TPTP Problem File: SLH0354^1.p

%------------------------------------------------------------------------------
% 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    : Quasi_Borel_Spaces/0003_Binary_CoProduct_QuasiBorel/prob_00492_019698__15223480_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1975 ( 606 unt; 693 typ;   0 def)
%            Number of atoms       : 3693 (1510 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 11294 ( 379   ~;  25   |; 180   &;9093   @)
%                                         (   0 <=>;1617  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   6 avg)
%            Number of types       :  126 ( 125 usr)
%            Number of type conns  : 1911 (1911   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  571 ( 568 usr;  79 con; 0-3 aty)
%            Number of variables   : 3163 ( 124   ^;2972   !;  67   ?;3163   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:09:12.026
%------------------------------------------------------------------------------
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% Explicit typings (568)
thf(sy_c_Basic__BNFs_Opred__sum_001_062_I_Eo_Mt__Real__Oreal_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    basic_4205938908511992687o_real: ( ( $o > real ) > $o ) > ( ( $o > real ) > $o ) > sum_su815935806896055909o_real > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001_062_I_Eo_Mt__Real__Oreal_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    basic_4510141698895607777t_real: ( ( $o > real ) > $o ) > ( ( nat > real ) > $o ) > sum_su7765403328973102337t_real > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001_062_I_Eo_Mt__Real__Oreal_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    basic_4697917730111463491real_a: ( ( $o > real ) > $o ) > ( ( real > a ) > $o ) > sum_su7886454506223791033real_a > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001_062_I_Eo_Mt__Real__Oreal_J_001_062_Itf__a_Mtf__b_J,type,
    basic_6216495006292779436al_a_b: ( ( $o > real ) > $o ) > ( ( a > b ) > $o ) > sum_sum_o_real_a_b > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001_062_I_Eo_Mt__Real__Oreal_J_001_062_Itf__c_Mtf__b_J,type,
    basic_641583842347427370al_c_b: ( ( $o > real ) > $o ) > ( ( c > b ) > $o ) > sum_sum_o_real_c_b > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001_062_It__Real__Oreal_Mtf__a_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    basic_2577378067801423003o_real: ( ( real > a ) > $o ) > ( ( $o > real ) > $o ) > sum_su2067798924538045457o_real > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    basic_7720748700303055285t_real: ( ( real > a ) > $o ) > ( ( nat > real ) > $o ) > sum_su5472343219575513685t_real > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    basic_8747522359897721711real_a: ( ( real > a ) > $o ) > ( ( real > a ) > $o ) > sum_su2571395965866611557real_a > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
    basic_3345994146885751680_a_a_b: ( ( real > a ) > $o ) > ( ( a > b ) > $o ) > sum_sum_real_a_a_b > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
    basic_6994455019795175422_a_c_b: ( ( real > a ) > $o ) > ( ( c > b ) > $o ) > sum_sum_real_a_c_b > $o ).

thf(sy_c_Basic__BNFs_Opred__sum_001tf__a_001tf__c,type,
    basic_pred_sum_a_c: ( a > $o ) > ( c > $o ) > sum_sum_a_c > $o ).

thf(sy_c_Basic__BNFs_Osetl_001_062_I_Eo_Mt__Real__Oreal_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    basic_1079876227406846053o_real: sum_su815935806896055909o_real > set_o_real ).

thf(sy_c_Basic__BNFs_Osetl_001_062_I_Eo_Mt__Real__Oreal_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    basic_343682719230182507t_real: sum_su7765403328973102337t_real > set_o_real ).

thf(sy_c_Basic__BNFs_Osetl_001_062_I_Eo_Mt__Real__Oreal_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    basic_5452494658869518905real_a: sum_su7886454506223791033real_a > set_o_real ).

thf(sy_c_Basic__BNFs_Osetl_001_062_I_Eo_Mt__Real__Oreal_J_001_062_Itf__a_Mtf__b_J,type,
    basic_1115715530398340662al_a_b: sum_sum_o_real_a_b > set_o_real ).

thf(sy_c_Basic__BNFs_Osetl_001_062_I_Eo_Mt__Real__Oreal_J_001_062_Itf__c_Mtf__b_J,type,
    basic_4764176403307764404al_c_b: sum_sum_o_real_c_b > set_o_real ).

thf(sy_c_Basic__BNFs_Osetl_001_062_It__Real__Oreal_Mtf__a_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    basic_3331954996559478417o_real: sum_su2067798924538045457o_real > set_real_a ).

thf(sy_c_Basic__BNFs_Osetl_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    basic_4771245887642665279t_real: sum_su5472343219575513685t_real > set_real_a ).

thf(sy_c_Basic__BNFs_Osetl_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    basic_1952671662426305125real_a: sum_su2571395965866611557real_a > set_real_a ).

thf(sy_c_Basic__BNFs_Osetl_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
    basic_8357132796964709130_a_a_b: sum_sum_real_a_a_b > set_real_a ).

thf(sy_c_Basic__BNFs_Osetl_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
    basic_2782221633019357064_a_c_b: sum_sum_real_a_c_b > set_real_a ).

thf(sy_c_Basic__BNFs_Osetl_001tf__a_001tf__c,type,
    basic_setl_a_c: sum_sum_a_c > set_a ).

thf(sy_c_Basic__BNFs_Osetlp_001tf__a_001tf__c,type,
    basic_setlp_a_c: sum_sum_a_c > a > $o ).

thf(sy_c_Basic__BNFs_Osetr_001_062_I_Eo_Mt__Real__Oreal_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    basic_3048226007810710379o_real: sum_su815935806896055909o_real > set_o_real ).

thf(sy_c_Basic__BNFs_Osetr_001_062_I_Eo_Mt__Real__Oreal_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    basic_8384512648665255653t_real: sum_su7765403328973102337t_real > set_nat_real ).

thf(sy_c_Basic__BNFs_Osetr_001_062_I_Eo_Mt__Real__Oreal_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    basic_9127930566137971263real_a: sum_su7886454506223791033real_a > set_real_a ).

thf(sy_c_Basic__BNFs_Osetr_001_062_I_Eo_Mt__Real__Oreal_J_001_062_Itf__a_Mtf__b_J,type,
    basic_6475791115876108464al_a_b: sum_sum_o_real_a_b > set_a_b ).

thf(sy_c_Basic__BNFs_Osetr_001_062_I_Eo_Mt__Real__Oreal_J_001_062_Itf__c_Mtf__b_J,type,
    basic_900879951930756398al_c_b: sum_sum_o_real_c_b > set_c_b ).

thf(sy_c_Basic__BNFs_Osetr_001_062_It__Real__Oreal_Mtf__a_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    basic_7007390903827930775o_real: sum_su2067798924538045457o_real > set_o_real ).

thf(sy_c_Basic__BNFs_Osetr_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    basic_7028784127217823417t_real: sum_su5472343219575513685t_real > set_nat_real ).

thf(sy_c_Basic__BNFs_Osetr_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    basic_2020826412209680235real_a: sum_su2571395965866611557real_a > set_real_a ).

thf(sy_c_Basic__BNFs_Osetr_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
    basic_6862289076754118788_a_a_b: sum_sum_real_a_a_b > set_a_b ).

thf(sy_c_Basic__BNFs_Osetr_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
    basic_1287377912808766722_a_c_b: sum_sum_real_a_c_b > set_c_b ).

thf(sy_c_Basic__BNFs_Osetr_001tf__a_001tf__c,type,
    basic_setr_a_c: sum_sum_a_c > set_c ).

thf(sy_c_Basic__BNFs_Osetrp_001tf__a_001tf__c,type,
    basic_setrp_a_c: sum_sum_a_c > c > $o ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001tf__a_001tf__c,type,
    binary8555328655094383375bs_a_c: quasi_borel_a > quasi_borel_c > quasi_4257511854121656471um_a_c ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx2_001tf__a_001tf__c,type,
    binary6242423198552412156x2_a_c: quasi_borel_a > quasi_borel_c > set_real_Sum_sum_a_c ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs__Mx_001tf__a_001tf__c,type,
    binary8286901584692334522Mx_a_c: quasi_borel_a > quasi_borel_c > set_real_Sum_sum_a_c ).

