf_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( inf_inf_set_o @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_321_Int__insert__right,axiom,
! [A2: a,A: set_a,B: set_a] :
( ( ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( insert_a @ A2 @ ( inf_inf_set_a @ A @ B ) ) ) )
& ( ~ ( member_a @ A2 @ A )
=> ( ( inf_inf_set_a @ A @ ( insert_a @ A2 @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_322_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B ) @ C2 )
= ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_323_Int__insert__left,axiom,
! [A2: $o,C2: set_o,B: set_o] :
( ( ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( insert_o @ A2 @ ( inf_inf_set_o @ B @ C2 ) ) ) )
& ( ~ ( member_o @ A2 @ C2 )
=> ( ( inf_inf_set_o @ ( insert_o @ A2 @ B ) @ C2 )
= ( inf_inf_set_o @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_324_Int__insert__left,axiom,
! [A2: a,C2: set_a,B: set_a] :
( ( ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( insert_a @ A2 @ ( inf_inf_set_a @ B @ C2 ) ) ) )
& ( ~ ( member_a @ A2 @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A2 @ B ) @ C2 )
= ( inf_inf_set_a @ B @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_325_insert__commute,axiom,
! [X: $o,Y: $o,A: set_o] :
( ( insert_o @ X @ ( insert_o @ Y @ A ) )
= ( insert_o @ Y @ ( insert_o @ X @ A ) ) ) ).
% insert_commute
thf(fact_326_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_327_insert__eq__iff,axiom,
! [A2: $o,A: set_o,B2: $o,B: set_o] :
( ~ ( member_o @ A2 @ A )
=> ( ~ ( member_o @ B2 @ B )
=> ( ( ( insert_o @ A2 @ A )
= ( insert_o @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 = (~ B2) )
=> ? [C3: set_o] :
( ( A
= ( insert_o @ B2 @ C3 ) )
& ~ ( member_o @ B2 @ C3 )
& ( B
= ( insert_o @ A2 @ C3 ) )
& ~ ( member_o @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_328_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B2 @ B )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_329_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_330_insert__absorb,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_331_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_332_insert__ident,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_333_insert__ident,axiom,
! [X: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X @ A )
=> ( ~ ( member_o @ X @ B )
=> ( ( ( insert_o @ X @ A )
= ( insert_o @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_334_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B )
=> ( ( ( insert_a @ X @ A )
= ( insert_a @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_335_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B7: set_nat] :
( ( A
= ( insert_nat @ X @ B7 ) )
=> ( member_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_336_Set_Oset__insert,axiom,
! [X: $o,A: set_o] :
( ( member_o @ X @ A )
=> ~ ! [B7: set_o] :
( ( A
= ( insert_o @ X @ B7 ) )
=> ( member_o @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_337_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B7: set_a] :
( ( A
= ( insert_a @ X @ B7 ) )
=> ( member_a @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_338_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat @ A2 @ B )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_339_insertI2,axiom,
! [A2: $o,B: set_o,B2: $o] :
( ( member_o @ A2 @ B )
=> ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% insertI2
thf(fact_340_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a @ A2 @ B )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% insertI2
thf(fact_341_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).
% insertI1
thf(fact_342_insertI1,axiom,
! [A2: $o,B: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B ) ) ).
% insertI1
thf(fact_343_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B ) ) ).
% insertI1
thf(fact_344_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_345_insertE,axiom,
! [A2: $o,B2: $o,A: set_o] :
( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
=> ( ( A2 = (~ B2) )
=> ( member_o @ A2 @ A ) ) ) ).
% insertE
thf(fact_346_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_347_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_348_IntD2,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C @ B ) ) ).
% IntD2
thf(fact_349_IntD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ B ) ) ).
% IntD2
thf(fact_350_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_351_IntD1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C @ A ) ) ).
% IntD1
thf(fact_352_IntD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_353_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_354_UnI2,axiom,
! [C: $o,B: set_o,A: set_o] :
( ( member_o @ C @ B )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI2
thf(fact_355_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a @ C @ B )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_356_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_357_UnI1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( member_o @ C @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI1
thf(fact_358_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( member_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_359_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_360_IntE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( inf_inf_set_o @ A @ B ) )
=> ~ ( ( member_o @ C @ A )
=> ~ ( member_o @ C @ B ) ) ) ).
% IntE
thf(fact_361_IntE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B ) ) ) ).
% IntE
thf(fact_362_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_363_UnE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( sup_sup_set_o @ A @ B ) )
=> ( ~ ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% UnE
thf(fact_364_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% UnE
thf(fact_365_insert__partition,axiom,
! [X: set_o,F: set_set_o] :
( ~ ( member_set_o @ X @ F )
=> ( ! [X3: set_o] :
( ( member_set_o @ X3 @ ( insert_set_o @ X @ F ) )
=> ! [Xa2: set_o] :
( ( member_set_o @ Xa2 @ ( insert_set_o @ X @ F ) )
=> ( ( X3 != Xa2 )
=> ( ( inf_inf_set_o @ X3 @ Xa2 )
= bot_bot_set_o ) ) ) )
=> ( ( inf_inf_set_o @ X @ ( comple90263536869209701_set_o @ F ) )
= bot_bot_set_o ) ) ) ).
% insert_partition
thf(fact_366_restrict__restrict__space,axiom,
! [A: set_a,M: sigma_measure_a,B: set_a] :
( ( member_set_a @ ( inf_inf_set_a @ A @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ ( inf_inf_set_a @ B @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( ( sigma_8692839461743104066pace_a @ ( sigma_8692839461743104066pace_a @ M @ A ) @ B )
= ( sigma_8692839461743104066pace_a @ M @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ).
% restrict_restrict_space
thf(fact_367_measure__pmf_Oemeasure__space__1,axiom,
! [M: probab3364570286911266904_pmf_a] :
( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
= one_on2969667320475766781nnreal ) ).
% measure_pmf.emeasure_space_1
thf(fact_368_prob__spaceI,axiom,
! [M: sigma_measure_a] :
( ( ( sigma_emeasure_a @ M @ ( sigma_space_a @ M ) )
= one_on2969667320475766781nnreal )
=> ( probab7247484486040049089pace_a @ M ) ) ).
% prob_spaceI
thf(fact_369_prob__space_Oemeasure__space__1,axiom,
! [M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( sigma_emeasure_a @ M @ ( sigma_space_a @ M ) )
= one_on2969667320475766781nnreal ) ) ).
% prob_space.emeasure_space_1
thf(fact_370_Un__empty__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% Un_empty_right
thf(fact_371_Un__empty__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B )
= B ) ).
% Un_empty_left
thf(fact_372_space__empty__eq__bot,axiom,
! [A2: sigma_measure_o] :
( ( ( sigma_space_o @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bo5758314138661044393sure_o ) ) ).
% space_empty_eq_bot
thf(fact_373_finite__UnI,axiom,
! [F: set_nat,G: set_nat] :
( ( finite_finite_nat @ F )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).
% finite_UnI
thf(fact_374_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_375_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_376_insert__subsetI,axiom,
! [X: nat,A: set_nat,X6: set_nat] :
( ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_377_insert__subsetI,axiom,
! [X: $o,A: set_o,X6: set_o] :
( ( member_o @ X @ A )
=> ( ( ord_less_eq_set_o @ X6 @ A )
=> ( ord_less_eq_set_o @ ( insert_o @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_378_insert__subsetI,axiom,
! [X: a,A: set_a,X6: set_a] :
( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_379_insert__mono,axiom,
! [C2: set_o,D: set_o,A2: $o] :
( ( ord_less_eq_set_o @ C2 @ D )
=> ( ord_less_eq_set_o @ ( insert_o @ A2 @ C2 ) @ ( insert_o @ A2 @ D ) ) ) ).
% insert_mono
thf(fact_380_subset__insert,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_381_subset__insert,axiom,
! [X: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X @ A )
=> ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B ) )
= ( ord_less_eq_set_o @ A @ B ) ) ) ).
% subset_insert
thf(fact_382_subset__insert,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_383_subset__insertI,axiom,
! [B: set_o,A2: $o] : ( ord_less_eq_set_o @ B @ ( insert_o @ A2 @ B ) ) ).
% subset_insertI
thf(fact_384_subset__insertI2,axiom,
! [A: set_o,B: set_o,B2: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_385_singleton__inject,axiom,
! [A2: $o,B2: $o] :
( ( ( insert_o @ A2 @ bot_bot_set_o )
= ( insert_o @ B2 @ bot_bot_set_o ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_386_insert__not__empty,axiom,
! [A2: $o,A: set_o] :
( ( insert_o @ A2 @ A )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_387_doubleton__eq__iff,axiom,
! [A2: $o,B2: $o,C: $o,D2: $o] :
( ( ( insert_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_388_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_389_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_390_singleton__iff,axiom,
! [B2: $o,A2: $o] :
( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_391_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_392_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_393_singletonD,axiom,
! [B2: $o,A2: $o] :
( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_394_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_395_Int__Collect__mono,axiom,
! [A: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_396_Int__Collect__mono,axiom,
! [A: set_a,B: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_397_disjoint__iff__not__equal,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ B )
=> ( X2 = (~ Y2) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_398_Int__empty__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_399_Int__empty__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_400_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_401_disjoint__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
= ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ~ ( member_a @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_402_disjoint__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ~ ( member_o @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_403_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_404_Int__emptyI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B ) )
=> ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_405_Int__emptyI,axiom,
! [A: set_o,B: set_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ A )
=> ~ ( member_o @ X3 @ B ) )
=> ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_406_finite_OinsertI,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( finite_finite_o @ ( insert_o @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_407_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_408_prob__space__axioms__def,axiom,
( probab8302655048591552734ioms_a
= ( ^ [M3: sigma_measure_a] :
( ( sigma_emeasure_a @ M3 @ ( sigma_space_a @ M3 ) )
= one_on2969667320475766781nnreal ) ) ) ).
% prob_space_axioms_def
thf(fact_409_prob__space__axioms_Ointro,axiom,
! [M: sigma_measure_a] :
( ( ( sigma_emeasure_a @ M @ ( sigma_space_a @ M ) )
= one_on2969667320475766781nnreal )
=> ( probab8302655048591552734ioms_a @ M ) ) ).
% prob_space_axioms.intro
thf(fact_410_sets__restrict__space__iff,axiom,
! [Omega: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( inf_inf_set_a @ Omega @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ Omega ) ) )
= ( ( ord_less_eq_set_a @ A @ Omega )
& ( member_set_a @ A @ ( sigma_sets_a @ M ) ) ) ) ) ).
% sets_restrict_space_iff
thf(fact_411_space__empty__iff,axiom,
! [N: sigma_measure_o] :
( ( ( sigma_space_o @ N )
= bot_bot_set_o )
= ( ( sigma_sets_o @ N )
= ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ) ).
% space_empty_iff
thf(fact_412_measure__pmf_Oemeasure__space__le__1,axiom,
! [M: probab3364570286911266904_pmf_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) ) @ one_on2969667320475766781nnreal ) ).
% measure_pmf.emeasure_space_le_1
thf(fact_413_emeasure__restrict__space,axiom,
! [Omega: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( inf_inf_set_a @ Omega @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ A @ Omega )
=> ( ( sigma_emeasure_a @ ( sigma_8692839461743104066pace_a @ M @ Omega ) @ A )
= ( sigma_emeasure_a @ M @ A ) ) ) ) ).
% emeasure_restrict_space
thf(fact_414_subset__singletonD,axiom,
! [A: set_o,X: $o] :
( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
=> ( ( A = bot_bot_set_o )
| ( A
= ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_415_subset__singleton__iff,axiom,
! [X6: set_o,A2: $o] :
( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A2 @ bot_bot_set_o ) )
= ( ( X6 = bot_bot_set_o )
| ( X6
= ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_416_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X3 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_417_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X3 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_418_infinite__finite__induct,axiom,
! [P: set_o > $o,A: set_o] :
( ! [A6: set_o] :
( ~ ( finite_finite_o @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,F4: set_o] :
( ( finite_finite_o @ F4 )
=> ( ~ ( member_o @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_o @ X3 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_419_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X3: a] : ( P @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_420_finite__ne__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( F != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_421_finite__ne__induct,axiom,
! [F: set_o,P: set_o > $o] :
( ( finite_finite_o @ F )
=> ( ( F != bot_bot_set_o )
=> ( ! [X3: $o] : ( P @ ( insert_o @ X3 @ bot_bot_set_o ) )
=> ( ! [X3: $o,F4: set_o] :
( ( finite_finite_o @ F4 )
=> ( ( F4 != bot_bot_set_o )
=> ( ~ ( member_o @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_o @ X3 @ F4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_422_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ X3 @ F4 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_423_finite__induct,axiom,
! [F: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X3 @ F4 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_424_finite__induct,axiom,
! [F: set_o,P: set_o > $o] :
( ( finite_finite_o @ F )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,F4: set_o] :
( ( finite_finite_o @ F4 )
=> ( ~ ( member_o @ X3 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_o @ X3 @ F4 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_425_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A3: set_nat,B4: nat] :
( ( A5
= ( insert_nat @ B4 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_426_finite_Osimps,axiom,
( finite_finite_o
= ( ^ [A5: set_o] :
( ( A5 = bot_bot_set_o )
| ? [A3: set_o,B4: $o] :
( ( A5
= ( insert_o @ B4 @ A3 ) )
& ( finite_finite_o @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_427_finite_Ocases,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A4: nat] :
( A2
= ( insert_nat @ A4 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_428_finite_Ocases,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( A2 != bot_bot_set_o )
=> ~ ! [A6: set_o] :
( ? [A4: $o] :
( A2
= ( insert_o @ A4 @ A6 ) )
=> ~ ( finite_finite_o @ A6 ) ) ) ) ).
% finite.cases
thf(fact_429_sets_Oinsert__in__sets,axiom,
! [X: $o,M: sigma_measure_o,A: set_o] :
( ( member_set_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) )
=> ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
=> ( member_set_o @ ( insert_o @ X @ A ) @ ( sigma_sets_o @ M ) ) ) ) ).
% sets.insert_in_sets
thf(fact_430_measure__pmf_Osubprob__not__empty,axiom,
! [M: probab1498759712122475378_pmf_o] :
( ( sigma_space_o @ ( probab7036721048548158344_pmf_o @ M ) )
!= bot_bot_set_o ) ).
% measure_pmf.subprob_not_empty
thf(fact_431_measure__pmf_Osubprob__not__empty,axiom,
! [M: probab3364570286911266904_pmf_a] :
( ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) )
!= bot_bot_set_a ) ).
% measure_pmf.subprob_not_empty
thf(fact_432_prob__space_Onot__empty,axiom,
! [M: sigma_measure_o] :
( ( probab1190487603588612059pace_o @ M )
=> ( ( sigma_space_o @ M )
!= bot_bot_set_o ) ) ).
% prob_space.not_empty
thf(fact_433_prob__space_Onot__empty,axiom,
! [M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( sigma_space_a @ M )
!= bot_bot_set_a ) ) ).
% prob_space.not_empty
thf(fact_434_sets__restrict__restrict__space,axiom,
! [M: sigma_measure_a,A: set_a,B: set_a] :
( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ ( sigma_8692839461743104066pace_a @ M @ A ) @ B ) )
= ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% sets_restrict_restrict_space
thf(fact_435_finite__Sup__in,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y4: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y4 @ A )
=> ( member_o @ ( sup_sup_o @ X3 @ Y4 ) @ A ) ) )
=> ( member_o @ ( complete_Sup_Sup_o @ A ) @ A ) ) ) ) ).
% finite_Sup_in
thf(fact_436_finite__subset__induct_H,axiom,
! [F: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A4 @ A )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ~ ( member_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_437_finite__subset__induct_H,axiom,
! [F: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A4 @ A )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ~ ( member_nat @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_438_finite__subset__induct_H,axiom,
! [F: set_o,A: set_o,P: set_o > $o] :
( ( finite_finite_o @ F )
=> ( ( ord_less_eq_set_o @ F @ A )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A4: $o,F4: set_o] :
( ( finite_finite_o @ F4 )
=> ( ( member_o @ A4 @ A )
=> ( ( ord_less_eq_set_o @ F4 @ A )
=> ( ~ ( member_o @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_o @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_439_finite__subset__induct,axiom,
! [F: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A4 @ A )
=> ( ~ ( member_a @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_a @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_440_finite__subset__induct,axiom,
! [F: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F )
=> ( ( ord_less_eq_set_nat @ F @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_441_finite__subset__induct,axiom,
! [F: set_o,A: set_o,P: set_o > $o] :
( ( finite_finite_o @ F )
=> ( ( ord_less_eq_set_o @ F @ A )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A4: $o,F4: set_o] :
( ( finite_finite_o @ F4 )
=> ( ( member_o @ A4 @ A )
=> ( ~ ( member_o @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_o @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_442_sets__eq__bot,axiom,
! [M: sigma_measure_o] :
( ( ( sigma_sets_o @ M )
= ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) )
= ( M = bot_bo5758314138661044393sure_o ) ) ).