thf(sy_c_Borel__Space_Ois__borel_001_Eo_001t__Real__Oreal,type,
    borel_2269360593130276488o_real: ( $o > real ) > sigma_measure_o > $o ).

thf(sy_c_Borel__Space_Ois__borel_001t__Nat__Onat_001t__Real__Oreal,type,
    borel_9213571707143006522t_real: ( nat > real ) > sigma_measure_nat > $o ).

thf(sy_c_Borel__Space_Ois__borel_001t__Real__Oreal_001_Eo,type,
    borel_3778424968171446062real_o: ( real > $o ) > sigma_measure_real > $o ).

thf(sy_c_Borel__Space_Ois__borel_001t__Real__Oreal_001t__Nat__Onat,type,
    borel_4557508243417129402al_nat: ( real > nat ) > sigma_measure_real > $o ).

thf(sy_c_Borel__Space_Ois__borel_001t__Real__Oreal_001t__Real__Oreal,type,
    borel_236569967776329622l_real: ( real > real ) > sigma_measure_real > $o ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001_Eo,type,
    borel_5500255247093592246orel_o: sigma_measure_o ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Extended____Nonnegative____Real__Oennreal,type,
    borel_6524799422816628122nnreal: sigma_7234349610311085201nnreal ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Nat__Onat,type,
    borel_8449730974584783410el_nat: sigma_measure_nat ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
    borel_5078946678739801102l_real: sigma_measure_real ).

thf(sy_c_Countable__Set_Ocountable_001_062_I_Eo_Mt__Real__Oreal_J,type,
    counta8783200249485735024o_real: set_o_real > $o ).

thf(sy_c_Countable__Set_Ocountable_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    counta2162411829015494944t_real: set_nat_real > $o ).

thf(sy_c_Countable__Set_Ocountable_001_062_It__Real__Oreal_Mtf__a_J,type,
    counta6639396083684174020real_a: set_real_a > $o ).

thf(sy_c_Countable__Set_Ocountable_001_062_Itf__a_Mtf__b_J,type,
    counta8232689092827506411le_a_b: set_a_b > $o ).

thf(sy_c_Countable__Set_Ocountable_001_062_Itf__c_Mtf__b_J,type,
    counta2657777928882154345le_c_b: set_c_b > $o ).

thf(sy_c_Countable__Set_Ocountable_001_Eo,type,
    counta5976203206615340371able_o: set_o > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Extended____Nonnegative____Real__Oennreal,type,
    counta8439243037236335165nnreal: set_Ex3793607809372303086nnreal > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Nat__Onat,type,
    counta1168086296615599829le_nat: set_nat > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Real__Oreal,type,
    counta7319604579010473777e_real: set_real > $o ).

thf(sy_c_Countable__Set_Ocountable_001tf__a,type,
    counta4098120917673242425able_a: set_a > $o ).

thf(sy_c_Countable__Set_Ocountable_001tf__c,type,
    counta4098120917673242427able_c: set_c > $o ).

thf(sy_c_Extended__Nonnegative__Real_Oennreal,type,
    extend7643940197134561352nnreal: real > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_Eo_001_Eo,type,
    comp_real_o_o: ( real > $o ) > ( $o > real ) > $o > $o ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    comp_r6281409797179841921nnreal: ( real > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > real ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_real_nat_nat: ( real > nat ) > ( nat > real ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_real_real_real: ( real > real ) > ( real > real ) > real > real ).

thf(sy_c_Fun_Oid_001t__Real__Oreal,type,
    id_real: real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
    minus_2870878895999678972o_real: set_o_real > set_o_real > set_o_real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    minus_3492551254948764970t_real: set_nat_real > set_nat_real > set_nat_real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
    minus_6532636778494125008real_a: set_real_a > set_real_a > set_real_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
    minus_minus_set_a_b: set_a_b > set_a_b > set_a_b ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__c_Mtf__b_J_J,type,
    minus_minus_set_c_b: set_c_b > set_c_b > set_c_b ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
    minus_minus_set_o: set_o > set_o > set_o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    minus_104578273773384135nnreal: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
    minus_minus_set_real: set_real > set_real > set_real ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
    zero_zero_real: real ).

thf(sy_c_If_001t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
    if_Sum_sum_a_c: $o > sum_sum_a_c > sum_sum_a_c > sum_sum_a_c ).

thf(sy_c_Lebesgue__Measure_Olborel_001t__Real__Oreal,type,
    lebesgue_lborel_real: sigma_measure_real ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001_Eo,type,
    measur2926627334652526644sure_o: quasi_borel_o > sigma_measure_o ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur7384687747506661788nnreal: quasi_9015997321629101608nnreal > sigma_7234349610311085201nnreal ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001t__Nat__Onat,type,
    measur7418878410283781684re_nat: quasi_borel_nat > sigma_measure_nat ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001t__Real__Oreal,type,
    measur1733462625046462224e_real: quasi_borel_real > sigma_measure_real ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001tf__a,type,
    measur7857763439677503898sure_a: quasi_borel_a > sigma_measure_a ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001tf__b,type,
    measur7857763439677503899sure_b: quasi_borel_b > sigma_measure_b ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Oqbs__to__measure_001tf__c,type,
    measur7857763439677503900sure_c: quasi_borel_c > sigma_measure_c ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Osigma__Mx_001_Eo,type,
    measur560158960551862217a_Mx_o: quasi_borel_o > set_set_o ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Osigma__Mx_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur4088046290863407431nnreal: quasi_9015997321629101608nnreal > set_se4580700918925141924nnreal ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Osigma__Mx_001t__Nat__Onat,type,
    measur4633199607246208479Mx_nat: quasi_borel_nat > set_set_nat ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Osigma__Mx_001t__Real__Oreal,type,
    measur7113710793995618619x_real: quasi_borel_real > set_set_real ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Osigma__Mx_001tf__a,type,
    measur1355555235234291375a_Mx_a: quasi_borel_a > set_set_a ).

thf(sy_c_Measure__QuasiBorel__Adjunction_Osigma__Mx_001tf__c,type,
    measur1355555235234291377a_Mx_c: quasi_borel_c > set_set_c ).

thf(sy_c_Measure__Space_OSup__measure_H_001_Eo,type,
    measur1214336222341667658sure_o: set_Sigma_measure_o > sigma_measure_o ).

thf(sy_c_Measure__Space_OSup__measure_H_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur1651139276328235014nnreal: set_Si97717610131227249nnreal > sigma_7234349610311085201nnreal ).

thf(sy_c_Measure__Space_OSup__measure_H_001t__Nat__Onat,type,
    measur3575099672463795358re_nat: set_Si3048223896905877257re_nat > sigma_measure_nat ).

thf(sy_c_Measure__Space_OSup__measure_H_001t__Real__Oreal,type,
    measur8657758558638653562e_real: set_Si6059263944882162789e_real > sigma_measure_real ).

thf(sy_c_Measure__Space_Osup__measure_H_001_Eo,type,
    measur4529518739368704874sure_o: sigma_measure_o > sigma_measure_o > sigma_measure_o ).

thf(sy_c_Measure__Space_Osup__measure_H_001t__Extended____Nonnegative____Real__Oennreal,type,
    measur4473656680840910822nnreal: sigma_7234349610311085201nnreal > sigma_7234349610311085201nnreal > sigma_7234349610311085201nnreal ).

thf(sy_c_Measure__Space_Osup__measure_H_001t__Nat__Onat,type,
    measur876423496291765374re_nat: sigma_measure_nat > sigma_measure_nat > sigma_measure_nat ).

thf(sy_c_Measure__Space_Osup__measure_H_001t__Real__Oreal,type,
    measur2147279183506585690e_real: sigma_measure_real > sigma_measure_real > sigma_measure_real ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001_Eo,type,
    nonneg5198678888045619090sure_o: set_o > sigma_measure_o ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001t__Extended____Nonnegative____Real__Oennreal,type,
    nonneg1394255657502361022nnreal: set_Ex3793607809372303086nnreal > sigma_7234349610311085201nnreal ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001t__Nat__Onat,type,
    nonneg7031465154080143958re_nat: set_nat > sigma_measure_nat ).

thf(sy_c_Nonnegative__Lebesgue__Integration_Ouniform__count__measure_001t__Real__Oreal,type,
    nonneg387815094551837234e_real: set_real > sigma_measure_real ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_I_Eo_Mt__Real__Oreal_J_M_Eo_J,type,
    bot_bot_o_real_o: ( $o > real ) > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
    bot_bot_nat_real_o: ( nat > real ) > $o ).