% sets_eq_bot
thf(fact_443_sets__eq__bot2,axiom,
! [M: sigma_measure_o] :
( ( ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o )
= ( sigma_sets_o @ M ) )
= ( M = bot_bo5758314138661044393sure_o ) ) ).
% sets_eq_bot2
thf(fact_444_emeasure__Int__set__pmf,axiom,
! [P2: probab1498759712122475378_pmf_o,A: set_o] :
( ( sigma_emeasure_o @ ( probab7036721048548158344_pmf_o @ P2 ) @ ( inf_inf_set_o @ A @ ( probab7458556812659319003_pmf_o @ P2 ) ) )
= ( sigma_emeasure_o @ ( probab7036721048548158344_pmf_o @ P2 ) @ A ) ) ).
% emeasure_Int_set_pmf
thf(fact_445_emeasure__Int__set__pmf,axiom,
! [P2: probab469873468395307276mf_nat,A: set_nat] :
( ( sigma_emeasure_nat @ ( probab1352011410425470944mf_nat @ P2 ) @ ( inf_inf_set_nat @ A @ ( probab3271515132085200205mf_nat @ P2 ) ) )
= ( sigma_emeasure_nat @ ( probab1352011410425470944mf_nat @ P2 ) @ A ) ) ).
% emeasure_Int_set_pmf
thf(fact_446_emeasure__Int__set__pmf,axiom,
! [P2: probab3364570286911266904_pmf_a,A: set_a] :
( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ P2 ) @ ( inf_inf_set_a @ A @ ( probab49036049091589825_pmf_a @ P2 ) ) )
= ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ P2 ) @ A ) ) ).
% emeasure_Int_set_pmf
thf(fact_447_measure__pmf_Oemeasure__le__1,axiom,
! [M: probab3364570286911266904_pmf_a,S: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ S ) @ one_on2969667320475766781nnreal ) ).
% measure_pmf.emeasure_le_1
thf(fact_448_measure__pmf_Oemeasure__ge__1__iff,axiom,
! [M: probab3364570286911266904_pmf_a,A: set_a] :
( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ A ) )
= ( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ A )
= one_on2969667320475766781nnreal ) ) ).
% measure_pmf.emeasure_ge_1_iff
thf(fact_449_measure__pmf_Osubprob__emeasure__le__1,axiom,
! [M: probab3364570286911266904_pmf_a,X6: set_a] : ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ X6 ) @ one_on2969667320475766781nnreal ) ).
% measure_pmf.subprob_emeasure_le_1
thf(fact_450_emeasure__pmf,axiom,
! [M: probab1498759712122475378_pmf_o] :
( ( sigma_emeasure_o @ ( probab7036721048548158344_pmf_o @ M ) @ ( probab7458556812659319003_pmf_o @ M ) )
= one_on2969667320475766781nnreal ) ).
% emeasure_pmf
thf(fact_451_emeasure__pmf,axiom,
! [M: probab469873468395307276mf_nat] :
( ( sigma_emeasure_nat @ ( probab1352011410425470944mf_nat @ M ) @ ( probab3271515132085200205mf_nat @ M ) )
= one_on2969667320475766781nnreal ) ).
% emeasure_pmf
thf(fact_452_emeasure__pmf,axiom,
! [M: probab3364570286911266904_pmf_a] :
( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ ( probab49036049091589825_pmf_a @ M ) )
= one_on2969667320475766781nnreal ) ).
% emeasure_pmf
thf(fact_453_prob__space_Omeasure__le__1,axiom,
! [M: sigma_measure_a,X6: set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ X6 ) @ one_on2969667320475766781nnreal ) ) ).
% prob_space.measure_le_1
thf(fact_454_prob__space_Oemeasure__le__1,axiom,
! [M: sigma_measure_a,S: set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_a @ M @ S ) @ one_on2969667320475766781nnreal ) ) ).
% prob_space.emeasure_le_1
thf(fact_455_prob__space_Oemeasure__ge__1__iff,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( ord_le3935885782089961368nnreal @ one_on2969667320475766781nnreal @ ( sigma_emeasure_a @ M @ A ) )
= ( ( sigma_emeasure_a @ M @ A )
= one_on2969667320475766781nnreal ) ) ) ).
% prob_space.emeasure_ge_1_iff
thf(fact_456_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_457_le__sup__iff,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_real @ X @ Z2 )
& ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_458_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_459_sup_Obounded__iff,axiom,
! [B2: real,C: real,A2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_real @ B2 @ A2 )
& ( ord_less_eq_real @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_460_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_461_le__inf__iff,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z2 ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_462_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_463_inf_Obounded__iff,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B2 @ C ) )
= ( ( ord_less_eq_real @ A2 @ B2 )
& ( ord_less_eq_real @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_464_vector__space__over__itself_Ofinite__Basis,axiom,
finite_finite_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ).
% vector_space_over_itself.finite_Basis
thf(fact_465_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F2: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_466_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_467_finite__ranking__induct,axiom,
! [S: set_o,P: set_o > $o,F2: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,S2: set_o] :
( ( finite_finite_o @ S2 )
=> ( ! [Y5: $o] :
( ( member_o @ Y5 @ S2 )
=> ( ord_less_eq_nat @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_o @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_468_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F2: a > real] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X3: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y5: a] :
( ( member_a @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_469_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F2: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_470_finite__ranking__induct,axiom,
! [S: set_o,P: set_o > $o,F2: $o > real] :
( ( finite_finite_o @ S )
=> ( ( P @ bot_bot_set_o )
=> ( ! [X3: $o,S2: set_o] :
( ( finite_finite_o @ S2 )
=> ( ! [Y5: $o] :
( ( member_o @ Y5 @ S2 )
=> ( ord_less_eq_real @ ( F2 @ Y5 ) @ ( F2 @ X3 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_o @ X3 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_471_main__part__null__part__Int,axiom,
! [S: set_o,M: sigma_measure_o] :
( ( member_set_o @ S @ ( sigma_sets_o @ ( comple48332195503990434tion_o @ M ) ) )
=> ( ( inf_inf_set_o @ ( complete_main_part_o @ M @ S ) @ ( complete_null_part_o @ M @ S ) )
= bot_bot_set_o ) ) ).
% main_part_null_part_Int
thf(fact_472_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_473_inf__sup__ord_I2_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_474_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_475_inf__sup__ord_I1_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_476_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_477_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_478_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_479_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_480_le__infE,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_481_le__infE,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_real @ X @ A2 )
=> ~ ( ord_less_eq_real @ X @ B2 ) ) ) ).
% le_infE
thf(fact_482_le__infI,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_483_le__infI,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less_eq_real @ X @ A2 )
=> ( ( ord_less_eq_real @ X @ B2 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_484_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_485_inf__mono,axiom,
! [A2: real,C: real,B2: real,D2: real] :
( ( ord_less_eq_real @ A2 @ C )
=> ( ( ord_less_eq_real @ B2 @ D2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B2 ) @ ( inf_inf_real @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_486_le__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_487_le__infI1,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less_eq_real @ A2 @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_488_le__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_489_le__infI2,axiom,
! [B2: real,X: real,A2: real] :
( ( ord_less_eq_real @ B2 @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_490_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_491_inf_OorderE,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( A2
= ( inf_inf_real @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_492_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_493_inf_OorderI,axiom,
! [A2: real,B2: real] :
( ( A2
= ( inf_inf_real @ A2 @ B2 ) )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_494_inf__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ( ord_less_eq_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ ( F2 @ Y4 @ Z ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_495_inf__unique,axiom,
! [F2: real > real > real,X: real,Y: real] :
( ! [X3: real,Y4: real] : ( ord_less_eq_real @ ( F2 @ X3 @ Y4 ) @ X3 )
=> ( ! [X3: real,Y4: real] : ( ord_less_eq_real @ ( F2 @ X3 @ Y4 ) @ Y4 )
=> ( ! [X3: real,Y4: real,Z: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ( ord_less_eq_real @ X3 @ Z )
=> ( ord_less_eq_real @ X3 @ ( F2 @ Y4 @ Z ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_496_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_497_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( inf_inf_real @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_498_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_499_inf_Oabsorb1,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( inf_inf_real @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_500_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_501_inf_Oabsorb2,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( inf_inf_real @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_502_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_503_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_504_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_505_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_506_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_507_inf_OboundedE,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B2 @ C ) )
=> ~ ( ( ord_less_eq_real @ A2 @ B2 )
=> ~ ( ord_less_eq_real @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_508_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_509_inf_OboundedI,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ A2 @ C )
=> ( ord_less_eq_real @ A2 @ ( inf_inf_real @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_510_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_511_inf__greatest,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z2 )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_512_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( A5
= ( inf_inf_nat @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_513_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( A5
= ( inf_inf_real @ A5 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_514_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_515_inf_Ocobounded1,axiom,
! [A2: real,B2: real] : ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_516_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_517_inf_Ocobounded2,axiom,
! [A2: real,B2: real] : ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_518_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( inf_inf_nat @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_519_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( inf_inf_real @ A5 @ B4 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_520_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_521_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( inf_inf_real @ A5 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_522_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_523_inf_OcoboundedI1,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ A2 @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_524_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_525_inf_OcoboundedI2,axiom,
! [B2: real,C: real,A2: real] :
( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_526_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_527_inf__sup__ord_I4_J,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_528_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_529_inf__sup__ord_I3_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_530_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_531_le__supE,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_real @ A2 @ X )
=> ~ ( ord_less_eq_real @ B2 @ X ) ) ) ).
% le_supE
thf(fact_532_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_533_le__supI,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less_eq_real @ A2 @ X )
=> ( ( ord_less_eq_real @ B2 @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_534_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_535_sup__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge1
thf(fact_536_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_537_sup__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge2
thf(fact_538_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_539_le__supI1,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less_eq_real @ X @ A2 )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_540_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_541_le__supI2,axiom,
! [X: real,B2: real,A2: real] :
( ( ord_less_eq_real @ X @ B2 )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_542_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_543_sup_Omono,axiom,
! [C: real,A2: real,D2: real,B2: real] :
( ( ord_less_eq_real @ C @ A2 )
=> ( ( ord_less_eq_real @ D2 @ B2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ C @ D2 ) @ ( sup_sup_real @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_544_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_545_sup__mono,axiom,
! [A2: real,C: real,B2: real,D2: real] :
( ( ord_less_eq_real @ A2 @ C )
=> ( ( ord_less_eq_real @ B2 @ D2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ A2 @ B2 ) @ ( sup_sup_real @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_546_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_547_sup__least,axiom,
! [Y: real,X: real,Z2: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ Z2 @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_548_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( sup_sup_nat @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_549_le__iff__sup,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( sup_sup_real @ X2 @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_550_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_551_sup_OorderE,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( A2
= ( sup_sup_real @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_552_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_553_sup_OorderI,axiom,
! [A2: real,B2: real] :
( ( A2
= ( sup_sup_real @ A2 @ B2 ) )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_554_sup__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y: nat] :
( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ X3 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ Y4 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( F2 @ Y4 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_555_sup__unique,axiom,
! [F2: real > real > real,X: real,Y: real] :
( ! [X3: real,Y4: real] : ( ord_less_eq_real @ X3 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: real,Y4: real] : ( ord_less_eq_real @ Y4 @ ( F2 @ X3 @ Y4 ) )
=> ( ! [X3: real,Y4: real,Z: real] :
( ( ord_less_eq_real @ Y4 @ X3 )
=> ( ( ord_less_eq_real @ Z @ X3 )
=> ( ord_less_eq_real @ ( F2 @ Y4 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_real @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_556_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_557_sup_Oabsorb1,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( sup_sup_real @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_558_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_559_sup_Oabsorb2,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( sup_sup_real @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_560_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_561_sup__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( sup_sup_real @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_562_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_563_sup__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( sup_sup_real @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_564_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_565_sup_OboundedE,axiom,
! [B2: real,C: real,A2: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_real @ B2 @ A2 )
=> ~ ( ord_less_eq_real @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_566_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_567_sup_OboundedI,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ A2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_568_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_569_sup_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( A5
= ( sup_sup_real @ A5 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_570_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_571_sup_Ocobounded1,axiom,
! [A2: real,B2: real] : ( ord_less_eq_real @ A2 @ ( sup_sup_real @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_572_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_573_sup_Ocobounded2,axiom,
! [B2: real,A2: real] : ( ord_less_eq_real @ B2 @ ( sup_sup_real @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_574_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_575_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( sup_sup_real @ A5 @ B4 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_576_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( sup_sup_nat @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_577_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( sup_sup_real @ A5 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_578_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_579_sup_OcoboundedI1,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_eq_real @ C @ A2 )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_580_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_581_sup_OcoboundedI2,axiom,
! [C: real,B2: real,A2: real] :
( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_582_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_583_distrib__inf__le,axiom,
! [X: real,Y: real,Z2: real] : ( ord_less_eq_real @ ( sup_sup_real @ ( inf_inf_real @ X @ Y ) @ ( inf_inf_real @ X @ Z2 ) ) @ ( inf_inf_real @ X @ ( sup_sup_real @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_584_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_585_distrib__sup__le,axiom,
! [X: real,Y: real,Z2: real] : ( ord_less_eq_real @ ( sup_sup_real @ X @ ( inf_inf_real @ Y @ Z2 ) ) @ ( inf_inf_real @ ( sup_sup_real @ X @ Y ) @ ( sup_sup_real @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_586_set__cond__pmf,axiom,
! [P2: probab3364570286911266904_pmf_a,S3: set_a] :
( ( ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 )
!= bot_bot_set_a )
=> ( ( probab49036049091589825_pmf_a @ ( probab4270644268839999083_pmf_a @ P2 @ S3 ) )
= ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 ) ) ) ).
% set_cond_pmf
thf(fact_587_set__cond__pmf,axiom,
! [P2: probab1498759712122475378_pmf_o,S3: set_o] :
( ( ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 )
!= bot_bot_set_o )
=> ( ( probab7458556812659319003_pmf_o @ ( probab8494970989125154181_pmf_o @ P2 @ S3 ) )
= ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 ) ) ) ).
% set_cond_pmf
thf(fact_588_set__cond__pmf,axiom,
! [P2: probab469873468395307276mf_nat,S3: set_nat] :
( ( ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 )
!= bot_bot_set_nat )
=> ( ( probab3271515132085200205mf_nat @ ( probab7431941403989380899mf_nat @ P2 @ S3 ) )
= ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 ) ) ) ).
% set_cond_pmf
thf(fact_589_the__elem__eq,axiom,
! [X: $o] :
( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
= X ) ).
% the_elem_eq
thf(fact_590_is__singletonI,axiom,
! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_591_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
( A3
= ( insert_o @ ( the_elem_o @ A3 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_592_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_593_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X3: a,Y4: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_594_is__singletonI_H,axiom,
! [A: set_o] :
( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y4: $o] :
( ( member_o @ X3 @ A )
=> ( ( member_o @ Y4 @ A )
=> ( X3 = Y4 ) ) )
=> ( is_singleton_o @ A ) ) ) ).
% is_singletonI'
thf(fact_595_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A3: set_o] :
? [X2: $o] :
( A3
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_596_is__singletonE,axiom,
! [A: set_o] :
( ( is_singleton_o @ A )
=> ~ ! [X3: $o] :
( A
!= ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_597_cond__pmf_Orep__eq,axiom,
! [P2: probab1498759712122475378_pmf_o,S3: set_o] :
( ( ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 )
!= bot_bot_set_o )
=> ( ( probab7036721048548158344_pmf_o @ ( probab8494970989125154181_pmf_o @ P2 @ S3 ) )
= ( nonneg5638544851443855887sure_o @ ( probab7036721048548158344_pmf_o @ P2 ) @ S3 ) ) ) ).
% cond_pmf.rep_eq
thf(fact_598_cond__pmf_Orep__eq,axiom,
! [P2: probab469873468395307276mf_nat,S3: set_nat] :
( ( ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 )
!= bot_bot_set_nat )
=> ( ( probab1352011410425470944mf_nat @ ( probab7431941403989380899mf_nat @ P2 @ S3 ) )
= ( nonneg5218579358776314137re_nat @ ( probab1352011410425470944mf_nat @ P2 ) @ S3 ) ) ) ).
% cond_pmf.rep_eq
thf(fact_599_cond__pmf_Orep__eq,axiom,
! [P2: probab3364570286911266904_pmf_a,S3: set_a] :
( ( ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 )
!= bot_bot_set_a )
=> ( ( probab7257411610070727406_pmf_a @ ( probab4270644268839999083_pmf_a @ P2 @ S3 ) )
= ( nonneg6757527617543859701sure_a @ ( probab7257411610070727406_pmf_a @ P2 ) @ S3 ) ) ) ).
% cond_pmf.rep_eq
thf(fact_600_arg__min__least,axiom,
! [S: set_a,Y: a,F2: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_nat @ ( F2 @ ( lattic6340287419671400565_a_nat @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_601_arg__min__least,axiom,
! [S: set_nat,Y: nat,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_602_arg__min__least,axiom,
! [S: set_o,Y: $o,F2: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ( ( member_o @ Y @ S )
=> ( ord_less_eq_nat @ ( F2 @ ( lattic2775856028456453135_o_nat @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_603_arg__min__least,axiom,
! [S: set_a,Y: a,F2: a > real] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y @ S )
=> ( ord_less_eq_real @ ( F2 @ ( lattic7288945864786915537a_real @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_604_arg__min__least,axiom,
! [S: set_nat,Y: nat,F2: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_real @ ( F2 @ ( lattic488527866317076247t_real @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_605_arg__min__least,axiom,
! [S: set_o,Y: $o,F2: $o > real] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ( ( member_o @ Y @ S )
=> ( ord_less_eq_real @ ( F2 @ ( lattic8697145971487455083o_real @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_606_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_607_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_608_cond__pmf_Otransfer,axiom,
! [P2: probab1498759712122475378_pmf_o,S3: set_o] :
( ( ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 )
!= bot_bot_set_o )
=> ( probab2081575740620332389_pmf_o @ ( nonneg5638544851443855887sure_o @ ( probab7036721048548158344_pmf_o @ P2 ) @ S3 ) @ ( probab8494970989125154181_pmf_o @ P2 @ S3 ) ) ) ).
% cond_pmf.transfer
thf(fact_609_cond__pmf_Otransfer,axiom,
! [P2: probab469873468395307276mf_nat,S3: set_nat] :
( ( ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 )
!= bot_bot_set_nat )
=> ( probab6917518385137753923mf_nat @ ( nonneg5218579358776314137re_nat @ ( probab1352011410425470944mf_nat @ P2 ) @ S3 ) @ ( probab7431941403989380899mf_nat @ P2 @ S3 ) ) ) ).
% cond_pmf.transfer
thf(fact_610_cond__pmf_Otransfer,axiom,
! [P2: probab3364570286911266904_pmf_a,S3: set_a] :
( ( ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 )
!= bot_bot_set_a )
=> ( probab451167891469848139_pmf_a @ ( nonneg6757527617543859701sure_a @ ( probab7257411610070727406_pmf_a @ P2 ) @ S3 ) @ ( probab4270644268839999083_pmf_a @ P2 @ S3 ) ) ) ).
% cond_pmf.transfer
thf(fact_611_cond__pmf_Oabs__eq,axiom,
! [P2: probab1498759712122475378_pmf_o,S3: set_o] :
( ( ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 )
!= bot_bot_set_o )
=> ( ( probab8494970989125154181_pmf_o @ P2 @ S3 )
= ( probab597269709993677180_pmf_o @ ( nonneg5638544851443855887sure_o @ ( probab7036721048548158344_pmf_o @ P2 ) @ S3 ) ) ) ) ).
% cond_pmf.abs_eq
thf(fact_612_cond__pmf_Oabs__eq,axiom,
! [P2: probab469873468395307276mf_nat,S3: set_nat] :
( ( ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 )
!= bot_bot_set_nat )
=> ( ( probab7431941403989380899mf_nat @ P2 @ S3 )
= ( probab5843134691084328684mf_nat @ ( nonneg5218579358776314137re_nat @ ( probab1352011410425470944mf_nat @ P2 ) @ S3 ) ) ) ) ).
% cond_pmf.abs_eq
thf(fact_613_cond__pmf_Oabs__eq,axiom,
! [P2: probab3364570286911266904_pmf_a,S3: set_a] :
( ( ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 )
!= bot_bot_set_a )
=> ( ( probab4270644268839999083_pmf_a @ P2 @ S3 )
= ( probab1189994150051702498_pmf_a @ ( nonneg6757527617543859701sure_a @ ( probab7257411610070727406_pmf_a @ P2 ) @ S3 ) ) ) ) ).
% cond_pmf.abs_eq
thf(fact_614_cond__pmf__def,axiom,
! [P2: probab1498759712122475378_pmf_o,S3: set_o] :
( ( ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 )
!= bot_bot_set_o )
=> ( ( probab8494970989125154181_pmf_o @ P2 @ S3 )
= ( probab597269709993677180_pmf_o @ ( nonneg5638544851443855887sure_o @ ( probab7036721048548158344_pmf_o @ P2 ) @ S3 ) ) ) ) ).
% cond_pmf_def
thf(fact_615_cond__pmf__def,axiom,
! [P2: probab469873468395307276mf_nat,S3: set_nat] :
( ( ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 )
!= bot_bot_set_nat )
=> ( ( probab7431941403989380899mf_nat @ P2 @ S3 )
= ( probab5843134691084328684mf_nat @ ( nonneg5218579358776314137re_nat @ ( probab1352011410425470944mf_nat @ P2 ) @ S3 ) ) ) ) ).
% cond_pmf_def
thf(fact_616_cond__pmf__def,axiom,
! [P2: probab3364570286911266904_pmf_a,S3: set_a] :
( ( ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 )
!= bot_bot_set_a )
=> ( ( probab4270644268839999083_pmf_a @ P2 @ S3 )
= ( probab1189994150051702498_pmf_a @ ( nonneg6757527617543859701sure_a @ ( probab7257411610070727406_pmf_a @ P2 ) @ S3 ) ) ) ) ).
% cond_pmf_def
thf(fact_617_unsigned__Hahn__decomposition,axiom,
! [N: sigma_measure_o,M: sigma_measure_o,A: set_o] :
( ( ( sigma_sets_o @ N )
= ( sigma_sets_o @ M ) )
=> ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
=> ( ( ( sigma_emeasure_o @ M @ A )
!= top_to1496364449551166952nnreal )
=> ( ( ( sigma_emeasure_o @ N @ A )
!= top_to1496364449551166952nnreal )
=> ? [X3: set_o] :
( ( member_set_o @ X3 @ ( sigma_sets_o @ M ) )
& ( ord_less_eq_set_o @ X3 @ A )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ ( sigma_sets_o @ M ) )
=> ( ( ord_less_eq_set_o @ Xa @ X3 )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ N @ Xa ) @ ( sigma_emeasure_o @ M @ Xa ) ) ) )
& ! [Xa: set_o] :
( ( member_set_o @ Xa @ ( sigma_sets_o @ M ) )
=> ( ( ord_less_eq_set_o @ Xa @ A )
=> ( ( ( inf_inf_set_o @ Xa @ X3 )
= bot_bot_set_o )
=> ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ M @ Xa ) @ ( sigma_emeasure_o @ N @ Xa ) ) ) ) ) ) ) ) ) ) ).
% unsigned_Hahn_decomposition
thf(fact_618_measure__pmf_Oemeasure__subprob__space__less__top,axiom,
! [M: probab3364570286911266904_pmf_a,A: set_a] :
( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ A )
!= top_to1496364449551166952nnreal ) ).
% measure_pmf.emeasure_subprob_space_less_top
thf(fact_619_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_620_top_Oextremum__uniqueI,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
=> ( A2 = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_621_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_622_top_Oextremum__unique,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
= ( A2 = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_623_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_624_top__greatest,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% top_greatest
thf(fact_625_measure__pmf__inverse,axiom,
! [X: probab3364570286911266904_pmf_a] :
( ( probab1189994150051702498_pmf_a @ ( probab7257411610070727406_pmf_a @ X ) )
= X ) ).
% measure_pmf_inverse
thf(fact_626_measure__pmf_Ofinite__emeasure__space,axiom,
! [M: probab3364570286911266904_pmf_a] :
( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
!= top_to1496364449551166952nnreal ) ).
% measure_pmf.finite_emeasure_space
thf(fact_627_emeasure__measure__pmf__not__zero,axiom,
! [P2: probab1498759712122475378_pmf_o,S3: set_o] :
( ( ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 )
!= bot_bot_set_o )
=> ( ( sigma_emeasure_o @ ( probab7036721048548158344_pmf_o @ P2 ) @ S3 )
!= zero_z7100319975126383169nnreal ) ) ).
% emeasure_measure_pmf_not_zero
thf(fact_628_emeasure__measure__pmf__not__zero,axiom,
! [P2: probab469873468395307276mf_nat,S3: set_nat] :
( ( ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 )
!= bot_bot_set_nat )
=> ( ( sigma_emeasure_nat @ ( probab1352011410425470944mf_nat @ P2 ) @ S3 )
!= zero_z7100319975126383169nnreal ) ) ).
% emeasure_measure_pmf_not_zero
thf(fact_629_emeasure__measure__pmf__not__zero,axiom,
! [P2: probab3364570286911266904_pmf_a,S3: set_a] :
( ( ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 )
!= bot_bot_set_a )
=> ( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ P2 ) @ S3 )
!= zero_z7100319975126383169nnreal ) ) ).
% emeasure_measure_pmf_not_zero
thf(fact_630_emeasure__pmf__single__eq__zero__iff,axiom,
! [M: probab1498759712122475378_pmf_o,Y: $o] :
( ( ( sigma_emeasure_o @ ( probab7036721048548158344_pmf_o @ M ) @ ( insert_o @ Y @ bot_bot_set_o ) )
= zero_z7100319975126383169nnreal )
= ( ~ ( member_o @ Y @ ( probab7458556812659319003_pmf_o @ M ) ) ) ) ).
% emeasure_pmf_single_eq_zero_iff
thf(fact_631_emeasure__pmf__single__eq__zero__iff,axiom,
! [M: probab469873468395307276mf_nat,Y: nat] :
( ( ( sigma_emeasure_nat @ ( probab1352011410425470944mf_nat @ M ) @ ( insert_nat @ Y @ bot_bot_set_nat ) )
= zero_z7100319975126383169nnreal )
= ( ~ ( member_nat @ Y @ ( probab3271515132085200205mf_nat @ M ) ) ) ) ).
% emeasure_pmf_single_eq_zero_iff
thf(fact_632_emeasure__pmf__single__eq__zero__iff,axiom,
! [M: probab3364570286911266904_pmf_a,Y: a] :
( ( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ ( insert_a @ Y @ bot_bot_set_a ) )
= zero_z7100319975126383169nnreal )
= ( ~ ( member_a @ Y @ ( probab49036049091589825_pmf_a @ M ) ) ) ) ).
% emeasure_pmf_single_eq_zero_iff
thf(fact_633_Sup__fin_Oinsert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_634_Sup__fin_Oinsert,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
= ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_635_Inf__fin_Oinsert,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_636_Inf__fin_Oinsert,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
= ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_637_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_638_finite__Plus__UNIV__iff,axiom,
( ( finite94888208985532392_nat_o @ top_to7120114879189831663_nat_o )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_639_finite__Plus__UNIV__iff,axiom,
( ( finite5809725721784815170_o_nat @ top_to6072511757011528009_o_nat )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_640_finite__Plus__UNIV__iff,axiom,
( ( finite6699802884135759036um_o_o @ top_to1686961084667892491um_o_o )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_641_Int__UNIV,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_642_Int__UNIV,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= top_top_set_o )
= ( ( A = top_top_set_o )
& ( B = top_top_set_o ) ) ) ).
% Int_UNIV
thf(fact_643_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_644_Pow__UNIV,axiom,
( ( pow_o @ top_top_set_o )
= top_top_set_set_o ) ).
% Pow_UNIV
thf(fact_645_sets__measure__pmf,axiom,
! [P2: probab3364570286911266904_pmf_a] :
( ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ P2 ) )
= top_top_set_set_a ) ).
% sets_measure_pmf
thf(fact_646_space__measure__pmf,axiom,
! [P2: probab469873468395307276mf_nat] :
( ( sigma_space_nat @ ( probab1352011410425470944mf_nat @ P2 ) )
= top_top_set_nat ) ).
% space_measure_pmf
thf(fact_647_space__measure__pmf,axiom,
! [P2: probab1498759712122475378_pmf_o] :
( ( sigma_space_o @ ( probab7036721048548158344_pmf_o @ P2 ) )
= top_top_set_o ) ).
% space_measure_pmf
thf(fact_648_space__measure__pmf,axiom,
! [P2: probab3364570286911266904_pmf_a] :
( ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ P2 ) )
= top_top_set_a ) ).
% space_measure_pmf
thf(fact_649_sets__restrict__UNIV,axiom,
! [M: sigma_measure_nat] :
( ( sigma_sets_nat @ ( sigma_744083341818469772ce_nat @ M @ top_top_set_nat ) )
= ( sigma_sets_nat @ M ) ) ).
% sets_restrict_UNIV
thf(fact_650_sets__restrict__UNIV,axiom,
! [M: sigma_measure_o] :
( ( sigma_sets_o @ ( sigma_8520893325391096540pace_o @ M @ top_top_set_o ) )
= ( sigma_sets_o @ M ) ) ).
% sets_restrict_UNIV
thf(fact_651_sets__restrict__UNIV,axiom,
! [M: sigma_measure_a] :
( ( sigma_sets_a @ ( sigma_8692839461743104066pace_a @ M @ top_top_set_a ) )
= ( sigma_sets_a @ M ) ) ).
% sets_restrict_UNIV
thf(fact_652_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_top_set_set_nat
= ( sigma_sets_nat @ ( nonneg7031465154080143958re_nat @ top_top_set_nat ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_653_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
( top_top_set_set_o
= ( sigma_sets_o @ ( nonneg5198678888045619090sure_o @ top_top_set_o ) ) ) ).
% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_654_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_sets_nat @ ( nonneg7031465154080143958re_nat @ top_top_set_nat ) )
= top_top_set_set_nat ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_655_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
( ( sigma_sets_o @ ( nonneg5198678888045619090sure_o @ top_top_set_o ) )
= top_top_set_set_o ) ).
% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_656_emeasure__empty,axiom,
! [M: sigma_measure_o] :
( ( sigma_emeasure_o @ M @ bot_bot_set_o )
= zero_z7100319975126383169nnreal ) ).
% emeasure_empty
thf(fact_657_inf__Sup__absorb,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ( ( inf_inf_o @ A2 @ ( lattic1508158080041050831_fin_o @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_658_inf__Sup__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) )
= A2 ) ) ) ).
% inf_Sup_absorb
thf(fact_659_emeasure__pmf__UNIV,axiom,
! [M: probab469873468395307276mf_nat] :
( ( sigma_emeasure_nat @ ( probab1352011410425470944mf_nat @ M ) @ top_top_set_nat )
= one_on2969667320475766781nnreal ) ).
% emeasure_pmf_UNIV
thf(fact_660_emeasure__pmf__UNIV,axiom,
! [M: probab1498759712122475378_pmf_o] :
( ( sigma_emeasure_o @ ( probab7036721048548158344_pmf_o @ M ) @ top_top_set_o )
= one_on2969667320475766781nnreal ) ).
% emeasure_pmf_UNIV
thf(fact_661_emeasure__pmf__UNIV,axiom,
! [M: probab3364570286911266904_pmf_a] :
( ( sigma_emeasure_a @ ( probab7257411610070727406_pmf_a @ M ) @ top_top_set_a )
= one_on2969667320475766781nnreal ) ).
% emeasure_pmf_UNIV
thf(fact_662_sup__Inf__absorb,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ( ( sup_sup_o @ ( lattic4107685809792843317_fin_o @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_663_sup__Inf__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_664_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_665_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_o @ top_top_set_set_o )
= ( finite_finite_o @ top_top_set_o ) ) ).
% Finite_Set.finite_set
thf(fact_666_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_667_finite__prod,axiom,
( ( finite5355008432043429460_nat_o @ top_to8070287629520841379_nat_o )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_668_finite__prod,axiom,
( ( finite1846473907987936430_o_nat @ top_to7022684507342537725_o_nat )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_669_finite__prod,axiom,
( ( finite6120865539452801872od_o_o @ top_to7721136755696657239od_o_o )
= ( ( finite_finite_o @ top_top_set_o )
& ( finite_finite_o @ top_top_set_o ) ) ) ).
% finite_prod
thf(fact_670_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_671_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite5355008432043429460_nat_o @ top_to8070287629520841379_nat_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_672_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite1846473907987936430_o_nat @ top_to7022684507342537725_o_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_673_finite__Prod__UNIV,axiom,
( ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_o @ top_top_set_o )
=> ( finite6120865539452801872od_o_o @ top_to7721136755696657239od_o_o ) ) ) ).
% finite_Prod_UNIV
thf(fact_674_Un__UNIV__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_675_Un__UNIV__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_676_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_677_Un__UNIV__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_678_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_679_Int__UNIV__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ top_top_set_o @ B )
= B ) ).
% Int_UNIV_left
thf(fact_680_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_681_Int__UNIV__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ top_top_set_o )
= A ) ).
% Int_UNIV_right
thf(fact_682_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_683_insert__UNIV,axiom,
! [X: $o] :
( ( insert_o @ X @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_684_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_685_ex__new__if__finite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A )
=> ? [A4: a] :
~ ( member_a @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_686_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A4: nat] :
~ ( member_nat @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_687_ex__new__if__finite,axiom,
! [A: set_o] :
( ~ ( finite_finite_o @ top_top_set_o )
=> ( ( finite_finite_o @ A )
=> ? [A4: $o] :
~ ( member_o @ A4 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_688_finite__class_Ofinite__UNIV,axiom,
finite_finite_o @ top_top_set_o ).