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thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__c_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__QuasiBorel__Oquasi____borel_Itf__a_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__c_J_J,type,
    ord_le5885474903713786379real_c: set_real_c > set_real_c > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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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__Real__Oreal_J,type,
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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__Extended____Nonnegative____Real__Oennreal_J_J,type,
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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_It__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Real__Oreal_M_Eo_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    top_to8895904057622651549l_real: set_Su3647026645378120685l_real ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001_Eo,type,
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thf(sy_c_QuasiBorel_Oempty__quasi__borel_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_QuasiBorel_Oempty__quasi__borel_001t__Nat__Onat,type,
    empty_8278123436611590770el_nat: quasi_borel_nat ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001t__Real__Oreal,type,
    empty_1876425439295802446l_real: quasi_borel_real ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001tf__a,type,
    empty_quasi_borel_a: quasi_borel_a ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001tf__c,type,
    empty_quasi_borel_c: quasi_borel_c ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_I_Eo_Mt__Real__Oreal_J,type,
    qbs_Mx_o_real: quasi_borel_o_real > set_real_o_real ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_Mx_nat_real: quasi_borel_nat_real > set_real_nat_real ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Real__Oreal_Mtf__a_J,type,
    qbs_Mx_real_a: quasi_borel_real_a > set_real_real_a ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_Itf__a_Mtf__b_J,type,
    qbs_Mx_a_b: quasi_borel_a_b > set_real_a_b ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_Itf__c_Mtf__b_J,type,
    qbs_Mx_c_b: quasi_borel_c_b > set_real_c_b ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_Eo,type,
    qbs_Mx_o: quasi_borel_o > set_real_o ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Extended____Nonnegative____Real__Oennreal,type,
    qbs_Mx6523938229262583809nnreal: quasi_9015997321629101608nnreal > set_re5328672808648366137nnreal ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Nat__Onat,type,
    qbs_Mx_nat: quasi_borel_nat > set_real_nat ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Real__Oreal,type,
    qbs_Mx_real: quasi_borel_real > set_real_real ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
    qbs_Mx_Sum_sum_a_c: quasi_4257511854121656471um_a_c > set_real_Sum_sum_a_c ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001tf__a,type,
    qbs_Mx_a: quasi_borel_a > set_real_a ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001tf__c,type,
    qbs_Mx_c: quasi_borel_c > set_real_c ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001_Eo,type,
    qbs_closed2_o: set_o > set_real_o > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001t__Extended____Nonnegative____Real__Oennreal,type,
    qbs_cl7346018279885218754nnreal: set_Ex3793607809372303086nnreal > set_re5328672808648366137nnreal > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001t__Nat__Onat,type,
    qbs_closed2_nat: set_nat > set_real_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001t__Real__Oreal,type,
    qbs_closed2_real: set_real > set_real_real > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
    qbs_cl8237352598534793137um_a_c: set_Sum_sum_a_c > set_real_Sum_sum_a_c > $o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__a_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    qbs_mo3511297643335615972o_real: quasi_borel_real_a > quasi_borel_o_real > set_real_a_o_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_mo6662696414006555820t_real: quasi_borel_real_a > quasi_borel_nat_real > set_real_a_nat_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    qbs_mo6715622035799359544real_a: quasi_borel_real_a > quasi_borel_real_a > set_real_a_real_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
    qbs_mo635784333843139191_a_a_b: quasi_borel_real_a > quasi_borel_a_b > set_real_a_a_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
    qbs_mo4284245206752562933_a_c_b: quasi_borel_real_a > quasi_borel_c_b > set_real_a_c_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_Eo_001t__Real__Oreal,type,
    qbs_morphism_o_real: quasi_borel_o > quasi_borel_real > set_o_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Nat__Onat_001t__Real__Oreal,type,
    qbs_mo2000642995705457910t_real: quasi_borel_nat > quasi_borel_real > set_nat_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001_Eo,type,
    qbs_morphism_real_o: quasi_borel_real > quasi_borel_o > set_real_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001t__Nat__Onat,type,
    qbs_mo6567951568834356598al_nat: quasi_borel_real > quasi_borel_nat > set_real_nat ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001t__Real__Oreal,type,
    qbs_mo5229770564518008146l_real: quasi_borel_real > quasi_borel_real > set_real_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Real__Oreal_001tf__a,type,
    qbs_morphism_real_a: quasi_borel_real > quasi_borel_a > set_real_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001_062_I_Eo_Mt__Real__Oreal_J,type,
    qbs_mo7370372776400040495o_real: quasi_borel_a > quasi_borel_o_real > set_a_o_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_mo5829272867083514145t_real: quasi_borel_a > quasi_borel_nat_real > set_a_nat_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001_062_It__Real__Oreal_Mtf__a_J,type,
    qbs_mo2545572719379674883real_a: quasi_borel_a > quasi_borel_real_a > set_a_real_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001_062_Itf__a_Mtf__b_J,type,
    qbs_morphism_a_a_b: quasi_borel_a > quasi_borel_a_b > set_a_a_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001_062_Itf__c_Mtf__b_J,type,
    qbs_morphism_a_c_b: quasi_borel_a > quasi_borel_c_b > set_a_c_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
    qbs_mo7250741323400969261um_a_c: quasi_borel_a > quasi_4257511854121656471um_a_c > set_a_Sum_sum_a_c ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001tf__b,type,
    qbs_morphism_a_b: quasi_borel_a > quasi_borel_b > set_a_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__c_001t__Sum____Type__Osum_Itf__a_Mtf__c_J,type,
    qbs_mo5084992033439934511um_a_c: quasi_borel_c > quasi_4257511854121656471um_a_c > set_c_Sum_sum_a_c ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__c_001tf__b,type,
    qbs_morphism_c_b: quasi_borel_c > quasi_borel_b > set_c_b ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_I_Eo_Mt__Real__Oreal_J,type,
    qbs_space_o_real: quasi_borel_o_real > set_o_real ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_space_nat_real: quasi_borel_nat_real > set_nat_real ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_It__Real__Oreal_Mtf__a_J,type,
    qbs_space_real_a: quasi_borel_real_a > set_real_a ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_Itf__a_Mtf__b_J,type,
    qbs_space_a_b: quasi_borel_a_b > set_a_b ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_Itf__c_Mtf__b_J,type,
    qbs_space_c_b: quasi_borel_c_b > set_c_b ).

thf(sy_c_QuasiBorel_Oqbs__space_001_Eo,type,
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thf(sy_c_QuasiBorel_Oqbs__space_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_QuasiBorel_Oqbs__space_001t__Nat__Onat,type,
    qbs_space_nat: quasi_borel_nat > set_nat ).

thf(sy_c_QuasiBorel_Oqbs__space_001t__Real__Oreal,type,
    qbs_space_real: quasi_borel_real > set_real ).

thf(sy_c_QuasiBorel_Oqbs__space_001tf__a,type,
    qbs_space_a: quasi_borel_a > set_a ).

thf(sy_c_QuasiBorel_Oqbs__space_001tf__b,type,
    qbs_space_b: quasi_borel_b > set_b ).

thf(sy_c_QuasiBorel_Oqbs__space_001tf__c,type,
    qbs_space_c: quasi_borel_c > set_c ).

thf(sy_c_Set_OBall_001_062_I_Eo_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OBall_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OBall_001_062_It__Real__Oreal_Mtf__a_J,type,
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thf(sy_c_Set_OBall_001_062_Itf__a_Mtf__b_J,type,
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thf(sy_c_Set_OBall_001_062_Itf__c_Mtf__b_J,type,
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thf(sy_c_Set_OCollect_001_062_I_Eo_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mtf__a_J,type,
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thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__b_J,type,
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thf(sy_c_Set_OCollect_001_062_Itf__c_Mtf__b_J,type,
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thf(sy_c_Set_OCollect_001_Eo,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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thf(sy_c_Set_OPow_001_062_I_Eo_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OPow_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OPow_001_062_It__Real__Oreal_Mtf__a_J,type,
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thf(sy_c_Set_OPow_001_062_Itf__a_Mtf__b_J,type,
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thf(sy_c_Set_OPow_001_062_Itf__c_Mtf__b_J,type,
    pow_c_b: set_c_b > set_set_c_b ).

thf(sy_c_Set_OPow_001_Eo,type,
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thf(sy_c_Set_OPow_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Set_OPow_001t__Nat__Onat,type,
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thf(sy_c_Set_OPow_001t__Real__Oreal,type,
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thf(sy_c_Set_Oinsert_001_062_I_Eo_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_Oinsert_001_062_It__Real__Oreal_Mtf__a_J,type,
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thf(sy_c_Set_Oinsert_001_062_Itf__a_Mtf__b_J,type,
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thf(sy_c_Set_Oinsert_001_062_Itf__c_Mtf__b_J,type,
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thf(sy_c_Set_Oinsert_001_Eo,type,
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thf(sy_c_Set_Oinsert_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Set_Oinsert_001tf__a,type,
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thf(sy_c_Set_Oinsert_001tf__c,type,
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thf(sy_c_Set_Ois__empty_001_Eo,type,
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thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
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thf(sy_c_Set_Ois__empty_001t__Real__Oreal,type,
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thf(sy_c_Set_Ois__singleton_001_062_I_Eo_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_Ois__singleton_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_Ois__singleton_001_062_Itf__a_Mtf__b_J,type,
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thf(sy_c_Set_Ois__singleton_001_062_Itf__c_Mtf__b_J,type,
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thf(sy_c_Set_Ois__singleton_001_Eo,type,
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thf(sy_c_Set_Ois__singleton_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
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thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
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thf(sy_c_Set_Oremove_001_062_I_Eo_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_Oremove_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_Oremove_001_062_It__Real__Oreal_Mtf__a_J,type,
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thf(sy_c_Set_Oremove_001_062_Itf__a_Mtf__b_J,type,
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thf(sy_c_Set_Oremove_001_062_Itf__c_Mtf__b_J,type,
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thf(sy_c_Set_Oremove_001_Eo,type,
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thf(sy_c_Set_Oremove_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
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thf(sy_c_Set_Oremove_001t__Real__Oreal,type,
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thf(sy_c_Set_Othe__elem_001_Eo,type,
    the_elem_o: set_o > $o ).

thf(sy_c_Set_Othe__elem_001t__Extended____Nonnegative____Real__Oennreal,type,
    the_el3795950934141317635nnreal: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: set_nat > nat ).

thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
    the_elem_real: set_real > real ).

thf(sy_c_Set_Ovimage_001t__Real__Oreal_001t__Nat__Onat,type,
    vimage_real_nat: ( real > nat ) > set_nat > set_real ).

thf(sy_c_Set_Ovimage_001t__Real__Oreal_001t__Real__Oreal,type,
    vimage_real_real: ( real > real ) > set_real > set_real ).

thf(sy_c_Set__Integral_Oset__integrable_001t__Real__Oreal_001t__Real__Oreal,type,
    set_se5970144800844511125l_real: sigma_measure_real > set_real > ( real > real ) > $o ).

thf(sy_c_Sigma__Algebra_Oalgebra_001_Eo,type,
    sigma_algebra_o: set_o > set_set_o > $o ).

thf(sy_c_Sigma__Algebra_Oalgebra_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_5981082695875523474nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > $o ).

thf(sy_c_Sigma__Algebra_Oalgebra_001t__Nat__Onat,type,
    sigma_algebra_nat: set_nat > set_set_nat > $o ).

thf(sy_c_Sigma__Algebra_Oalgebra_001t__Real__Oreal,type,
    sigma_algebra_real: set_real > set_set_real > $o ).

thf(sy_c_Sigma__Algebra_Oclosed__cdi_001_Eo,type,
    sigma_closed_cdi_o: set_o > set_set_o > $o ).

thf(sy_c_Sigma__Algebra_Oclosed__cdi_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_114563780369365222nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > $o ).

thf(sy_c_Sigma__Algebra_Oclosed__cdi_001t__Nat__Onat,type,
    sigma_closed_cdi_nat: set_nat > set_set_nat > $o ).

thf(sy_c_Sigma__Algebra_Oclosed__cdi_001t__Real__Oreal,type,
    sigma_227922725797042522i_real: set_real > set_set_real > $o ).

thf(sy_c_Sigma__Algebra_Ocount__space_001_Eo,type,
    sigma_count_space_o: set_o > sigma_measure_o ).

thf(sy_c_Sigma__Algebra_Ocount__space_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_7204664791115113951nnreal: set_Ex3793607809372303086nnreal > sigma_7234349610311085201nnreal ).

thf(sy_c_Sigma__Algebra_Ocount__space_001t__Nat__Onat,type,
    sigma_7685844798829912695ce_nat: set_nat > sigma_measure_nat ).

thf(sy_c_Sigma__Algebra_Ocount__space_001t__Real__Oreal,type,
    sigma_8508918144308765139e_real: set_real > sigma_measure_real ).

thf(sy_c_Sigma__Algebra_Ocount__space_001tf__a,type,
    sigma_count_space_a: set_a > sigma_measure_a ).

thf(sy_c_Sigma__Algebra_Ocount__space_001tf__c,type,
    sigma_count_space_c: set_c > sigma_measure_c ).

thf(sy_c_Sigma__Algebra_Oemeasure_001_062_I_Eo_Mt__Real__Oreal_J,type,
    sigma_4433523422001307788o_real: sigma_measure_o_real > set_o_real > extend8495563244428889912nnreal ).

thf(sy_c_Sigma__Algebra_Oemeasure_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    sigma_2433462726372594692t_real: sigma_3396294578489551860t_real > set_nat_real > extend8495563244428889912nnreal ).

thf(sy_c_Sigma__Algebra_Oemeasure_001_062_It__Real__Oreal_Mtf__a_J,type,
    sigma_6502373073922819808real_a: sigma_measure_real_a > set_real_a > extend8495563244428889912nnreal ).

thf(sy_c_Sigma__Algebra_Oemeasure_001_062_Itf__a_Mtf__b_J,type,
    sigma_emeasure_a_b: sigma_measure_a_b > set_a_b > extend8495563244428889912nnreal ).

thf(sy_c_Sigma__Algebra_Oemeasure_001_062_Itf__c_Mtf__b_J,type,
    sigma_emeasure_c_b: sigma_measure_c_b > set_c_b > extend8495563244428889912nnreal ).