% finite_class.finite_UNIV
thf(fact_689_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_690_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_691_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_692_subset__UNIV,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% subset_UNIV
thf(fact_693_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_694_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_695_Inf__fin__le__Sup__fin,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_696_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_697_Inf__fin__le__Sup__fin,axiom,
! [A: set_real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ord_less_eq_real @ ( lattic2677971596711400399n_real @ A ) @ ( lattic8928443293348198069n_real @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_698_Inf__fin_OcoboundedI,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_699_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_700_Inf__fin_OcoboundedI,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ( ord_less_eq_real @ ( lattic2677971596711400399n_real @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_701_Sup__fin_OcoboundedI,axiom,
! [A: set_o,A2: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ A2 @ A )
=> ( ord_less_eq_o @ A2 @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_702_Sup__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_703_Sup__fin_OcoboundedI,axiom,
! [A: set_real,A2: real] :
( ( finite_finite_real @ A )
=> ( ( member_real @ A2 @ A )
=> ( ord_less_eq_real @ A2 @ ( lattic8928443293348198069n_real @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_704_Inf__fin_Oin__idem,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
= ( lattic4107685809792843317_fin_o @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_705_Inf__fin_Oin__idem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_706_Sup__fin_Oin__idem,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) )
= ( lattic1508158080041050831_fin_o @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_707_Sup__fin_Oin__idem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_708_Inf__fin_OboundedE,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
=> ! [A7: $o] :
( ( member_o @ A7 @ A )
=> ( ord_less_eq_o @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_709_Inf__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A7: nat] :
( ( member_nat @ A7 @ A )
=> ( ord_less_eq_nat @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_710_Inf__fin_OboundedE,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic2677971596711400399n_real @ A ) )
=> ! [A7: real] :
( ( member_real @ A7 @ A )
=> ( ord_less_eq_real @ X @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_711_Inf__fin_OboundedI,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ord_less_eq_o @ X @ A4 ) )
=> ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_712_Inf__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_713_Inf__fin_OboundedI,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ord_less_eq_real @ X @ A4 ) )
=> ( ord_less_eq_real @ X @ ( lattic2677971596711400399n_real @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_714_Sup__fin_OboundedE,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
=> ! [A7: $o] :
( ( member_o @ A7 @ A )
=> ( ord_less_eq_o @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_715_Sup__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
=> ! [A7: nat] :
( ( member_nat @ A7 @ A )
=> ( ord_less_eq_nat @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_716_Sup__fin_OboundedE,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic8928443293348198069n_real @ A ) @ X )
=> ! [A7: real] :
( ( member_real @ A7 @ A )
=> ( ord_less_eq_real @ A7 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_717_Sup__fin_OboundedI,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [A4: $o] :
( ( member_o @ A4 @ A )
=> ( ord_less_eq_o @ A4 @ X ) )
=> ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_718_Sup__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_719_Sup__fin_OboundedI,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ! [A4: real] :
( ( member_real @ A4 @ A )
=> ( ord_less_eq_real @ A4 @ X ) )
=> ( ord_less_eq_real @ ( lattic8928443293348198069n_real @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_720_Inf__fin_Obounded__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A ) )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_721_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_722_Inf__fin_Obounded__iff,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_eq_real @ X @ ( lattic2677971596711400399n_real @ A ) )
= ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_723_Sup__fin_Obounded__iff,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ X )
= ( ! [X2: $o] :
( ( member_o @ X2 @ A )
=> ( ord_less_eq_o @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_724_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_725_Sup__fin_Obounded__iff,axiom,
! [A: set_real,X: real] :
( ( finite_finite_real @ A )
=> ( ( A != bot_bot_set_real )
=> ( ( ord_less_eq_real @ ( lattic8928443293348198069n_real @ A ) @ X )
= ( ! [X2: real] :
( ( member_real @ X2 @ A )
=> ( ord_less_eq_real @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_726_Sup__fin__Sup,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic1508158080041050831_fin_o @ A )
= ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_fin_Sup
thf(fact_727_cSup__eq__Sup__fin,axiom,
! [X6: set_nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat @ X6 )
= ( lattic1093996805478795353in_nat @ X6 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_728_cSup__eq__Sup__fin,axiom,
! [X6: set_o] :
( ( finite_finite_o @ X6 )
=> ( ( X6 != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o @ X6 )
= ( lattic1508158080041050831_fin_o @ X6 ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_729_Inf__fin_Osubset__imp,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B )
=> ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ B ) @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_730_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_731_Inf__fin_Osubset__imp,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( A != bot_bot_set_real )
=> ( ( finite_finite_real @ B )
=> ( ord_less_eq_real @ ( lattic2677971596711400399n_real @ B ) @ ( lattic2677971596711400399n_real @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_732_Sup__fin_Osubset__imp,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B )
=> ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_733_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_734_Sup__fin_Osubset__imp,axiom,
! [A: set_real,B: set_real] :
( ( ord_less_eq_set_real @ A @ B )
=> ( ( A != bot_bot_set_real )
=> ( ( finite_finite_real @ B )
=> ( ord_less_eq_real @ ( lattic8928443293348198069n_real @ A ) @ ( lattic8928443293348198069n_real @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_735_emeasure__single__in__space,axiom,
! [M: sigma_measure_nat,X: nat] :
( ( ( sigma_emeasure_nat @ M @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= zero_z7100319975126383169nnreal )
=> ( member_nat @ X @ ( sigma_space_nat @ M ) ) ) ).
% emeasure_single_in_space
thf(fact_736_emeasure__single__in__space,axiom,
! [M: sigma_measure_a,X: a] :
( ( ( sigma_emeasure_a @ M @ ( insert_a @ X @ bot_bot_set_a ) )
!= zero_z7100319975126383169nnreal )
=> ( member_a @ X @ ( sigma_space_a @ M ) ) ) ).
% emeasure_single_in_space
thf(fact_737_emeasure__single__in__space,axiom,
! [M: sigma_measure_o,X: $o] :
( ( ( sigma_emeasure_o @ M @ ( insert_o @ X @ bot_bot_set_o ) )
!= zero_z7100319975126383169nnreal )
=> ( member_o @ X @ ( sigma_space_o @ M ) ) ) ).
% emeasure_single_in_space
thf(fact_738_Inf__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) )
= ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_739_Inf__fin_Osubset,axiom,
! [A: set_o,B: set_o] :
( ( finite_finite_o @ A )
=> ( ( B != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ B @ A )
=> ( ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ B ) @ ( lattic4107685809792843317_fin_o @ A ) )
= ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_740_Sup__fin_Osubset,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_741_Sup__fin_Osubset,axiom,
! [A: set_o,B: set_o] :
( ( finite_finite_o @ A )
=> ( ( B != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ B @ A )
=> ( ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ B ) @ ( lattic1508158080041050831_fin_o @ A ) )
= ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_742_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_743_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
= ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_744_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y4 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_745_Inf__fin_Oclosed,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y4: $o] : ( member_o @ ( inf_inf_o @ X3 @ Y4 ) @ ( insert_o @ X3 @ ( insert_o @ Y4 @ bot_bot_set_o ) ) )
=> ( member_o @ ( lattic4107685809792843317_fin_o @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_746_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_747_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ~ ( member_o @ X @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
= ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_748_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y4: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y4 ) @ ( insert_nat @ X3 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_749_Sup__fin_Oclosed,axiom,
! [A: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ! [X3: $o,Y4: $o] : ( member_o @ ( sup_sup_o @ X3 @ Y4 ) @ ( insert_o @ X3 @ ( insert_o @ Y4 @ bot_bot_set_o ) ) )
=> ( member_o @ ( lattic1508158080041050831_fin_o @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_750_Inf__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_751_Inf__fin_Ounion,axiom,
! [A: set_o,B: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B )
=> ( ( B != bot_bot_set_o )
=> ( ( lattic4107685809792843317_fin_o @ ( sup_sup_set_o @ A @ B ) )
= ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ A ) @ ( lattic4107685809792843317_fin_o @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_752_Sup__fin_Ounion,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_753_Sup__fin_Ounion,axiom,
! [A: set_o,B: set_o] :
( ( finite_finite_o @ A )
=> ( ( A != bot_bot_set_o )
=> ( ( finite_finite_o @ B )
=> ( ( B != bot_bot_set_o )
=> ( ( lattic1508158080041050831_fin_o @ ( sup_sup_set_o @ A @ B ) )
= ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ A ) @ ( lattic1508158080041050831_fin_o @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_754_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_755_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_756_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_757_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_758_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_759_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_760_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_761_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_762_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_763_UNIV__I,axiom,
! [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_I
thf(fact_764_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_765_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_766_UNIV__witness,axiom,
? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).
% UNIV_witness
thf(fact_767_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_768_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_769_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_770_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_771_UNIV__eq__I,axiom,
! [A: set_o] :
( ! [X3: $o] : ( member_o @ X3 @ A )
=> ( top_top_set_o = A ) ) ).
% UNIV_eq_I
thf(fact_772_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_773_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_774_finite__option__UNIV,axiom,
( ( finite4093902646404507527tion_o @ top_top_set_option_o )
= ( finite_finite_o @ top_top_set_o ) ) ).
% finite_option_UNIV
thf(fact_775_DiffI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_776_DiffI,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ A )
=> ( ~ ( member_o @ C @ B )
=> ( member_o @ C @ ( minus_minus_set_o @ A @ B ) ) ) ) ).
% DiffI
thf(fact_777_DiffI,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B ) ) ) ) ).
% DiffI
thf(fact_778_Diff__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_779_Diff__iff,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
= ( ( member_o @ C @ A )
& ~ ( member_o @ C @ B ) ) ) ).
% Diff_iff
thf(fact_780_Diff__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B ) ) ) ).
% Diff_iff
thf(fact_781_Diff__cancel,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ A )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_782_empty__Diff,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_783_Diff__empty,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ bot_bot_set_o )
= A ) ).
% Diff_empty
thf(fact_784_finite__Diff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff
thf(fact_785_finite__Diff2,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_786_insert__Diff1,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_787_insert__Diff1,axiom,
! [X: $o,B: set_o,A: set_o] :
( ( member_o @ X @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B )
= ( minus_minus_set_o @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_788_insert__Diff1,axiom,
! [X: a,B: set_a,A: set_a] :
( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_789_Diff__insert0,axiom,
! [X: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( minus_minus_set_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_790_Diff__insert0,axiom,
! [X: $o,A: set_o,B: set_o] :
( ~ ( member_o @ X @ A )
=> ( ( minus_minus_set_o @ A @ ( insert_o @ X @ B ) )
= ( minus_minus_set_o @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_791_Diff__insert0,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X @ B ) )
= ( minus_minus_set_a @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_792_diff__ge__0__iff__ge,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_793_Diff__UNIV,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ top_top_set_nat )
= bot_bot_set_nat ) ).
% Diff_UNIV
thf(fact_794_Diff__UNIV,axiom,
! [A: set_o] :
( ( minus_minus_set_o @ A @ top_top_set_o )
= bot_bot_set_o ) ).
% Diff_UNIV
thf(fact_795_Diff__eq__empty__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( minus_minus_set_o @ A @ B )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_796_insert__Diff__single,axiom,
! [A2: $o,A: set_o] :
( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= ( insert_o @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_797_Diff__disjoint,axiom,
! [A: set_o,B: set_o] :
( ( inf_inf_set_o @ A @ ( minus_minus_set_o @ B @ A ) )
= bot_bot_set_o ) ).
% Diff_disjoint
thf(fact_798_finite__Diff__insert,axiom,
! [A: set_o,A2: $o,B: set_o] :
( ( finite_finite_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) ) )
= ( finite_finite_o @ ( minus_minus_set_o @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_799_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B ) ) ) ).
% finite_Diff_insert
thf(fact_800_diff__mono,axiom,
! [A2: real,B2: real,D2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ D2 @ C )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B2 @ D2 ) ) ) ) ).
% diff_mono
thf(fact_801_diff__left__mono,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_802_diff__right__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_803_diff__eq__diff__less__eq,axiom,
! [A2: real,B2: real,C: real,D2: real] :
( ( ( minus_minus_real @ A2 @ B2 )
= ( minus_minus_real @ C @ D2 ) )
=> ( ( ord_less_eq_real @ A2 @ B2 )
= ( ord_less_eq_real @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_804_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_805_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A: set_nat] :
( ( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( minus_minus_set_nat @ A @ B ) ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_806_insert__Diff__if,axiom,
! [X: $o,B: set_o,A: set_o] :
( ( ( member_o @ X @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B )
= ( minus_minus_set_o @ A @ B ) ) )
& ( ~ ( member_o @ X @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ B )
= ( insert_o @ X @ ( minus_minus_set_o @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_807_insert__Diff__if,axiom,
! [X: a,B: set_a,A: set_a] :
( ( ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( minus_minus_set_a @ A @ B ) ) )
& ( ~ ( member_a @ X @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ B )
= ( insert_a @ X @ ( minus_minus_set_a @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_808_DiffE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_809_DiffE,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( ( member_o @ C @ A )
=> ( member_o @ C @ B ) ) ) ).
% DiffE
thf(fact_810_DiffE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B ) ) ) ).
% DiffE
thf(fact_811_DiffD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_812_DiffD1,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
=> ( member_o @ C @ A ) ) ).
% DiffD1
thf(fact_813_DiffD1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_814_DiffD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_815_DiffD2,axiom,
! [C: $o,A: set_o,B: set_o] :
( ( member_o @ C @ ( minus_minus_set_o @ A @ B ) )
=> ~ ( member_o @ C @ B ) ) ).
% DiffD2
thf(fact_816_DiffD2,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B ) )
=> ~ ( member_a @ C @ B ) ) ).
% DiffD2
thf(fact_817_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_818_subset__Diff__insert,axiom,
! [A: set_nat,B: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B @ C2 ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_819_subset__Diff__insert,axiom,
! [A: set_o,B: set_o,X: $o,C2: set_o] :
( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ ( insert_o @ X @ C2 ) ) )
= ( ( ord_less_eq_set_o @ A @ ( minus_minus_set_o @ B @ C2 ) )
& ~ ( member_o @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_820_subset__Diff__insert,axiom,
! [A: set_a,B: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B @ C2 ) )
& ~ ( member_a @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_821_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_822_Diff__insert__absorb,axiom,
! [X: a,A: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_823_Diff__insert__absorb,axiom,
! [X: $o,A: set_o] :
( ~ ( member_o @ X @ A )
=> ( ( minus_minus_set_o @ ( insert_o @ X @ A ) @ ( insert_o @ X @ bot_bot_set_o ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_824_Diff__insert2,axiom,
! [A: set_o,A2: $o,B: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B ) ) ).
% Diff_insert2
thf(fact_825_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_826_insert__Diff,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A ) ) ).
% insert_Diff
thf(fact_827_insert__Diff,axiom,
! [A2: $o,A: set_o] :
( ( member_o @ A2 @ A )
=> ( ( insert_o @ A2 @ ( minus_minus_set_o @ A @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
= A ) ) ).
% insert_Diff
thf(fact_828_Diff__insert,axiom,
! [A: set_o,A2: $o,B: set_o] :
( ( minus_minus_set_o @ A @ ( insert_o @ A2 @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A @ B ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_829_Int__Diff__disjoint,axiom,
! [A: set_o,B: set_o] :
( ( inf_inf_set_o @ ( inf_inf_set_o @ A @ B ) @ ( minus_minus_set_o @ A @ B ) )
= bot_bot_set_o ) ).
% Int_Diff_disjoint
thf(fact_830_Diff__triv,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
=> ( ( minus_minus_set_o @ A @ B )
= A ) ) ).
% Diff_triv
thf(fact_831_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_832_subset__insert__iff,axiom,
! [A: set_a,X: a,B: set_a] :
( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B ) )
= ( ( ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X @ A )
=> ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_833_subset__insert__iff,axiom,
! [A: set_o,X: $o,B: set_o] :
( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B ) )
= ( ( ( member_o @ X @ A )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B ) )
& ( ~ ( member_o @ X @ A )
=> ( ord_less_eq_set_o @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_834_Diff__single__insert,axiom,
! [A: set_o,X: $o,B: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) @ B )
=> ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_835_infinite__remove,axiom,
! [S: set_nat,A2: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_836_infinite__remove,axiom,
! [S: set_o,A2: $o] :
( ~ ( finite_finite_o @ S )
=> ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% infinite_remove
thf(fact_837_infinite__coinduct,axiom,
! [X6: set_nat > $o,A: set_nat] :
( ( X6 @ A )
=> ( ! [A6: set_nat] :
( ( X6 @ A6 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A6 )
& ( ( X6 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A ) ) ) ).
% infinite_coinduct
thf(fact_838_infinite__coinduct,axiom,
! [X6: set_o > $o,A: set_o] :
( ( X6 @ A )
=> ( ! [A6: set_o] :
( ( X6 @ A6 )
=> ? [X4: $o] :
( ( member_o @ X4 @ A6 )
& ( ( X6 @ ( minus_minus_set_o @ A6 @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
| ~ ( finite_finite_o @ ( minus_minus_set_o @ A6 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) )
=> ~ ( finite_finite_o @ A ) ) ) ).