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__Extended____Nonnegative____Real__Oennreal,type,
    sigma_6589832970846575905nnreal: sigma_7234349610311085201nnreal > set_Ex3793607809372303086nnreal > 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_001t__Real__Oreal,type,
    sigma_emeasure_real: sigma_measure_real > set_real > extend8495563244428889912nnreal ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001_Eo,type,
    sigma_measurable_o_o: sigma_measure_o > sigma_measure_o > set_o_o ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_6459699357617223168nnreal: sigma_measure_o > sigma_7234349610311085201nnreal > set_o_3823858056404284705nnreal ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Nat__Onat,type,
    sigma_1999164137574644376_o_nat: sigma_measure_o > sigma_measure_nat > set_o_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_Eo_001t__Real__Oreal,type,
    sigma_2430008634441611636o_real: sigma_measure_o > sigma_measure_real > set_o_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Extended____Nonnegative____Real__Oennreal_001_Eo,type,
    sigma_6279906219187228174real_o: sigma_7234349610311085201nnreal > sigma_measure_o > set_Ex70502500924464887real_o ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_7926153774531450434nnreal: sigma_7234349610311085201nnreal > sigma_7234349610311085201nnreal > set_Ex7514979451064110021nnreal ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Extended____Nonnegative____Real__Oennreal_001t__Nat__Onat,type,
    sigma_1856489715609627354al_nat: sigma_7234349610311085201nnreal > sigma_measure_nat > set_Ex1875306066257539165al_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    sigma_7049758200512112822l_real: sigma_7234349610311085201nnreal > sigma_measure_real > set_Ex5658717452565810105l_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001_Eo,type,
    sigma_5101835498682829686_nat_o: sigma_measure_nat > sigma_measure_o > set_nat_o ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_6306161311797543642nnreal: sigma_measure_nat > sigma_7234349610311085201nnreal > set_na7716847989478749277nnreal ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001t__Nat__Onat,type,
    sigma_4350458207664084850at_nat: sigma_measure_nat > sigma_measure_nat > set_nat_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001t__Real__Oreal,type,
    sigma_1747752005702207822t_real: sigma_measure_nat > sigma_measure_real > set_nat_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001_Eo,type,
    sigma_3939073009482781210real_o: sigma_measure_real > sigma_measure_o > set_real_o ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_9017504469962657078nnreal: sigma_measure_real > sigma_7234349610311085201nnreal > set_re5328672808648366137nnreal ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Nat__Onat,type,
    sigma_6315060578831106510al_nat: sigma_measure_real > sigma_measure_nat > set_real_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Real__Oreal,type,
    sigma_5267869275261027754l_real: sigma_measure_real > sigma_measure_real > set_real_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001tf__a,type,
    sigma_523072396149930112real_a: sigma_measure_real > sigma_measure_a > set_real_a ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001tf__c,type,
    sigma_523072396149930114real_c: sigma_measure_real > sigma_measure_c > set_real_c ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__a_001tf__b,type,
    sigma_measurable_a_b: sigma_measure_a > sigma_measure_b > set_a_b ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__c_001tf__b,type,
    sigma_measurable_c_b: sigma_measure_c > sigma_measure_b > set_c_b ).

thf(sy_c_Sigma__Algebra_Omeasure__of_001_Eo,type,
    sigma_measure_of_o: set_o > set_set_o > ( set_o > extend8495563244428889912nnreal ) > sigma_measure_o ).

thf(sy_c_Sigma__Algebra_Omeasure__of_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_8167827323036178527nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > ( set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal ) > sigma_7234349610311085201nnreal ).

thf(sy_c_Sigma__Algebra_Omeasure__of_001t__Nat__Onat,type,
    sigma_measure_of_nat: set_nat > set_set_nat > ( set_nat > extend8495563244428889912nnreal ) > sigma_measure_nat ).

thf(sy_c_Sigma__Algebra_Omeasure__of_001t__Real__Oreal,type,
    sigma_2693083824694760531f_real: set_real > set_set_real > ( set_real > extend8495563244428889912nnreal ) > sigma_measure_real ).

thf(sy_c_Sigma__Algebra_Osets_001_062_I_Eo_Mt__Real__Oreal_J,type,
    sigma_sets_o_real: sigma_measure_o_real > set_set_o_real ).

thf(sy_c_Sigma__Algebra_Osets_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    sigma_sets_nat_real: sigma_3396294578489551860t_real > set_set_nat_real ).

thf(sy_c_Sigma__Algebra_Osets_001_062_It__Real__Oreal_Mtf__a_J,type,
    sigma_sets_real_a: sigma_measure_real_a > set_set_real_a ).

thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__a_Mtf__b_J,type,
    sigma_sets_a_b: sigma_measure_a_b > set_set_a_b ).

thf(sy_c_Sigma__Algebra_Osets_001_062_Itf__c_Mtf__b_J,type,
    sigma_sets_c_b: sigma_measure_c_b > set_set_c_b ).

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__Extended____Nonnegative____Real__Oennreal,type,
    sigma_5465916536984168985nnreal: sigma_7234349610311085201nnreal > set_se4580700918925141924nnreal ).

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_001t__Real__Oreal,type,
    sigma_sets_real: sigma_measure_real > set_set_real ).

thf(sy_c_Sigma__Algebra_Osets_001tf__a,type,
    sigma_sets_a: sigma_measure_a > set_set_a ).

thf(sy_c_Sigma__Algebra_Osets_001tf__c,type,
    sigma_sets_c: sigma_measure_c > set_set_c ).

thf(sy_c_Sigma__Algebra_Osigma__algebra_001_062_I_Eo_Mt__Real__Oreal_J,type,
    sigma_1431946479552111010o_real: set_o_real > set_set_o_real > $o ).

thf(sy_c_Sigma__Algebra_Osigma__algebra_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    sigma_5696473160036025454t_real: set_nat_real > set_set_nat_real > $o ).

thf(sy_c_Sigma__Algebra_Osigma__algebra_001_062_It__Real__Oreal_Mtf__a_J,type,
    sigma_6829682388519410934real_a: set_real_a > set_set_real_a > $o ).

thf(sy_c_Sigma__Algebra_Osigma__algebra_001_062_Itf__a_Mtf__b_J,type,
    sigma_17020134428762681ra_a_b: set_a_b > set_set_a_b > $o ).

thf(sy_c_Sigma__Algebra_Osigma__algebra_001_062_Itf__c_Mtf__b_J,type,
    sigma_3665481007338186423ra_c_b: set_c_b > set_set_c_b > $o ).

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_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_2413694886200424843nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > $o ).

thf(sy_c_Sigma__Algebra_Osigma__algebra_001t__Nat__Onat,type,
    sigma_8817008012692346403ra_nat: set_nat > set_set_nat > $o ).

thf(sy_c_Sigma__Algebra_Osigma__algebra_001t__Real__Oreal,type,
    sigma_1481383337440427903a_real: set_real > set_set_real > $o ).

thf(sy_c_Sigma__Algebra_Osigma__sets_001_Eo,type,
    sigma_sigma_sets_o: set_o > set_set_o > set_set_o ).

thf(sy_c_Sigma__Algebra_Osigma__sets_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_7808855514367478112nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > set_se4580700918925141924nnreal ).

thf(sy_c_Sigma__Algebra_Osigma__sets_001t__Nat__Onat,type,
    sigma_sigma_sets_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Sigma__Algebra_Osigma__sets_001t__Real__Oreal,type,
    sigma_7195353284648819924s_real: set_real > set_set_real > set_set_real ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_I_Eo_Mt__Real__Oreal_J,type,
    sigma_space_o_real: sigma_measure_o_real > set_o_real ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    sigma_space_nat_real: sigma_3396294578489551860t_real > set_nat_real ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_It__Real__Oreal_Mtf__a_J,type,
    sigma_space_real_a: sigma_measure_real_a > set_real_a ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__a_Mtf__b_J,type,
    sigma_space_a_b: sigma_measure_a_b > set_a_b ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_Itf__c_Mtf__b_J,type,
    sigma_space_c_b: sigma_measure_c_b > set_c_b ).

thf(sy_c_Sigma__Algebra_Ospace_001_Eo,type,
    sigma_space_o: sigma_measure_o > set_o ).

thf(sy_c_Sigma__Algebra_Ospace_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_3147302497200244656nnreal: sigma_7234349610311085201nnreal > set_Ex3793607809372303086nnreal ).

thf(sy_c_Sigma__Algebra_Ospace_001t__Nat__Onat,type,
    sigma_space_nat: sigma_measure_nat > set_nat ).

thf(sy_c_Sigma__Algebra_Ospace_001t__Real__Oreal,type,
    sigma_space_real: sigma_measure_real > set_real ).

thf(sy_c_Sigma__Algebra_Ospace_001tf__a,type,
    sigma_space_a: sigma_measure_a > set_a ).

thf(sy_c_Sigma__Algebra_Ospace_001tf__b,type,
    sigma_space_b: sigma_measure_b > set_b ).

thf(sy_c_Sigma__Algebra_Ospace_001tf__c,type,
    sigma_space_c: sigma_measure_c > set_c ).

thf(sy_c_StandardBorel_Obiexp01__well__formed,type,
    biexp01_well_formed: ( nat > nat ) > $o ).

thf(sy_c_StandardBorel_Opair__standard__borel_001t__Nat__Onat_001t__Real__Oreal,type,
    pair_s8264832550775477520t_real: sigma_measure_nat > sigma_measure_real > $o ).

thf(sy_c_StandardBorel_Opair__standard__borel__space__UNIV_001t__Nat__Onat_001t__Real__Oreal,type,
    pair_s5107880421860391064t_real: sigma_measure_nat > sigma_measure_real > $o ).

thf(sy_c_StandardBorel_Or01__binary__expansion_H,type,
    r01_binary_expansion: real > nat > nat ).

thf(sy_c_StandardBorel_Or01__r01__to__r01_H,type,
    r01_r01_to_r01: produc2422161461964618553l_real > nat > nat ).

thf(sy_c_StandardBorel_Or01__to__r01__r01__fst,type,
    r01_to_r01_r01_fst: real > real ).

thf(sy_c_StandardBorel_Or01__to__r01__r01__fst_H,type,
    r01_to_r01_r01_fst2: real > nat > nat ).

thf(sy_c_StandardBorel_Or01__to__r01__r01__snd,type,
    r01_to_r01_r01_snd: real > real ).

thf(sy_c_StandardBorel_Or01__to__r01__r01__snd_H,type,
    r01_to_r01_r01_snd2: real > nat > nat ).

thf(sy_c_StandardBorel_Ostandard__borel_001t__Real__Oreal,type,
    standard_borel_real: sigma_measure_real > $o ).

thf(sy_c_StandardBorel_Ostandard__borel_Of_001_Eo,type,
    standard_f_o: sigma_measure_o > $o > real ).

thf(sy_c_StandardBorel_Ostandard__borel_Of_001t__Extended____Nonnegative____Real__Oennreal,type,
    standa4501783974915749827nnreal: sigma_7234349610311085201nnreal > extend8495563244428889912nnreal > real ).

thf(sy_c_StandardBorel_Ostandard__borel_Of_001t__Nat__Onat,type,
    standard_f_nat: sigma_measure_nat > nat > real ).

thf(sy_c_StandardBorel_Ostandard__borel_Of_001t__Real__Oreal,type,
    standard_f_real: sigma_measure_real > real > real ).

thf(sy_c_StandardBorel_Ostandard__borel_Og_001_Eo,type,
    standard_g_o: sigma_measure_o > real > $o ).

thf(sy_c_StandardBorel_Ostandard__borel_Og_001t__Extended____Nonnegative____Real__Oennreal,type,
    standa1398259892199664580nnreal: sigma_7234349610311085201nnreal > real > extend8495563244428889912nnreal ).

thf(sy_c_StandardBorel_Ostandard__borel_Og_001t__Nat__Onat,type,
    standard_g_nat: sigma_measure_nat > real > nat ).

thf(sy_c_StandardBorel_Ostandard__borel_Og_001t__Real__Oreal,type,
    standard_g_real: sigma_measure_real > real > real ).

thf(sy_c_StandardBorel_Ostandard__borel__space__UNIV_001t__Real__Oreal,type,
    standa1306199911732814765V_real: sigma_measure_real > $o ).

thf(sy_c_StandardBorel_Ostandard__borel__space__UNIV__axioms_001_Eo,type,
    standa4575222554423029108ioms_o: sigma_measure_o > $o ).

thf(sy_c_StandardBorel_Ostandard__borel__space__UNIV__axioms_001t__Extended____Nonnegative____Real__Oennreal,type,
    standa602082540683807836nnreal: sigma_7234349610311085201nnreal > $o ).

thf(sy_c_StandardBorel_Ostandard__borel__space__UNIV__axioms_001t__Nat__Onat,type,
    standa4898135366436483316ms_nat: sigma_measure_nat > $o ).

thf(sy_c_StandardBorel_Ostandard__borel__space__UNIV__axioms_001t__Real__Oreal,type,
    standa1498722272452280784s_real: sigma_measure_real > $o ).

thf(sy_c_Sum__Type_OInl_001_062_I_Eo_Mt__Real__Oreal_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    sum_In8123802879950721198o_real: ( $o > real ) > sum_su815935806896055909o_real ).

thf(sy_c_Sum__Type_OInl_001_062_I_Eo_Mt__Real__Oreal_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    sum_In4218321019262890082t_real: ( $o > real ) > sum_su7765403328973102337t_real ).

thf(sy_c_Sum__Type_OInl_001_062_I_Eo_Mt__Real__Oreal_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    sum_In1930673884277835266real_a: ( $o > real ) > sum_su7886454506223791033real_a ).

thf(sy_c_Sum__Type_OInl_001_062_I_Eo_Mt__Real__Oreal_J_001_062_Itf__c_Mtf__b_J,type,
    sum_Inl_o_real_c_b: ( $o > real ) > sum_sum_o_real_c_b ).

thf(sy_c_Sum__Type_OInl_001_062_It__Real__Oreal_Mtf__a_J_001_062_I_Eo_Mt__Real__Oreal_J,type,
    sum_In9033506258822570586o_real: ( real > a ) > sum_su2067798924538045457o_real ).

thf(sy_c_Sum__Type_OInl_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    sum_In5343589694070989238t_real: ( real > a ) > sum_su5472343219575513685t_real ).

thf(sy_c_Sum__Type_OInl_001_062_It__Real__Oreal_Mtf__a_J_001_062_It__Real__Oreal_Mtf__a_J,type,
    sum_In8182254581923447214real_a: ( real > a ) > sum_su2571395965866611557real_a ).

thf(sy_c_Sum__Type_OInl_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
    sum_Inl_real_a_a_b: ( real > a ) > sum_sum_real_a_a_b ).

thf(sy_c_Sum__Type_OInl_001_062_It__Real__Oreal_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
    sum_Inl_real_a_c_b: ( real > a ) > sum_sum_real_a_c_b ).

thf(sy_c_Sum__Type_OInl_001tf__a_001tf__c,type,
    sum_Inl_a_c: a > sum_sum_a_c ).

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thf(sy_v_Z,type,
    z: quasi_borel_b ).

thf(sy_v__092_060alpha_062____,type,
    alpha: real > sum_sum_a_c ).

thf(sy_v_f,type,
    f: a > b ).

thf(sy_v_g,type,
    g: c > b ).

thf(sy_v_thesis____,type,
    thesis: $o ).

% Relevant facts (1277)
thf(fact_0__092_060open_062_092_060alpha_062_A_092_060in_062_Acopair__qbs__Mx_AX_AY_092_060close_062,axiom,
    member2264291325230826761um_a_c @ alpha @ ( binary8286901584692334522Mx_a_c @ x @ y ) ).

% \<open>\<alpha> \<in> copair_qbs_Mx X Y\<close>
thf(fact_1_hs,axiom,
    ( ( member_set_real @ s @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
    & ( ( s = bot_bot_set_real )
     => ? [X: real > a] :
          ( ( member_real_a @ X @ ( qbs_Mx_a @ x ) )
          & ( alpha
            = ( ^ [R: real] : ( sum_Inl_a_c @ ( X @ R ) ) ) ) ) )
    & ( ( s = top_top_set_real )
     => ? [X: real > c] :
          ( ( member_real_c @ X @ ( qbs_Mx_c @ y ) )
          & ( alpha
            = ( ^ [R: real] : ( sum_Inr_c_a @ ( X @ R ) ) ) ) ) )
    & ( ( ( s != bot_bot_set_real )
        & ( s != top_top_set_real ) )
     => ? [X: real > a] :
          ( ( member_real_a @ X @ ( qbs_Mx_a @ x ) )
          & ? [Xa: real > c] :
              ( ( member_real_c @ Xa @ ( qbs_Mx_c @ y ) )
              & ( alpha
                = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ s ) @ ( sum_Inl_a_c @ ( X @ R ) ) @ ( sum_Inr_c_a @ ( Xa @ R ) ) ) ) ) ) ) ) ) ).