% infinite_coinduct
thf(fact_839_finite__empty__induct,axiom,
! [A: set_a,P: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ( P @ A )
=> ( ! [A4: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A4 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_840_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A4: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A4 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_841_finite__empty__induct,axiom,
! [A: set_o,P: set_o > $o] :
( ( finite_finite_o @ A )
=> ( ( P @ A )
=> ( ! [A4: $o,A6: set_o] :
( ( finite_finite_o @ A6 )
=> ( ( member_o @ A4 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_o @ A6 @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ) )
=> ( P @ bot_bot_set_o ) ) ) ) ).
% finite_empty_induct
thf(fact_842_remove__induct,axiom,
! [P: set_a > $o,B: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B )
=> ( P @ B ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_843_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_844_remove__induct,axiom,
! [P: set_o > $o,B: set_o] :
( ( P @ bot_bot_set_o )
=> ( ( ~ ( finite_finite_o @ B )
=> ( P @ B ) )
=> ( ! [A6: set_o] :
( ( finite_finite_o @ A6 )
=> ( ( A6 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ A6 @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A6 )
=> ( P @ ( minus_minus_set_o @ A6 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_845_finite__remove__induct,axiom,
! [B: set_a,P: set_a > $o] :
( ( finite_finite_a @ B )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( P @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_846_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_847_finite__remove__induct,axiom,
! [B: set_o,P: set_o > $o] :
( ( finite_finite_o @ B )
=> ( ( P @ bot_bot_set_o )
=> ( ! [A6: set_o] :
( ( finite_finite_o @ A6 )
=> ( ( A6 != bot_bot_set_o )
=> ( ( ord_less_eq_set_o @ A6 @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A6 )
=> ( P @ ( minus_minus_set_o @ A6 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_848_Inf__fin_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_849_Inf__fin_Oinsert__remove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A ) )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_850_Inf__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_851_Inf__fin_Oremove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( lattic4107685809792843317_fin_o @ A )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_852_Sup__fin_Oinsert__remove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_853_Sup__fin_Oinsert__remove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A ) )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_854_Sup__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_855_Sup__fin_Oremove,axiom,
! [A: set_o,X: $o] :
( ( finite_finite_o @ A )
=> ( ( member_o @ X @ A )
=> ( ( lattic1508158080041050831_fin_o @ A )
= ( ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
= bot_bot_set_o )
=> X )
& ( ( ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
!= bot_bot_set_o )
=> ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_856_diff__shunt__var,axiom,
! [X: set_o,Y: set_o] :
( ( ( minus_minus_set_o @ X @ Y )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_857_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A5 @ B4 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_858_remove__def,axiom,
( remove_o
= ( ^ [X2: $o,A3: set_o] : ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% remove_def
thf(fact_859_algebra__single__set,axiom,
! [X6: set_o,S: set_o] :
( ( ord_less_eq_set_o @ X6 @ S )
=> ( sigma_algebra_o @ S @ ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ X6 @ ( insert_set_o @ ( minus_minus_set_o @ S @ X6 ) @ ( insert_set_o @ S @ bot_bot_set_set_o ) ) ) ) ) ) ).
% algebra_single_set
thf(fact_860_member__remove,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
= ( ( member_nat @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_861_member__remove,axiom,
! [X: $o,Y: $o,A: set_o] :
( ( member_o @ X @ ( remove_o @ Y @ A ) )
= ( ( member_o @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_862_member__remove,axiom,
! [X: a,Y: a,A: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A ) )
= ( ( member_a @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_863_algebra__iff__Int,axiom,
( sigma_algebra_o
= ( ^ [Omega2: set_o,M3: set_set_o] :
( ( ord_le4374716579403074808_set_o @ M3 @ ( pow_o @ Omega2 ) )
& ( member_set_o @ bot_bot_set_o @ M3 )
& ! [X2: set_o] :
( ( member_set_o @ X2 @ M3 )
=> ( member_set_o @ ( minus_minus_set_o @ Omega2 @ X2 ) @ M3 ) )
& ! [X2: set_o] :
( ( member_set_o @ X2 @ M3 )
=> ! [Y2: set_o] :
( ( member_set_o @ Y2 @ M3 )
=> ( member_set_o @ ( inf_inf_set_o @ X2 @ Y2 ) @ M3 ) ) ) ) ) ) ).
% algebra_iff_Int
thf(fact_864_algebra__iff__Un,axiom,
( sigma_algebra_o
= ( ^ [Omega2: set_o,M3: set_set_o] :
( ( ord_le4374716579403074808_set_o @ M3 @ ( pow_o @ Omega2 ) )
& ( member_set_o @ bot_bot_set_o @ M3 )
& ! [X2: set_o] :
( ( member_set_o @ X2 @ M3 )
=> ( member_set_o @ ( minus_minus_set_o @ Omega2 @ X2 ) @ M3 ) )
& ! [X2: set_o] :
( ( member_set_o @ X2 @ M3 )
=> ! [Y2: set_o] :
( ( member_set_o @ Y2 @ M3 )
=> ( member_set_o @ ( sup_sup_set_o @ X2 @ Y2 ) @ M3 ) ) ) ) ) ) ).
% algebra_iff_Un
thf(fact_865_ring__of__setsI,axiom,
! [M: set_set_o,Omega: set_o] :
( ( ord_le4374716579403074808_set_o @ M @ ( pow_o @ Omega ) )
=> ( ( member_set_o @ bot_bot_set_o @ M )
=> ( ! [A4: set_o,B3: set_o] :
( ( member_set_o @ A4 @ M )
=> ( ( member_set_o @ B3 @ M )
=> ( member_set_o @ ( sup_sup_set_o @ A4 @ B3 ) @ M ) ) )
=> ( ! [A4: set_o,B3: set_o] :
( ( member_set_o @ A4 @ M )
=> ( ( member_set_o @ B3 @ M )
=> ( member_set_o @ ( minus_minus_set_o @ A4 @ B3 ) @ M ) ) )
=> ( sigma_ring_of_sets_o @ Omega @ M ) ) ) ) ) ).
% ring_of_setsI
thf(fact_866_ring__of__sets__iff,axiom,
( sigma_ring_of_sets_o
= ( ^ [Omega2: set_o,M3: set_set_o] :
( ( ord_le4374716579403074808_set_o @ M3 @ ( pow_o @ Omega2 ) )
& ( member_set_o @ bot_bot_set_o @ M3 )
& ! [X2: set_o] :
( ( member_set_o @ X2 @ M3 )
=> ! [Y2: set_o] :
( ( member_set_o @ Y2 @ M3 )
=> ( member_set_o @ ( sup_sup_set_o @ X2 @ Y2 ) @ M3 ) ) )
& ! [X2: set_o] :
( ( member_set_o @ X2 @ M3 )
=> ! [Y2: set_o] :
( ( member_set_o @ Y2 @ M3 )
=> ( member_set_o @ ( minus_minus_set_o @ X2 @ Y2 ) @ M3 ) ) ) ) ) ) ).
% ring_of_sets_iff
thf(fact_867_sets_Osmallest__ccdi__sets__Un,axiom,
! [A: set_o,M: sigma_measure_o,B: set_o] :
( ( member_set_o @ A @ ( sigma_7164253587203589747sets_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) )
=> ( ( member_set_o @ B @ ( sigma_7164253587203589747sets_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) )
=> ( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
=> ( member_set_o @ ( sup_sup_set_o @ A @ B ) @ ( sigma_7164253587203589747sets_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) ) ) ) ) ).
% sets.smallest_ccdi_sets_Un
thf(fact_868_sigma__algebra__single__set,axiom,
! [X6: set_o,S: set_o] :
( ( ord_less_eq_set_o @ X6 @ S )
=> ( sigma_3687534776968752773ebra_o @ S @ ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ X6 @ ( insert_set_o @ ( minus_minus_set_o @ S @ X6 ) @ ( insert_set_o @ S @ bot_bot_set_set_o ) ) ) ) ) ) ).
% sigma_algebra_single_set
thf(fact_869_prob__space_Osigma__algebra__tail__events,axiom,
! [M: sigma_measure_a,A: nat > set_set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ! [I2: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ M ) @ ( A @ I2 ) )
=> ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ M ) @ ( indepe7538416700049374166_a_nat @ M @ A ) ) ) ) ).
% prob_space.sigma_algebra_tail_events
thf(fact_870_zero__le__divide__1__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% zero_le_divide_1_iff
thf(fact_871_divide__le__0__1__iff,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% divide_le_0_1_iff
thf(fact_872_divide__le__0__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% divide_le_0_iff
thf(fact_873_divide__right__mono__neg,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B2 @ C ) @ ( divide_divide_real @ A2 @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_874_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_875_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_876_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_877_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_878_zero__le__divide__iff,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_879_divide__right__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) ) ) ) ).
% divide_right_mono
thf(fact_880_plus__emeasure,axiom,
! [A2: set_o,M: sigma_measure_o,B2: set_o] :
( ( member_set_o @ A2 @ ( sigma_sets_o @ M ) )
=> ( ( member_set_o @ B2 @ ( sigma_sets_o @ M ) )
=> ( ( ( inf_inf_set_o @ A2 @ B2 )
= bot_bot_set_o )
=> ( ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_o @ M @ A2 ) @ ( sigma_emeasure_o @ M @ B2 ) )
= ( sigma_emeasure_o @ M @ ( sup_sup_set_o @ A2 @ B2 ) ) ) ) ) ) ).
% plus_emeasure
thf(fact_881_prob__space_Okolmogorov__0__1__law,axiom,
! [M: sigma_measure_a,A: nat > set_set_a,X6: set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ! [I2: nat] : ( sigma_4968961713055010667ebra_a @ ( sigma_space_a @ M ) @ ( A @ I2 ) )
=> ( ( indepe6267730027088848354_a_nat @ M @ A @ top_top_set_nat )
=> ( ( member_set_a @ X6 @ ( indepe7538416700049374166_a_nat @ M @ A ) )
=> ( ( ( sigma_measure_a2 @ M @ X6 )
= zero_zero_real )
| ( ( sigma_measure_a2 @ M @ X6 )
= one_one_real ) ) ) ) ) ) ).
% prob_space.kolmogorov_0_1_law
thf(fact_882_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_883_add__le__cancel__left,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_884_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_885_add__le__cancel__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_886_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_887_add__le__same__cancel1,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_888_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_889_add__le__same__cancel2,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_890_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_891_le__add__same__cancel1,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ A2 @ B2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_892_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_893_le__add__same__cancel2,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ B2 @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_894_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
= ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_895_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
= ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_896_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_897_le__add__diff__inverse2,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_898_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_899_le__add__diff__inverse,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( plus_plus_real @ B2 @ ( minus_minus_real @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_900_measure__empty,axiom,
! [M: sigma_measure_o] :
( ( sigma_measure_o2 @ M @ bot_bot_set_o )
= zero_zero_real ) ).
% measure_empty
thf(fact_901_measure__pmf_Oprob__le__1,axiom,
! [M: probab3364570286911266904_pmf_a,A: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) @ one_one_real ) ).
% measure_pmf.prob_le_1
thf(fact_902_measure__pmf_Osubprob__measure__le__1,axiom,
! [M: probab3364570286911266904_pmf_a,X6: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ X6 ) @ one_one_real ) ).
% measure_pmf.subprob_measure_le_1
thf(fact_903_measure__pmf__UNIV,axiom,
! [P2: probab469873468395307276mf_nat] :
( ( sigma_measure_nat2 @ ( probab1352011410425470944mf_nat @ P2 ) @ top_top_set_nat )
= one_one_real ) ).
% measure_pmf_UNIV
thf(fact_904_measure__pmf__UNIV,axiom,
! [P2: probab1498759712122475378_pmf_o] :
( ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ P2 ) @ top_top_set_o )
= one_one_real ) ).
% measure_pmf_UNIV
thf(fact_905_measure__pmf__UNIV,axiom,
! [P2: probab3364570286911266904_pmf_a] :
( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ P2 ) @ top_top_set_a )
= one_one_real ) ).
% measure_pmf_UNIV
thf(fact_906_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_907_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_908_measure__pmf_Omeasure__ge__1__iff,axiom,
! [M: probab3364570286911266904_pmf_a,A: set_a] :
( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) )
= ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A )
= one_one_real ) ) ).
% measure_pmf.measure_ge_1_iff
thf(fact_909_prob__space_Omeasure__ge__1__iff,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( ord_less_eq_real @ one_one_real @ ( sigma_measure_a2 @ M @ A ) )
= ( ( sigma_measure_a2 @ M @ A )
= one_one_real ) ) ) ).
% prob_space.measure_ge_1_iff
thf(fact_910_prob__space_Oprob__le__1,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ M @ A ) @ one_one_real ) ) ).
% prob_space.prob_le_1
thf(fact_911_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_912_add__le__imp__le__right,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_913_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_914_add__le__imp__le__left,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_915_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C4: nat] :
( B4
= ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_916_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_917_add__right__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_918_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C5: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C5 ) ) ) ).
% less_eqE
thf(fact_919_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_920_add__left__mono,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_921_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_922_add__mono,axiom,
! [A2: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_923_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J3 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_924_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: real,J3: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J3 )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K2 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_925_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J3: nat,K2: nat,L: nat] :
( ( ( I3 = J3 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_926_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: real,J3: real,K2: real,L: real] :
( ( ( I3 = J3 )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K2 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_927_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J3 )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_928_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: real,J3: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J3 )
& ( K2 = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K2 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_929_verit__sum__simplify,axiom,
! [A2: real] :
( ( plus_plus_real @ A2 @ zero_zero_real )
= A2 ) ).
% verit_sum_simplify
thf(fact_930_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_931_add__nonpos__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_932_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_933_add__nonneg__eq__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ( plus_plus_real @ X @ Y )
= zero_zero_real )
= ( ( X = zero_zero_real )
& ( Y = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_934_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_935_add__nonpos__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_936_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_937_add__nonneg__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_938_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_939_add__increasing2,axiom,
! [C: real,B2: real,A2: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B2 @ A2 )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_940_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_941_add__decreasing2,axiom,
! [C: real,A2: real,B2: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_942_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_943_add__increasing,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_944_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_945_add__decreasing,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_946_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_947_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_948_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_949_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_950_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_951_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_952_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_953_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_954_le__add__diff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_955_add__le__add__imp__diff__le,axiom,
! [I3: nat,K2: nat,N2: nat,J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J3 @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J3 @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K2 ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_956_add__le__add__imp__diff__le,axiom,
! [I3: real,K2: real,N2: real,J3: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K2 ) @ N2 )
=> ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J3 @ K2 ) )
=> ( ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K2 ) @ N2 )
=> ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J3 @ K2 ) )
=> ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K2 ) @ J3 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_957_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_958_add__le__imp__le__diff,axiom,
! [I3: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ N2 @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_959_add__le__imp__le__diff,axiom,
! [I3: real,K2: real,N2: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ I3 @ K2 ) @ N2 )
=> ( ord_less_eq_real @ I3 @ ( minus_minus_real @ N2 @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_960_le__diff__eq,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ A2 @ ( minus_minus_real @ C @ B2 ) )
= ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B2 ) @ C ) ) ).
% le_diff_eq
thf(fact_961_diff__le__eq,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B2 ) @ C )
= ( ord_less_eq_real @ A2 @ ( plus_plus_real @ C @ B2 ) ) ) ).
% diff_le_eq
thf(fact_962_measure__pmf_Omeasure__increasing,axiom,
! [M: probab3364570286911266904_pmf_a] : ( measur1776380161843274167a_real @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) ) ) ).
% measure_pmf.measure_increasing
thf(fact_963_measure__pmf_Obounded__measure,axiom,
! [M: probab3364570286911266904_pmf_a,A: set_a] : ( ord_less_eq_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) ) ) ).
% measure_pmf.bounded_measure
thf(fact_964_measure__pmf_Oprob__space,axiom,
! [M: probab3364570286911266904_pmf_a] :
( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
= one_one_real ) ).
% measure_pmf.prob_space
thf(fact_965_prob__space_Oprob__space,axiom,
! [M: sigma_measure_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( sigma_measure_a2 @ M @ ( sigma_space_a @ M ) )
= one_one_real ) ) ).
% prob_space.prob_space
thf(fact_966_measure__pmf_Ofinite__measure__mono,axiom,
! [A: set_a,B: set_a,M: probab3364570286911266904_pmf_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ B ) ) ) ) ).
% measure_pmf.finite_measure_mono
thf(fact_967_measure__pmf_Omeasure__zero__union,axiom,
! [S3: set_a,M: probab3364570286911266904_pmf_a,T3: set_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ T3 )
= zero_zero_real )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( sup_sup_set_a @ S3 @ T3 ) )
= ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ S3 ) ) ) ) ) ).
% measure_pmf.measure_zero_union
thf(fact_968_measure__Int__set__pmf,axiom,
! [P2: probab1498759712122475378_pmf_o,A: set_o] :
( ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ P2 ) @ ( inf_inf_set_o @ A @ ( probab7458556812659319003_pmf_o @ P2 ) ) )
= ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ P2 ) @ A ) ) ).