% hs
thf(fact_2_assms_I1_J,axiom,
    member_a_b @ f @ ( qbs_morphism_a_b @ x @ z ) ).

% assms(1)
thf(fact_3_sum_Oinject_I1_J,axiom,
    ! [X1: a,Y1: a] :
      ( ( ( sum_Inl_a_c @ X1 )
        = ( sum_Inl_a_c @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% sum.inject(1)
thf(fact_4_old_Osum_Oinject_I1_J,axiom,
    ! [A: a,A2: a] :
      ( ( ( sum_Inl_a_c @ A )
        = ( sum_Inl_a_c @ A2 ) )
      = ( A = A2 ) ) ).

% old.sum.inject(1)
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062S_O_AS_A_092_060in_062_Asets_Areal__borel_A_092_060and_062_A_IS_A_061_A_123_125_A_092_060longrightarrow_062_A_I_092_060exists_062_092_060alpha_0621_092_060in_062qbs__Mx_AX_O_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_AInl_A_I_092_060alpha_0621_Ar_J_J_J_J_A_092_060and_062_A_IS_A_061_AUNIV_A_092_060longrightarrow_062_A_I_092_060exists_062_092_060alpha_0622_092_060in_062qbs__Mx_AY_O_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_AInr_A_I_092_060alpha_0622_Ar_J_J_J_J_A_092_060and_062_A_IS_A_092_060noteq_062_A_123_125_A_092_060and_062_AS_A_092_060noteq_062_AUNIV_A_092_060longrightarrow_062_A_I_092_060exists_062_092_060alpha_0621_092_060in_062qbs__Mx_AX_O_A_092_060exists_062_092_060alpha_0622_092_060in_062qbs__Mx_AY_O_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_Aif_Ar_A_092_060in_062_AS_Athen_AInl_A_I_092_060alpha_0621_Ar_J_Aelse_AInr_A_I_092_060alpha_0622_Ar_J_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [S: set_real] :
        ~ ( ( member_set_real @ S @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
          & ( ( S = bot_bot_set_real )
           => ? [X: real > a] :
                ( ( member_real_a @ X @ ( qbs_Mx_a @ x ) )
                & ( alpha
                  = ( ^ [R: real] : ( sum_Inl_a_c @ ( X @ R ) ) ) ) ) )
          & ( ( S = top_top_set_real )
           => ? [X: real > c] :
                ( ( member_real_c @ X @ ( qbs_Mx_c @ y ) )
                & ( alpha
                  = ( ^ [R: real] : ( sum_Inr_c_a @ ( X @ R ) ) ) ) ) )
          & ( ( ( S != bot_bot_set_real )
              & ( S != top_top_set_real ) )
           => ? [X: real > a] :
                ( ( member_real_a @ X @ ( qbs_Mx_a @ x ) )
                & ? [Xa: real > c] :
                    ( ( member_real_c @ Xa @ ( qbs_Mx_c @ y ) )
                    & ( alpha
                      = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ S ) @ ( sum_Inl_a_c @ ( X @ R ) ) @ ( sum_Inr_c_a @ ( Xa @ R ) ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>S. S \<in> sets real_borel \<and> (S = {} \<longrightarrow> (\<exists>\<alpha>1\<in>qbs_Mx X. \<alpha> = (\<lambda>r. Inl (\<alpha>1 r)))) \<and> (S = UNIV \<longrightarrow> (\<exists>\<alpha>2\<in>qbs_Mx Y. \<alpha> = (\<lambda>r. Inr (\<alpha>2 r)))) \<and> (S \<noteq> {} \<and> S \<noteq> UNIV \<longrightarrow> (\<exists>\<alpha>1\<in>qbs_Mx X. \<exists>\<alpha>2\<in>qbs_Mx Y. \<alpha> = (\<lambda>r. if r \<in> S then Inl (\<alpha>1 r) else Inr (\<alpha>2 r)))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062S_A_061_A_123_125_A_092_060Longrightarrow_062_Athesis_059_AS_A_061_AUNIV_A_092_060Longrightarrow_062_Athesis_059_AS_A_092_060noteq_062_A_123_125_A_092_060and_062_AS_A_092_060noteq_062_AUNIV_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ( ( s != bot_bot_set_real )
   => ( ( s != top_top_set_real )
     => ( ( s != bot_bot_set_real )
        & ( s != top_top_set_real ) ) ) ) ).

% \<open>\<And>thesis. \<lbrakk>S = {} \<Longrightarrow> thesis; S = UNIV \<Longrightarrow> thesis; S \<noteq> {} \<and> S \<noteq> UNIV \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_7_qbs__eqI,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X2 )
        = ( qbs_Mx_a @ Y ) )
     => ( X2 = Y ) ) ).

% qbs_eqI
thf(fact_8_qbs__eqI,axiom,
    ! [X2: quasi_borel_c,Y: quasi_borel_c] :
      ( ( ( qbs_Mx_c @ X2 )
        = ( qbs_Mx_c @ Y ) )
     => ( X2 = Y ) ) ).

% qbs_eqI
thf(fact_9_Inl__inject,axiom,
    ! [X3: a,Y2: a] :
      ( ( ( sum_Inl_a_c @ X3 )
        = ( sum_Inl_a_c @ Y2 ) )
     => ( X3 = Y2 ) ) ).

% Inl_inject
thf(fact_10_Inl__Inr__False,axiom,
    ! [X3: a,Y2: c] :
      ( ( sum_Inl_a_c @ X3 )
     != ( sum_Inr_c_a @ Y2 ) ) ).

% Inl_Inr_False
thf(fact_11_Inr__Inl__False,axiom,
    ! [X3: c,Y2: a] :
      ( ( sum_Inr_c_a @ X3 )
     != ( sum_Inl_a_c @ Y2 ) ) ).

% Inr_Inl_False
thf(fact_12_sum_Odistinct_I1_J,axiom,
    ! [X1: a,X22: c] :
      ( ( sum_Inl_a_c @ X1 )
     != ( sum_Inr_c_a @ X22 ) ) ).

% sum.distinct(1)
thf(fact_13__092_060open_062S_A_061_A_123_125_092_060close_062,axiom,
    s = bot_bot_set_real ).

% \<open>S = {}\<close>
thf(fact_14_sum_Oinject_I2_J,axiom,
    ! [X22: c,Y22: c] :
      ( ( ( sum_Inr_c_a @ X22 )
        = ( sum_Inr_c_a @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% sum.inject(2)
thf(fact_15_old_Osum_Oinject_I2_J,axiom,
    ! [B: c,B2: c] :
      ( ( ( sum_Inr_c_a @ B )
        = ( sum_Inr_c_a @ B2 ) )
      = ( B = B2 ) ) ).

% old.sum.inject(2)
thf(fact_16_assms_I2_J,axiom,
    member_c_b @ g @ ( qbs_morphism_c_b @ y @ z ) ).

% assms(2)
thf(fact_17_Inr__inject,axiom,
    ! [X3: c,Y2: c] :
      ( ( ( sum_Inr_c_a @ X3 )
        = ( sum_Inr_c_a @ Y2 ) )
     => ( X3 = Y2 ) ) ).

% Inr_inject
thf(fact_18_split__sum__all,axiom,
    ( ( ^ [P: sum_sum_a_c > $o] :
        ! [X4: sum_sum_a_c] : ( P @ X4 ) )
    = ( ^ [P2: sum_sum_a_c > $o] :
          ( ! [X5: a] : ( P2 @ ( sum_Inl_a_c @ X5 ) )
          & ! [X5: c] : ( P2 @ ( sum_Inr_c_a @ X5 ) ) ) ) ) ).

% split_sum_all
thf(fact_19_split__sum__ex,axiom,
    ( ( ^ [P: sum_sum_a_c > $o] :
        ? [X4: sum_sum_a_c] : ( P @ X4 ) )
    = ( ^ [P2: sum_sum_a_c > $o] :
          ( ? [X5: a] : ( P2 @ ( sum_Inl_a_c @ X5 ) )
          | ? [X5: c] : ( P2 @ ( sum_Inr_c_a @ X5 ) ) ) ) ) ).