% measure_Int_set_pmf
thf(fact_969_measure__Int__set__pmf,axiom,
! [P2: probab469873468395307276mf_nat,A: set_nat] :
( ( sigma_measure_nat2 @ ( probab1352011410425470944mf_nat @ P2 ) @ ( inf_inf_set_nat @ A @ ( probab3271515132085200205mf_nat @ P2 ) ) )
= ( sigma_measure_nat2 @ ( probab1352011410425470944mf_nat @ P2 ) @ A ) ) ).
% measure_Int_set_pmf
thf(fact_970_measure__Int__set__pmf,axiom,
! [P2: probab3364570286911266904_pmf_a,A: set_a] :
( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ P2 ) @ ( inf_inf_set_a @ A @ ( probab49036049091589825_pmf_a @ P2 ) ) )
= ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ P2 ) @ A ) ) ).
% measure_Int_set_pmf
thf(fact_971_measure__pmf_Ofinite__measure__Diff,axiom,
! [A: set_a,M: probab3364570286911266904_pmf_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ B ) ) ) ) ) ) ).
% measure_pmf.finite_measure_Diff
thf(fact_972_measure__pmf_Omeasure__space__inter,axiom,
! [S3: set_a,M: probab3364570286911266904_pmf_a,T3: set_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ T3 )
= ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) ) )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( inf_inf_set_a @ S3 @ T3 ) )
= ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ S3 ) ) ) ) ) ).
% measure_pmf.measure_space_inter
thf(fact_973_measure__pmf_Ofinite__measure__Diff_H,axiom,
! [A: set_a,M: probab3364570286911266904_pmf_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( minus_minus_set_a @ A @ B ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( inf_inf_set_a @ A @ B ) ) ) ) ) ) ).
% measure_pmf.finite_measure_Diff'
thf(fact_974_measure__pmf_Oprob__compl,axiom,
! [A: set_a,M: probab3364570286911266904_pmf_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( minus_minus_set_a @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) @ A ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) ) ) ) ).
% measure_pmf.prob_compl
thf(fact_975_measure__pmf_Ofinite__measure__compl,axiom,
! [S: set_a,M: probab3364570286911266904_pmf_a] :
( ( member_set_a @ S @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( minus_minus_set_a @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) @ S ) )
= ( minus_minus_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ S ) ) ) ) ).
% measure_pmf.finite_measure_compl
thf(fact_976_measure__pmf_Omeasure__eq__compl,axiom,
! [S3: set_a,M: probab3364570286911266904_pmf_a,T3: set_a] :
( ( member_set_a @ S3 @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( member_set_a @ T3 @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( minus_minus_set_a @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) @ S3 ) )
= ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( minus_minus_set_a @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) @ T3 ) ) )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ S3 )
= ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ T3 ) ) ) ) ) ).
% measure_pmf.measure_eq_compl
thf(fact_977_prob__space_Oprob__compl,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( probab7247484486040049089pace_a @ M )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ( sigma_measure_a2 @ M @ ( minus_minus_set_a @ ( sigma_space_a @ M ) @ A ) )
= ( minus_minus_real @ one_one_real @ ( sigma_measure_a2 @ M @ A ) ) ) ) ) ).
% prob_space.prob_compl
thf(fact_978_measure__measure__pmf__not__zero,axiom,
! [P2: probab1498759712122475378_pmf_o,S3: set_o] :
( ( ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 )
!= bot_bot_set_o )
=> ( ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ P2 ) @ S3 )
!= zero_zero_real ) ) ).
% measure_measure_pmf_not_zero
thf(fact_979_measure__measure__pmf__not__zero,axiom,
! [P2: probab469873468395307276mf_nat,S3: set_nat] :
( ( ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 )
!= bot_bot_set_nat )
=> ( ( sigma_measure_nat2 @ ( probab1352011410425470944mf_nat @ P2 ) @ S3 )
!= zero_zero_real ) ) ).
% measure_measure_pmf_not_zero
thf(fact_980_measure__measure__pmf__not__zero,axiom,
! [P2: probab3364570286911266904_pmf_a,S3: set_a] :
( ( ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 )
!= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ P2 ) @ S3 )
!= zero_zero_real ) ) ).
% measure_measure_pmf_not_zero
thf(fact_981_measure__pmf__zero__iff,axiom,
! [P2: probab1498759712122475378_pmf_o,S3: set_o] :
( ( ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ P2 ) @ S3 )
= zero_zero_real )
= ( ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 )
= bot_bot_set_o ) ) ).
% measure_pmf_zero_iff
thf(fact_982_measure__pmf__zero__iff,axiom,
! [P2: probab469873468395307276mf_nat,S3: set_nat] :
( ( ( sigma_measure_nat2 @ ( probab1352011410425470944mf_nat @ P2 ) @ S3 )
= zero_zero_real )
= ( ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 )
= bot_bot_set_nat ) ) ).
% measure_pmf_zero_iff
thf(fact_983_measure__pmf__zero__iff,axiom,
! [P2: probab3364570286911266904_pmf_a,S3: set_a] :
( ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ P2 ) @ S3 )
= zero_zero_real )
= ( ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 )
= bot_bot_set_a ) ) ).
% measure_pmf_zero_iff
thf(fact_984_emeasure__insert__ne,axiom,
! [A: set_nat,X: nat,M: sigma_measure_nat] :
( ( A != bot_bot_set_nat )
=> ( ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) )
=> ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
=> ( ~ ( member_nat @ X @ A )
=> ( ( sigma_emeasure_nat @ M @ ( insert_nat @ X @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_nat @ M @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( sigma_emeasure_nat @ M @ A ) ) ) ) ) ) ) ).
% emeasure_insert_ne
thf(fact_985_emeasure__insert__ne,axiom,
! [A: set_a,X: a,M: sigma_measure_a] :
( ( A != bot_bot_set_a )
=> ( ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ~ ( member_a @ X @ A )
=> ( ( sigma_emeasure_a @ M @ ( insert_a @ X @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( sigma_emeasure_a @ M @ A ) ) ) ) ) ) ) ).
% emeasure_insert_ne
thf(fact_986_emeasure__insert__ne,axiom,
! [A: set_o,X: $o,M: sigma_measure_o] :
( ( A != bot_bot_set_o )
=> ( ( member_set_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) )
=> ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
=> ( ~ ( member_o @ X @ A )
=> ( ( sigma_emeasure_o @ M @ ( insert_o @ X @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_o @ M @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( sigma_emeasure_o @ M @ A ) ) ) ) ) ) ) ).
% emeasure_insert_ne
thf(fact_987_emeasure__insert,axiom,
! [X: nat,M: sigma_measure_nat,A: set_nat] :
( ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) )
=> ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
=> ( ~ ( member_nat @ X @ A )
=> ( ( sigma_emeasure_nat @ M @ ( insert_nat @ X @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_nat @ M @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( sigma_emeasure_nat @ M @ A ) ) ) ) ) ) ).
% emeasure_insert
thf(fact_988_emeasure__insert,axiom,
! [X: a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) )
=> ( ( member_set_a @ A @ ( sigma_sets_a @ M ) )
=> ( ~ ( member_a @ X @ A )
=> ( ( sigma_emeasure_a @ M @ ( insert_a @ X @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_a @ M @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( sigma_emeasure_a @ M @ A ) ) ) ) ) ) ).
% emeasure_insert
thf(fact_989_emeasure__insert,axiom,
! [X: $o,M: sigma_measure_o,A: set_o] :
( ( member_set_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) )
=> ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
=> ( ~ ( member_o @ X @ A )
=> ( ( sigma_emeasure_o @ M @ ( insert_o @ X @ A ) )
= ( plus_p1859984266308609217nnreal @ ( sigma_emeasure_o @ M @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( sigma_emeasure_o @ M @ A ) ) ) ) ) ) ).
% emeasure_insert
thf(fact_990_measure__pmf_Omeasure__exclude,axiom,
! [A: set_o,M: probab1498759712122475378_pmf_o,B: set_o] :
( ( member_set_o @ A @ ( sigma_sets_o @ ( probab7036721048548158344_pmf_o @ M ) ) )
=> ( ( member_set_o @ B @ ( sigma_sets_o @ ( probab7036721048548158344_pmf_o @ M ) ) )
=> ( ( ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ M ) @ A )
= ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ M ) @ ( sigma_space_o @ ( probab7036721048548158344_pmf_o @ M ) ) ) )
=> ( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
=> ( ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ M ) @ B )
= zero_zero_real ) ) ) ) ) ).
% measure_pmf.measure_exclude
thf(fact_991_measure__pmf_Omeasure__exclude,axiom,
! [A: set_a,M: probab3364570286911266904_pmf_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A )
= ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( sigma_space_a @ ( probab7257411610070727406_pmf_a @ M ) ) ) )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ B )
= zero_zero_real ) ) ) ) ) ).
% measure_pmf.measure_exclude
thf(fact_992_measure__restrict__space,axiom,
! [Omega: set_a,M: sigma_measure_a,A: set_a] :
( ( member_set_a @ ( inf_inf_set_a @ Omega @ ( sigma_space_a @ M ) ) @ ( sigma_sets_a @ M ) )
=> ( ( ord_less_eq_set_a @ A @ Omega )
=> ( ( sigma_measure_a2 @ ( sigma_8692839461743104066pace_a @ M @ Omega ) @ A )
= ( sigma_measure_a2 @ M @ A ) ) ) ) ).
% measure_restrict_space
thf(fact_993_finite__measure_Omeasure__exclude,axiom,
! [M: sigma_measure_o,A: set_o,B: set_o] :
( ( measur2447921437955784316sure_o @ M )
=> ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
=> ( ( member_set_o @ B @ ( sigma_sets_o @ M ) )
=> ( ( ( sigma_measure_o2 @ M @ A )
= ( sigma_measure_o2 @ M @ ( sigma_space_o @ M ) ) )
=> ( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
=> ( ( sigma_measure_o2 @ M @ B )
= zero_zero_real ) ) ) ) ) ) ).
% finite_measure.measure_exclude
thf(fact_994_subadditiveD,axiom,
! [M: set_set_o,F2: set_o > nat,X: set_o,Y: set_o] :
( ( measur8239940624735150288_o_nat @ M @ F2 )
=> ( ( ( inf_inf_set_o @ X @ Y )
= bot_bot_set_o )
=> ( ( member_set_o @ X @ M )
=> ( ( member_set_o @ Y @ M )
=> ( ord_less_eq_nat @ ( F2 @ ( sup_sup_set_o @ X @ Y ) ) @ ( plus_plus_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ) ) ).
% subadditiveD
thf(fact_995_subadditiveD,axiom,
! [M: set_set_o,F2: set_o > real,X: set_o,Y: set_o] :
( ( measur9098857796376888236o_real @ M @ F2 )
=> ( ( ( inf_inf_set_o @ X @ Y )
= bot_bot_set_o )
=> ( ( member_set_o @ X @ M )
=> ( ( member_set_o @ Y @ M )
=> ( ord_less_eq_real @ ( F2 @ ( sup_sup_set_o @ X @ Y ) ) @ ( plus_plus_real @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ) ) ) ) ).
% subadditiveD
thf(fact_996_subadditive__def,axiom,
( measur8239940624735150288_o_nat
= ( ^ [M3: set_set_o,F5: set_o > nat] :
! [X2: set_o] :
( ( member_set_o @ X2 @ M3 )
=> ! [Y2: set_o] :
( ( member_set_o @ Y2 @ M3 )
=> ( ( ( inf_inf_set_o @ X2 @ Y2 )
= bot_bot_set_o )
=> ( ord_less_eq_nat @ ( F5 @ ( sup_sup_set_o @ X2 @ Y2 ) ) @ ( plus_plus_nat @ ( F5 @ X2 ) @ ( F5 @ Y2 ) ) ) ) ) ) ) ) ).
% subadditive_def
thf(fact_997_subadditive__def,axiom,
( measur9098857796376888236o_real
= ( ^ [M3: set_set_o,F5: set_o > real] :
! [X2: set_o] :
( ( member_set_o @ X2 @ M3 )
=> ! [Y2: set_o] :
( ( member_set_o @ Y2 @ M3 )
=> ( ( ( inf_inf_set_o @ X2 @ Y2 )
= bot_bot_set_o )
=> ( ord_less_eq_real @ ( F5 @ ( sup_sup_set_o @ X2 @ Y2 ) ) @ ( plus_plus_real @ ( F5 @ X2 ) @ ( F5 @ Y2 ) ) ) ) ) ) ) ) ).
% subadditive_def
thf(fact_998_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M6: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M6 @ N3 )
& ( member_nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_999_measure__pmf_Ofinite__measure__subadditive,axiom,
! [A: set_a,M: probab3364570286911266904_pmf_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ord_less_eq_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( sup_sup_set_a @ A @ B ) ) @ ( plus_plus_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ B ) ) ) ) ) ).
% measure_pmf.finite_measure_subadditive
thf(fact_1000_measure__pmf_Ofinite__measure__Union_H,axiom,
! [A: set_a,M: probab3364570286911266904_pmf_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( sup_sup_set_a @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( minus_minus_set_a @ B @ A ) ) ) ) ) ) ).
% measure_pmf.finite_measure_Union'
thf(fact_1001_measure__pmf_Ofinite__measure__Union,axiom,
! [A: set_o,M: probab1498759712122475378_pmf_o,B: set_o] :
( ( member_set_o @ A @ ( sigma_sets_o @ ( probab7036721048548158344_pmf_o @ M ) ) )
=> ( ( member_set_o @ B @ ( sigma_sets_o @ ( probab7036721048548158344_pmf_o @ M ) ) )
=> ( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
=> ( ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ M ) @ ( sup_sup_set_o @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ M ) @ A ) @ ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ M ) @ B ) ) ) ) ) ) ).
% measure_pmf.finite_measure_Union
thf(fact_1002_measure__pmf_Ofinite__measure__Union,axiom,
! [A: set_a,M: probab3364570286911266904_pmf_a,B: set_a] :
( ( member_set_a @ A @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( member_set_a @ B @ ( sigma_sets_a @ ( probab7257411610070727406_pmf_a @ M ) ) )
=> ( ( ( inf_inf_set_a @ A @ B )
= bot_bot_set_a )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( sup_sup_set_a @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ A ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ B ) ) ) ) ) ) ).
% measure_pmf.finite_measure_Union
thf(fact_1003_finite__measure_Ofinite__measure__Union,axiom,
! [M: sigma_measure_o,A: set_o,B: set_o] :
( ( measur2447921437955784316sure_o @ M )
=> ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
=> ( ( member_set_o @ B @ ( sigma_sets_o @ M ) )
=> ( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
=> ( ( sigma_measure_o2 @ M @ ( sup_sup_set_o @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_o2 @ M @ A ) @ ( sigma_measure_o2 @ M @ B ) ) ) ) ) ) ) ).
% finite_measure.finite_measure_Union
thf(fact_1004_finite__set__plus,axiom,
! [S3: set_nat,T3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ( finite_finite_nat @ T3 )
=> ( finite_finite_nat @ ( plus_plus_set_nat @ S3 @ T3 ) ) ) ) ).
% finite_set_plus
thf(fact_1005_prob__space__uniform__measure,axiom,
! [M: sigma_measure_a,A: set_a] :
( ( ( sigma_emeasure_a @ M @ A )
!= zero_z7100319975126383169nnreal )
=> ( ( ( sigma_emeasure_a @ M @ A )
!= extend2057119558705770725nnreal )
=> ( probab7247484486040049089pace_a @ ( nonneg6757527617543859701sure_a @ M @ A ) ) ) ) ).
% prob_space_uniform_measure
thf(fact_1006_set__zero__plus2,axiom,
! [A: set_nat,B: set_nat] :
( ( member_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_set_nat @ B @ ( plus_plus_set_nat @ A @ B ) ) ) ).
% set_zero_plus2
thf(fact_1007_set__zero__plus2,axiom,
! [A: set_real,B: set_real] :
( ( member_real @ zero_zero_real @ A )
=> ( ord_less_eq_set_real @ B @ ( plus_plus_set_real @ A @ B ) ) ) ).
% set_zero_plus2
thf(fact_1008_measure__Union,axiom,
! [M: sigma_measure_o,A: set_o,B: set_o] :
( ( ( sigma_emeasure_o @ M @ A )
!= extend2057119558705770725nnreal )
=> ( ( ( sigma_emeasure_o @ M @ B )
!= extend2057119558705770725nnreal )
=> ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
=> ( ( member_set_o @ B @ ( sigma_sets_o @ M ) )
=> ( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
=> ( ( sigma_measure_o2 @ M @ ( sup_sup_set_o @ A @ B ) )
= ( plus_plus_real @ ( sigma_measure_o2 @ M @ A ) @ ( sigma_measure_o2 @ M @ B ) ) ) ) ) ) ) ) ).
% measure_Union
thf(fact_1009_pmf__cond,axiom,
! [P2: probab1498759712122475378_pmf_o,S3: set_o,X: $o] :
( ( ( inf_inf_set_o @ ( probab7458556812659319003_pmf_o @ P2 ) @ S3 )
!= bot_bot_set_o )
=> ( ( ( member_o @ X @ S3 )
=> ( ( probab7541796623121487107_pmf_o @ ( probab8494970989125154181_pmf_o @ P2 @ S3 ) @ X )
= ( divide_divide_real @ ( probab7541796623121487107_pmf_o @ P2 @ X ) @ ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ P2 ) @ S3 ) ) ) )
& ( ~ ( member_o @ X @ S3 )
=> ( ( probab7541796623121487107_pmf_o @ ( probab8494970989125154181_pmf_o @ P2 @ S3 ) @ X )
= zero_zero_real ) ) ) ) ).
% pmf_cond
thf(fact_1010_pmf__cond,axiom,
! [P2: probab469873468395307276mf_nat,S3: set_nat,X: nat] :
( ( ( inf_inf_set_nat @ ( probab3271515132085200205mf_nat @ P2 ) @ S3 )
!= bot_bot_set_nat )
=> ( ( ( member_nat @ X @ S3 )
=> ( ( probab2040650700041456421mf_nat @ ( probab7431941403989380899mf_nat @ P2 @ S3 ) @ X )
= ( divide_divide_real @ ( probab2040650700041456421mf_nat @ P2 @ X ) @ ( sigma_measure_nat2 @ ( probab1352011410425470944mf_nat @ P2 ) @ S3 ) ) ) )
& ( ~ ( member_nat @ X @ S3 )
=> ( ( probab2040650700041456421mf_nat @ ( probab7431941403989380899mf_nat @ P2 @ S3 ) @ X )
= zero_zero_real ) ) ) ) ).
% pmf_cond
thf(fact_1011_pmf__cond,axiom,
! [P2: probab3364570286911266904_pmf_a,S3: set_a,X: a] :
( ( ( inf_inf_set_a @ ( probab49036049091589825_pmf_a @ P2 ) @ S3 )
!= bot_bot_set_a )
=> ( ( ( member_a @ X @ S3 )
=> ( ( probab3485170606694471401_pmf_a @ ( probab4270644268839999083_pmf_a @ P2 @ S3 ) @ X )
= ( divide_divide_real @ ( probab3485170606694471401_pmf_a @ P2 @ X ) @ ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ P2 ) @ S3 ) ) ) )
& ( ~ ( member_a @ X @ S3 )
=> ( ( probab3485170606694471401_pmf_a @ ( probab4270644268839999083_pmf_a @ P2 @ S3 ) @ X )
= zero_zero_real ) ) ) ) ).
% pmf_cond
thf(fact_1012_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M6: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_eq_nat @ X2 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1013_pmf__le__0__iff,axiom,
! [M: probab1498759712122475378_pmf_o,P2: $o] :
( ( ord_less_eq_real @ ( probab7541796623121487107_pmf_o @ M @ P2 ) @ zero_zero_real )
= ( ( probab7541796623121487107_pmf_o @ M @ P2 )
= zero_zero_real ) ) ).
% pmf_le_0_iff
thf(fact_1014_pmf__False__conv__True,axiom,
! [P2: probab1498759712122475378_pmf_o] :
( ( probab7541796623121487107_pmf_o @ P2 @ $false )
= ( minus_minus_real @ one_one_real @ ( probab7541796623121487107_pmf_o @ P2 @ $true ) ) ) ).
% pmf_False_conv_True
thf(fact_1015_pmf__True__conv__False,axiom,
! [P2: probab1498759712122475378_pmf_o] :
( ( probab7541796623121487107_pmf_o @ P2 @ $true )
= ( minus_minus_real @ one_one_real @ ( probab7541796623121487107_pmf_o @ P2 @ $false ) ) ) ).
% pmf_True_conv_False
thf(fact_1016_set__pmf__iff,axiom,
! [X: a,M: probab3364570286911266904_pmf_a] :
( ( member_a @ X @ ( probab49036049091589825_pmf_a @ M ) )
= ( ( probab3485170606694471401_pmf_a @ M @ X )
!= zero_zero_real ) ) ).
% set_pmf_iff
thf(fact_1017_set__pmf__iff,axiom,
! [X: $o,M: probab1498759712122475378_pmf_o] :
( ( member_o @ X @ ( probab7458556812659319003_pmf_o @ M ) )
= ( ( probab7541796623121487107_pmf_o @ M @ X )
!= zero_zero_real ) ) ).
% set_pmf_iff
thf(fact_1018_set__pmf__iff,axiom,
! [X: nat,M: probab469873468395307276mf_nat] :
( ( member_nat @ X @ ( probab3271515132085200205mf_nat @ M ) )
= ( ( probab2040650700041456421mf_nat @ M @ X )
!= zero_zero_real ) ) ).
% set_pmf_iff
thf(fact_1019_pmf__eq__0__set__pmf,axiom,
! [M: probab3364570286911266904_pmf_a,P2: a] :
( ( ( probab3485170606694471401_pmf_a @ M @ P2 )
= zero_zero_real )
= ( ~ ( member_a @ P2 @ ( probab49036049091589825_pmf_a @ M ) ) ) ) ).
% pmf_eq_0_set_pmf
thf(fact_1020_pmf__eq__0__set__pmf,axiom,
! [M: probab1498759712122475378_pmf_o,P2: $o] :
( ( ( probab7541796623121487107_pmf_o @ M @ P2 )
= zero_zero_real )
= ( ~ ( member_o @ P2 @ ( probab7458556812659319003_pmf_o @ M ) ) ) ) ).
% pmf_eq_0_set_pmf
thf(fact_1021_pmf__eq__0__set__pmf,axiom,
! [M: probab469873468395307276mf_nat,P2: nat] :
( ( ( probab2040650700041456421mf_nat @ M @ P2 )
= zero_zero_real )
= ( ~ ( member_nat @ P2 @ ( probab3271515132085200205mf_nat @ M ) ) ) ) ).
% pmf_eq_0_set_pmf
thf(fact_1022_pmf__eq__iff,axiom,
( ( ^ [Y3: probab1498759712122475378_pmf_o,Z3: probab1498759712122475378_pmf_o] : ( Y3 = Z3 ) )
= ( ^ [M3: probab1498759712122475378_pmf_o,N4: probab1498759712122475378_pmf_o] :
! [I4: $o] :
( ( probab7541796623121487107_pmf_o @ M3 @ I4 )
= ( probab7541796623121487107_pmf_o @ N4 @ I4 ) ) ) ) ).
% pmf_eq_iff
thf(fact_1023_pmf__eqI,axiom,
! [M: probab1498759712122475378_pmf_o,N: probab1498759712122475378_pmf_o] :
( ! [I2: $o] :
( ( probab7541796623121487107_pmf_o @ M @ I2 )
= ( probab7541796623121487107_pmf_o @ N @ I2 ) )
=> ( M = N ) ) ).
% pmf_eqI
thf(fact_1024_pmf__nonneg,axiom,
! [P2: probab1498759712122475378_pmf_o,X: $o] : ( ord_less_eq_real @ zero_zero_real @ ( probab7541796623121487107_pmf_o @ P2 @ X ) ) ).
% pmf_nonneg
thf(fact_1025_pmf__le__1,axiom,
! [P2: probab1498759712122475378_pmf_o,X: $o] : ( ord_less_eq_real @ ( probab7541796623121487107_pmf_o @ P2 @ X ) @ one_one_real ) ).
% pmf_le_1
thf(fact_1026_pmf_Orep__eq,axiom,
( probab7541796623121487107_pmf_o
= ( ^ [X2: probab1498759712122475378_pmf_o,Y2: $o] : ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ X2 ) @ ( insert_o @ Y2 @ bot_bot_set_o ) ) ) ) ).
% pmf.rep_eq
thf(fact_1027_pmf_Orep__eq,axiom,
( probab3485170606694471401_pmf_a
= ( ^ [X2: probab3364570286911266904_pmf_a,Y2: a] : ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ X2 ) @ ( insert_a @ Y2 @ bot_bot_set_a ) ) ) ) ).
% pmf.rep_eq
thf(fact_1028_measure__pmf__single,axiom,
! [M: probab1498759712122475378_pmf_o,X: $o] :
( ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ M ) @ ( insert_o @ X @ bot_bot_set_o ) )
= ( probab7541796623121487107_pmf_o @ M @ X ) ) ).
% measure_pmf_single
thf(fact_1029_measure__pmf__single,axiom,
! [M: probab3364570286911266904_pmf_a,X: a] :
( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ M ) @ ( insert_a @ X @ bot_bot_set_a ) )
= ( probab3485170606694471401_pmf_a @ M @ X ) ) ).
% measure_pmf_single
thf(fact_1030_measure__prob__cong__0,axiom,
! [A: set_nat,B: set_nat,P2: probab469873468395307276mf_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ A @ B ) )
=> ( ( probab2040650700041456421mf_nat @ P2 @ X3 )
= zero_zero_real ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( minus_minus_set_nat @ B @ A ) )
=> ( ( probab2040650700041456421mf_nat @ P2 @ X3 )
= zero_zero_real ) )
=> ( ( sigma_measure_nat2 @ ( probab1352011410425470944mf_nat @ P2 ) @ A )
= ( sigma_measure_nat2 @ ( probab1352011410425470944mf_nat @ P2 ) @ B ) ) ) ) ).
% measure_prob_cong_0
thf(fact_1031_measure__prob__cong__0,axiom,
! [A: set_o,B: set_o,P2: probab1498759712122475378_pmf_o] :
( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ A @ B ) )
=> ( ( probab7541796623121487107_pmf_o @ P2 @ X3 )
= zero_zero_real ) )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ ( minus_minus_set_o @ B @ A ) )
=> ( ( probab7541796623121487107_pmf_o @ P2 @ X3 )
= zero_zero_real ) )
=> ( ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ P2 ) @ A )
= ( sigma_measure_o2 @ ( probab7036721048548158344_pmf_o @ P2 ) @ B ) ) ) ) ).
% measure_prob_cong_0
thf(fact_1032_measure__prob__cong__0,axiom,
! [A: set_a,B: set_a,P2: probab3364570286911266904_pmf_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ A @ B ) )
=> ( ( probab3485170606694471401_pmf_a @ P2 @ X3 )
= zero_zero_real ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ B @ A ) )
=> ( ( probab3485170606694471401_pmf_a @ P2 @ X3 )
= zero_zero_real ) )
=> ( ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ P2 ) @ A )
= ( sigma_measure_a2 @ ( probab7257411610070727406_pmf_a @ P2 ) @ B ) ) ) ) ).
% measure_prob_cong_0
thf(fact_1033_pmf__bernoulli__False,axiom,
! [P2: real] :
( ( ord_less_eq_real @ zero_zero_real @ P2 )
=> ( ( ord_less_eq_real @ P2 @ one_one_real )
=> ( ( probab7541796623121487107_pmf_o @ ( probab6844364797682710202li_pmf @ P2 ) @ $false )
= ( minus_minus_real @ one_one_real @ P2 ) ) ) ) ).
% pmf_bernoulli_False
thf(fact_1034_divide__le__eq__1__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1035_divide__le__eq__1__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ord_less_eq_real @ B2 @ A2 ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1036_le__divide__eq__1__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ord_less_eq_real @ B2 @ A2 ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1037_le__divide__eq__1__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ord_less_eq_real @ A2 @ B2 ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1038_pmf__bernoulli__True,axiom,
! [P2: real] :
( ( ord_less_eq_real @ zero_zero_real @ P2 )
=> ( ( ord_less_eq_real @ P2 @ one_one_real )
=> ( ( probab7541796623121487107_pmf_o @ ( probab6844364797682710202li_pmf @ P2 ) @ $true )
= P2 ) ) ) ).
% pmf_bernoulli_True
thf(fact_1039_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_1040_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_1041_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_1042_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_1043_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1044_nless__le,axiom,
! [A2: real,B2: real] :
( ( ~ ( ord_less_real @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1045_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1046_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_1047_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1048_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_1049_dense__ge,axiom,
! [Z2: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z2 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_ge
thf(fact_1050_dense__le,axiom,
! [Y: real,Z2: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z2 ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ).
% dense_le
thf(fact_1051_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1052_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ~ ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1053_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1054_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_1055_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1056_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_real @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1057_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1058_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1059_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1060_order_Ostrict__trans1,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_1061_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1062_order_Ostrict__trans2,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_1063_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1064_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A5: real,B4: real] :
( ( ord_less_eq_real @ A5 @ B4 )
& ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1065_dense__ge__bounded,axiom,
! [Z2: real,X: real,Y: real] :
( ( ord_less_real @ Z2 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z2 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_ge_bounded
thf(fact_1066_dense__le__bounded,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z2 ) ) )
=> ( ord_less_eq_real @ Y @ Z2 ) ) ) ).
% dense_le_bounded
thf(fact_1067_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1068_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_real @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1069_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1070_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1071_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1072_dual__order_Ostrict__trans1,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_eq_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1073_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1074_dual__order_Ostrict__trans2,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1075_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1076_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B4: real,A5: real] :
( ( ord_less_eq_real @ B4 @ A5 )
& ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1077_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1078_order_Ostrict__implies__order,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ord_less_eq_real @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1079_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1080_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1081_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_1082_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_1083_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_1084_order__less__le,axiom,
( ord_less_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_1085_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1086_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_1087_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1088_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_1089_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1090_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_1091_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1092_order__le__neq__trans,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1093_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1094_order__neq__le__trans,axiom,
! [A2: real,B2: real] :
( ( A2 != B2 )
=> ( ( ord_less_eq_real @ A2 @ B2 )
=> ( ord_less_real @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1095_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1096_order__le__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1097_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1098_order__less__le__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1099_order__le__less__subst1,axiom,
! [A2: nat,F2: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1100_order__le__less__subst1,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1101_order__le__less__subst1,axiom,
! [A2: real,F2: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1102_order__le__less__subst1,axiom,
! [A2: real,F2: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1103_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1104_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > real,C: real] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1105_order__le__less__subst2,axiom,
! [A2: real,B2: real,F2: real > nat,C: nat] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1106_order__le__less__subst2,axiom,
! [A2: real,B2: real,F2: real > real,C: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F2 @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1107_order__less__le__subst1,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1108_order__less__le__subst1,axiom,
! [A2: real,F2: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1109_order__less__le__subst1,axiom,
! [A2: nat,F2: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1110_order__less__le__subst1,axiom,
! [A2: real,F2: real > real,B2: real,C: real] :
( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_eq_real @ X3 @ Y4 )
=> ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1111_order__less__le__subst2,axiom,
! [A2: real,B2: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1112_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1113_order__less__le__subst2,axiom,
! [A2: real,B2: real,F2: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F2 @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1114_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_real @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_1115_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1116_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_1117_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1118_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1119_verit__comp__simplify1_I3_J,axiom,
! [B8: nat,A8: nat] :
( ( ~ ( ord_less_eq_nat @ B8 @ A8 ) )
= ( ord_less_nat @ A8 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1120_verit__comp__simplify1_I3_J,axiom,
! [B8: real,A8: real] :
( ( ~ ( ord_less_eq_real @ B8 @ A8 ) )
= ( ord_less_real @ A8 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1121_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A2 @ C5 )
& ( ord_less_eq_nat @ C5 @ B2 )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ C5 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1122_complete__interval,axiom,
! [A2: real,B2: real,P: real > $o] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C5: real] :
( ( ord_less_eq_real @ A2 @ C5 )
& ( ord_less_eq_real @ C5 @ B2 )
& ! [X4: real] :
( ( ( ord_less_eq_real @ A2 @ X4 )
& ( ord_less_real @ X4 @ C5 ) )
=> ( P @ X4 ) )
& ! [D3: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A2 @ X3 )
& ( ord_less_real @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1123_top_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != top_top_set_nat )
= ( ord_less_set_nat @ A2 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_1124_top_Onot__eq__extremum,axiom,
! [A2: set_o] :
( ( A2 != top_top_set_o )
= ( ord_less_set_o @ A2 @ top_top_set_o ) ) ).
% top.not_eq_extremum
thf(fact_1125_top_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A2 ) ).
% top.extremum_strict
thf(fact_1126_top_Oextremum__strict,axiom,
! [A2: set_o] :
~ ( ord_less_set_o @ top_top_set_o @ A2 ) ).
% top.extremum_strict
thf(fact_1127_lt__ex,axiom,
! [X: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% lt_ex
thf(fact_1128_gt__ex,axiom,
! [X: real] :
? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).
% gt_ex
thf(fact_1129_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_1130_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_1131_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1132_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_1133_order_Oasym,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order.asym
thf(fact_1134_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_1135_ord__eq__less__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( A2 = B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1136_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_1137_ord__less__eq__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1138_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_1139_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_1140_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1141_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_1142_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1143_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_1144_dual__order_Oasym,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ~ ( ord_less_real @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_1145_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_1146_dual__order_Oirrefl,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_1147_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_1148_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ~ ( P3 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1149_linorder__less__wlog,axiom,
! [P: real > real > $o,A2: real,B2: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_1150_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_1151_order_Ostrict__trans,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_1152_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_1153_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1154_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1155_dual__order_Ostrict__trans,axiom,
! [B2: real,A2: real,C: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_1156_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_1157_order_Ostrict__implies__not__eq,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_1158_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_1159_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A2: real] :
( ( ord_less_real @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1160_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1161_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1162_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_1163_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_1164_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_1165_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1166_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_1167_order__less__asym_H,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ~ ( ord_less_real @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_1168_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_1169_order__less__trans,axiom,
! [X: real,Y: real,Z2: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z2 )
=> ( ord_less_real @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_1170_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_1171_ord__eq__less__subst,axiom,
! [A2: real,F2: real > real,B2: real,C: real] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1172_ord__eq__less__subst,axiom,
! [A2: nat,F2: real > nat,B2: real,C: real] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1173_ord__eq__less__subst,axiom,
! [A2: real,F2: nat > real,B2: nat,C: nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1174_ord__eq__less__subst,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1175_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F2: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1176_ord__less__eq__subst,axiom,
! [A2: real,B2: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1177_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1178_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1179_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_1180_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_1181_order__less__subst1,axiom,
! [A2: real,F2: real > real,B2: real,C: real] :
( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1182_order__less__subst1,axiom,
! [A2: real,F2: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1183_order__less__subst1,axiom,
! [A2: nat,F2: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1184_order__less__subst1,axiom,
! [A2: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_1185_order__less__subst2,axiom,
! [A2: real,B2: real,F2: real > real,C: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_real @ ( F2 @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1186_order__less__subst2,axiom,
! [A2: real,B2: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: real,Y4: real] :
( ( ord_less_real @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1187_order__less__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_real @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_real @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1188_order__less__subst2,axiom,
! [A2: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_1189_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_1190_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_1191_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_1192_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_1193_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_1194_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_1195_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1196_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1197_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1198_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_1199_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_1200_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_1201_verit__comp__simplify1_I1_J,axiom,
! [A2: real] :
~ ( ord_less_real @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_1202_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_1203_bot_Onot__eq__extremum,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
= ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1204_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1205_bot_Oextremum__strict,axiom,
! [A2: set_o] :
~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% bot.extremum_strict
thf(fact_1206_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1207_finite__imp__Sup__less,axiom,
! [X6: set_real,X: real,A2: real] :
( ( finite_finite_real @ X6 )
=> ( ( member_real @ X @ X6 )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ( ord_less_real @ X3 @ A2 ) )
=> ( ord_less_real @ ( comple1385675409528146559p_real @ X6 ) @ A2 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_1208_finite__imp__Sup__less,axiom,
! [X6: set_nat,X: nat,A2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( member_nat @ X @ X6 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ( ord_less_nat @ X3 @ A2 ) )
=> ( ord_less_nat @ ( complete_Sup_Sup_nat @ X6 ) @ A2 ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_1209_ex__min__if__finite,axiom,
! [S: set_o] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ? [X3: $o] :
( ( member_o @ X3 @ S )
& ~ ? [Xa: $o] :
( ( member_o @ Xa @ S )
& ( ord_less_o @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1210_ex__min__if__finite,axiom,
! [S: set_real] :
( ( finite_finite_real @ S )
=> ( ( S != bot_bot_set_real )
=> ? [X3: real] :
( ( member_real @ X3 @ S )
& ~ ? [Xa: real] :
( ( member_real @ Xa @ S )
& ( ord_less_real @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1211_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1212_infinite__growing,axiom,
! [X6: set_o] :
( ( X6 != bot_bot_set_o )
=> ( ! [X3: $o] :
( ( member_o @ X3 @ X6 )
=> ? [Xa: $o] :
( ( member_o @ Xa @ X6 )
& ( ord_less_o @ X3 @ Xa ) ) )
=> ~ ( finite_finite_o @ X6 ) ) ) ).
% infinite_growing
thf(fact_1213_infinite__growing,axiom,
! [X6: set_real] :
( ( X6 != bot_bot_set_real )
=> ( ! [X3: real] :
( ( member_real @ X3 @ X6 )
=> ? [Xa: real] :
( ( member_real @ Xa @ X6 )
& ( ord_less_real @ X3 @ Xa ) ) )
=> ~ ( finite_finite_real @ X6 ) ) ) ).
% infinite_growing
thf(fact_1214_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X6 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_1215_add__less__le__mono,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1216_add__less__le__mono,axiom,
! [A2: real,B2: real,C: real,D2: real] :
( ( ord_less_real @ A2 @ B2 )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1217_add__le__less__mono,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1218_add__le__less__mono,axiom,
! [A2: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_real @ A2 @ B2 )
=> ( ( ord_less_real @ C @ D2 )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B2 @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1219_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J3 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1220_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: real,J3: real,K2: real,L: real] :
( ( ( ord_less_real @ I3 @ J3 )
& ( ord_less_eq_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K2 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1221_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: nat,J3: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J3 )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1222_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: real,J3: real,K2: real,L: real] :
( ( ( ord_less_eq_real @ I3 @ J3 )
& ( ord_less_real @ K2 @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I3 @ K2 ) @ ( plus_plus_real @ J3 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1223_le__sup__lexord,axiom,
! [K2: nat > nat,A: nat,B: nat,Ca: nat,C: nat,S3: nat] :
( ( ( ord_less_nat @ ( K2 @ A ) @ ( K2 @ B ) )
=> ( ord_less_eq_nat @ Ca @ B ) )
=> ( ( ( ord_less_nat @ ( K2 @ B ) @ ( K2 @ A ) )
=> ( ord_less_eq_nat @ Ca @ A ) )
=> ( ( ( ( K2 @ A )
= ( K2 @ B ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K2 @ B ) @ ( K2 @ A ) )
=> ( ~ ( ord_less_eq_nat @ ( K2 @ A ) @ ( K2 @ B ) )
=> ( ord_less_eq_nat @ Ca @ S3 ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur4601247141005857854at_nat @ A @ B @ K2 @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_1224_le__sup__lexord,axiom,
! [K2: nat > real,A: nat,B: nat,Ca: nat,C: nat,S3: nat] :
( ( ( ord_less_real @ ( K2 @ A ) @ ( K2 @ B ) )
=> ( ord_less_eq_nat @ Ca @ B ) )
=> ( ( ( ord_less_real @ ( K2 @ B ) @ ( K2 @ A ) )
=> ( ord_less_eq_nat @ Ca @ A ) )
=> ( ( ( ( K2 @ A )
= ( K2 @ B ) )
=> ( ord_less_eq_nat @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_real @ ( K2 @ B ) @ ( K2 @ A ) )
=> ( ~ ( ord_less_eq_real @ ( K2 @ A ) @ ( K2 @ B ) )
=> ( ord_less_eq_nat @ Ca @ S3 ) ) )
=> ( ord_less_eq_nat @ Ca @ ( measur8600355784167071770t_real @ A @ B @ K2 @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_1225_le__sup__lexord,axiom,
! [K2: real > nat,A: real,B: real,Ca: real,C: real,S3: real] :
( ( ( ord_less_nat @ ( K2 @ A ) @ ( K2 @ B ) )
=> ( ord_less_eq_real @ Ca @ B ) )
=> ( ( ( ord_less_nat @ ( K2 @ B ) @ ( K2 @ A ) )
=> ( ord_less_eq_real @ Ca @ A ) )
=> ( ( ( ( K2 @ A )
= ( K2 @ B ) )
=> ( ord_less_eq_real @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_nat @ ( K2 @ B ) @ ( K2 @ A ) )
=> ( ~ ( ord_less_eq_nat @ ( K2 @ A ) @ ( K2 @ B ) )
=> ( ord_less_eq_real @ Ca @ S3 ) ) )
=> ( ord_less_eq_real @ Ca @ ( measur3944292320441194650al_nat @ A @ B @ K2 @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_1226_le__sup__lexord,axiom,
! [K2: real > real,A: real,B: real,Ca: real,C: real,S3: real] :
( ( ( ord_less_real @ ( K2 @ A ) @ ( K2 @ B ) )
=> ( ord_less_eq_real @ Ca @ B ) )
=> ( ( ( ord_less_real @ ( K2 @ B ) @ ( K2 @ A ) )
=> ( ord_less_eq_real @ Ca @ A ) )
=> ( ( ( ( K2 @ A )
= ( K2 @ B ) )
=> ( ord_less_eq_real @ Ca @ C ) )
=> ( ( ~ ( ord_less_eq_real @ ( K2 @ B ) @ ( K2 @ A ) )
=> ( ~ ( ord_less_eq_real @ ( K2 @ A ) @ ( K2 @ B ) )
=> ( ord_less_eq_real @ Ca @ S3 ) ) )
=> ( ord_less_eq_real @ Ca @ ( measur6875964016165910134l_real @ A @ B @ K2 @ S3 @ C ) ) ) ) ) ) ).
% le_sup_lexord
thf(fact_1227_add__strict__increasing2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1228_add__strict__increasing2,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1229_add__strict__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1230_add__strict__increasing,axiom,
! [A2: real,B2: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ B2 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1231_add__pos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1232_add__pos__nonneg,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1233_add__nonpos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1234_add__nonpos__neg,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ A2 @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_1235_add__nonneg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1236_add__nonneg__pos,axiom,
! [A2: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1237_add__neg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1238_add__neg__nonpos,axiom,
! [A2: real,B2: real] :
( ( ord_less_real @ A2 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A2 @ B2 ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_1239_field__le__epsilon,axiom,
! [X: real,Y: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E ) ) )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% field_le_epsilon
thf(fact_1240_divide__nonpos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_pos
thf(fact_1241_divide__nonpos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1242_divide__nonneg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1243_divide__nonneg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_neg
thf(fact_1244_divide__le__cancel,axiom,
! [A2: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B2 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A2 @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% divide_le_cancel
thf(fact_1245_frac__less2,axiom,
! [X: real,Y: real,W2: real,Z2: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less2
thf(fact_1246_frac__less,axiom,
! [X: real,Y: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_less
thf(fact_1247_frac__le,axiom,
! [Y: real,X: real,W2: real,Z2: real] :
( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ zero_zero_real @ W2 )
=> ( ( ord_less_eq_real @ W2 @ Z2 )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Z2 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).
% frac_le
thf(fact_1248_finite__linorder__max__induct,axiom,
! [A: set_o,P: set_o > $o] :
( ( finite_finite_o @ A )
=> ( ( P @ bot_bot_set_o )
=> ( ! [B3: $o,A6: set_o] :
( ( finite_finite_o @ A6 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A6 )
=> ( ord_less_o @ X4 @ B3 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_o @ B3 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1249_finite__linorder__max__induct,axiom,
! [A: set_real,P: set_real > $o] :
( ( finite_finite_real @ A )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B3: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( ord_less_real @ X4 @ B3 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_real @ B3 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1250_finite__linorder__max__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( ord_less_nat @ X4 @ B3 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B3 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1251_finite__linorder__min__induct,axiom,
! [A: set_o,P: set_o > $o] :
( ( finite_finite_o @ A )
=> ( ( P @ bot_bot_set_o )
=> ( ! [B3: $o,A6: set_o] :
( ( finite_finite_o @ A6 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A6 )
=> ( ord_less_o @ B3 @ X4 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_o @ B3 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1252_finite__linorder__min__induct,axiom,
! [A: set_real,P: set_real > $o] :
( ( finite_finite_real @ A )
=> ( ( P @ bot_bot_set_real )
=> ( ! [B3: real,A6: set_real] :
( ( finite_finite_real @ A6 )
=> ( ! [X4: real] :
( ( member_real @ X4 @ A6 )
=> ( ord_less_real @ B3 @ X4 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_real @ B3 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1253_finite__linorder__min__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( ord_less_nat @ B3 @ X4 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat @ B3 @ A6 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1254_finite__Sup__less__iff,axiom,
! [X6: set_real,A2: real] :
( ( finite_finite_real @ X6 )
=> ( ( X6 != bot_bot_set_real )
=> ( ( ord_less_real @ ( comple1385675409528146559p_real @ X6 ) @ A2 )
= ( ! [X2: real] :
( ( member_real @ X2 @ X6 )
=> ( ord_less_real @ X2 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_1255_finite__Sup__less__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( finite_finite_nat @ X6 )
=> ( ( X6 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( complete_Sup_Sup_nat @ X6 ) @ A2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ X6 )
=> ( ord_less_nat @ X2 @ A2 ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_1256_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F2: nat > real] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ ( lattic488527866317076247t_real @ F2 @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1257_arg__min__if__finite_I2_J,axiom,
! [S: set_o,F2: $o > real] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ~ ? [X4: $o] :
( ( member_o @ X4 @ S )
& ( ord_less_real @ ( F2 @ X4 ) @ ( F2 @ ( lattic8697145971487455083o_real @ F2 @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1258_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat @ X4 @ S )
& ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1259_arg__min__if__finite_I2_J,axiom,
! [S: set_o,F2: $o > nat] :
( ( finite_finite_o @ S )
=> ( ( S != bot_bot_set_o )
=> ~ ? [X4: $o] :
( ( member_o @ X4 @ S )
& ( ord_less_nat @ ( F2 @ X4 ) @ ( F2 @ ( lattic2775856028456453135_o_nat @ F2 @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1260_divide__le__eq__1,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ B2 @ A2 ) @ one_one_real )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ B2 @ A2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ A2 @ B2 ) )
| ( A2 = zero_zero_real ) ) ) ).
% divide_le_eq_1
thf(fact_1261_le__divide__eq__1,axiom,
! [B2: real,A2: real] :
( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B2 @ A2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A2 )
& ( ord_less_eq_real @ A2 @ B2 ) )
| ( ( ord_less_real @ A2 @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ A2 ) ) ) ) ).
% le_divide_eq_1
thf(fact_1262_set__pmf__bernoulli,axiom,
! [P2: real] :
( ( ord_less_real @ zero_zero_real @ P2 )
=> ( ( ord_less_real @ P2 @ one_one_real )
=> ( ( probab7458556812659319003_pmf_o @ ( probab6844364797682710202li_pmf @ P2 ) )
= top_top_set_o ) ) ) ).
% set_pmf_bernoulli
thf(fact_1263_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M6: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N4 )
=> ( ord_less_nat @ X2 @ M6 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1264_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ord_less_nat @ X3 @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_1265_pmf__neq__exists__less,axiom,
! [M: probab1498759712122475378_pmf_o,N: probab1498759712122475378_pmf_o] :
( ( M != N )
=> ? [X3: $o] : ( ord_less_real @ ( probab7541796623121487107_pmf_o @ M @ X3 ) @ ( probab7541796623121487107_pmf_o @ N @ X3 ) ) ) ).
% pmf_neq_exists_less
thf(fact_1266_pmf__pos,axiom,
! [P2: probab1498759712122475378_pmf_o,X: $o] :
( ( ( probab7541796623121487107_pmf_o @ P2 @ X )
!= zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( probab7541796623121487107_pmf_o @ P2 @ X ) ) ) ).
% pmf_pos
thf(fact_1267_pmf__not__neg,axiom,
! [P2: probab1498759712122475378_pmf_o,X: $o] :
~ ( ord_less_real @ ( probab7541796623121487107_pmf_o @ P2 @ X ) @ zero_zero_real ) ).
% pmf_not_neg
thf(fact_1268_not__psubset__empty,axiom,
! [A: set_o] :
~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).
% not_psubset_empty
thf(fact_1269_psubset__imp__ex__mem,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1270_psubset__imp__ex__mem,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_set_o @ A @ B )
=> ? [B3: $o] : ( member_o @ B3 @ ( minus_minus_set_o @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1271_psubset__imp__ex__mem,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1272_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M6: nat] :
? [N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
& ( member_nat @ N3 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1273_unbounded__k__infinite,axiom,
! [K2: nat,S: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K2 @ M4 )
=> ? [N5: nat] :
( ( ord_less_nat @ M4 @ N5 )
& ( member_nat @ N5 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1274_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B9: set_nat] :
( ( ord_less_set_nat @ B9 @ A6 )
=> ( P @ B9 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_1275_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1276_set__pmf__poisson,axiom,
! [Rate: real] :
( ( ord_less_real @ zero_zero_real @ Rate )
=> ( ( probab3271515132085200205mf_nat @ ( probab4011777617282711093on_pmf @ Rate ) )
= top_top_set_nat ) ) ).
% set_pmf_poisson
thf(fact_1277_set__pmf__geometric,axiom,
! [P2: real] :
( ( ord_less_real @ zero_zero_real @ P2 )
=> ( ( ord_less_real @ P2 @ one_one_real )
=> ( ( probab3271515132085200205mf_nat @ ( probab1729510186672448573ic_pmf @ P2 ) )
= top_top_set_nat ) ) ) ).
% set_pmf_geometric
% Conjectures (1)
thf(conj_0,conjecture,
probab7247484486040049089pace_a @ ( sigma_8692839461743104066pace_a @ m @ ( probab49036049091589825_pmf_a @ p ) ) ).
%------------------------------------------------------------------------------