% split_sum_ex
thf(fact_20_Inr__not__Inl,axiom,
    ! [B: c,A: a] :
      ( ( sum_Inr_c_a @ B )
     != ( sum_Inl_a_c @ A ) ) ).

% Inr_not_Inl
thf(fact_21_sumE,axiom,
    ! [S2: sum_sum_a_c] :
      ( ! [X: a] :
          ( S2
         != ( sum_Inl_a_c @ X ) )
     => ~ ! [Y3: c] :
            ( S2
           != ( sum_Inr_c_a @ Y3 ) ) ) ).

% sumE
thf(fact_22_old_Osum_Oexhaust,axiom,
    ! [Y2: sum_sum_a_c] :
      ( ! [A3: a] :
          ( Y2
         != ( sum_Inl_a_c @ A3 ) )
     => ~ ! [B3: c] :
            ( Y2
           != ( sum_Inr_c_a @ B3 ) ) ) ).

% old.sum.exhaust
thf(fact_23_old_Osum_Odistinct_I1_J,axiom,
    ! [A: a,B2: c] :
      ( ( sum_Inl_a_c @ A )
     != ( sum_Inr_c_a @ B2 ) ) ).

% old.sum.distinct(1)
thf(fact_24_old_Osum_Odistinct_I2_J,axiom,
    ! [B2: c,A: a] :
      ( ( sum_Inr_c_a @ B2 )
     != ( sum_Inl_a_c @ A ) ) ).

% old.sum.distinct(2)
thf(fact_25_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_real] : ( member_set_real @ bot_bot_set_real @ ( sigma_sets_real @ M ) ) ).

% sets.empty_sets
thf(fact_26_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_nat] : ( member_set_nat @ bot_bot_set_nat @ ( sigma_sets_nat @ M ) ) ).

% sets.empty_sets
thf(fact_27_sets_Oempty__sets,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.empty_sets
thf(fact_28_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_o] : ( member_set_o @ bot_bot_set_o @ ( sigma_sets_o @ M ) ) ).

% sets.empty_sets
thf(fact_29_space__in__borel,axiom,
    member_set_real @ top_top_set_real @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).

% space_in_borel
thf(fact_30_space__in__borel,axiom,
    member603777416030116741nnreal @ top_to7994903218803871134nnreal @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ).

% space_in_borel
thf(fact_31_space__in__borel,axiom,
    member_set_o @ top_top_set_o @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ).

% space_in_borel
thf(fact_32_space__in__borel,axiom,
    member_set_nat @ top_top_set_nat @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ).

% space_in_borel
thf(fact_33_empty__iff,axiom,
    ! [C: real > a] :
      ~ ( member_real_a @ C @ bot_bot_set_real_a ) ).

% empty_iff
thf(fact_34_empty__iff,axiom,
    ! [C: $o > real] :
      ~ ( member_o_real @ C @ bot_bot_set_o_real ) ).

% empty_iff
thf(fact_35_empty__iff,axiom,
    ! [C: nat > real] :
      ~ ( member_nat_real @ C @ bot_bot_set_nat_real ) ).

% empty_iff
thf(fact_36_empty__iff,axiom,
    ! [C: c > b] :
      ~ ( member_c_b @ C @ bot_bot_set_c_b ) ).

% empty_iff
thf(fact_37_empty__iff,axiom,
    ! [C: a > b] :
      ~ ( member_a_b @ C @ bot_bot_set_a_b ) ).

% empty_iff
thf(fact_38_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_39_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_40_empty__iff,axiom,
    ! [C: extend8495563244428889912nnreal] :
      ~ ( member7908768830364227535nnreal @ C @ bot_bo4854962954004695426nnreal ) ).

% empty_iff
thf(fact_41_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_42_all__not__in__conv,axiom,
    ! [A4: set_real_a] :
      ( ( ! [X5: real > a] :
            ~ ( member_real_a @ X5 @ A4 ) )
      = ( A4 = bot_bot_set_real_a ) ) ).

% all_not_in_conv
thf(fact_43_all__not__in__conv,axiom,
    ! [A4: set_o_real] :
      ( ( ! [X5: $o > real] :
            ~ ( member_o_real @ X5 @ A4 ) )
      = ( A4 = bot_bot_set_o_real ) ) ).

% all_not_in_conv
thf(fact_44_all__not__in__conv,axiom,
    ! [A4: set_nat_real] :
      ( ( ! [X5: nat > real] :
            ~ ( member_nat_real @ X5 @ A4 ) )
      = ( A4 = bot_bot_set_nat_real ) ) ).

% all_not_in_conv
thf(fact_45_all__not__in__conv,axiom,
    ! [A4: set_c_b] :
      ( ( ! [X5: c > b] :
            ~ ( member_c_b @ X5 @ A4 ) )
      = ( A4 = bot_bot_set_c_b ) ) ).

% all_not_in_conv
thf(fact_46_all__not__in__conv,axiom,
    ! [A4: set_a_b] :
      ( ( ! [X5: a > b] :
            ~ ( member_a_b @ X5 @ A4 ) )
      = ( A4 = bot_bot_set_a_b ) ) ).

% all_not_in_conv
thf(fact_47_all__not__in__conv,axiom,
    ! [A4: set_real] :
      ( ( ! [X5: real] :
            ~ ( member_real @ X5 @ A4 ) )
      = ( A4 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_48_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ! [X5: nat] :
            ~ ( member_nat @ X5 @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_49_all__not__in__conv,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( ! [X5: extend8495563244428889912nnreal] :
            ~ ( member7908768830364227535nnreal @ X5 @ A4 ) )
      = ( A4 = bot_bo4854962954004695426nnreal ) ) ).

% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
    ! [A4: set_o] :
      ( ( ! [X5: $o] :
            ~ ( member_o @ X5 @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_51_Collect__empty__eq,axiom,
    ! [P3: real > $o] :
      ( ( ( collect_real @ P3 )
        = bot_bot_set_real )
      = ( ! [X5: real] :
            ~ ( P3 @ X5 ) ) ) ).

% Collect_empty_eq
thf(fact_52_Collect__empty__eq,axiom,
    ! [P3: nat > $o] :
      ( ( ( collect_nat @ P3 )
        = bot_bot_set_nat )
      = ( ! [X5: nat] :
            ~ ( P3 @ X5 ) ) ) ).

% Collect_empty_eq
thf(fact_53_Collect__empty__eq,axiom,
    ! [P3: extend8495563244428889912nnreal > $o] :
      ( ( ( collec6648975593938027277nnreal @ P3 )
        = bot_bo4854962954004695426nnreal )
      = ( ! [X5: extend8495563244428889912nnreal] :
            ~ ( P3 @ X5 ) ) ) ).

% Collect_empty_eq
thf(fact_54_Collect__empty__eq,axiom,
    ! [P3: $o > $o] :
      ( ( ( collect_o @ P3 )
        = bot_bot_set_o )
      = ( ! [X5: $o] :
            ~ ( P3 @ X5 ) ) ) ).

% Collect_empty_eq
thf(fact_55_empty__Collect__eq,axiom,
    ! [P3: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P3 ) )
      = ( ! [X5: real] :
            ~ ( P3 @ X5 ) ) ) ).

% empty_Collect_eq
thf(fact_56_empty__Collect__eq,axiom,
    ! [P3: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P3 ) )
      = ( ! [X5: nat] :
            ~ ( P3 @ X5 ) ) ) ).

% empty_Collect_eq
thf(fact_57_empty__Collect__eq,axiom,
    ! [P3: extend8495563244428889912nnreal > $o] :
      ( ( bot_bo4854962954004695426nnreal
        = ( collec6648975593938027277nnreal @ P3 ) )
      = ( ! [X5: extend8495563244428889912nnreal] :
            ~ ( P3 @ X5 ) ) ) ).

% empty_Collect_eq
thf(fact_58_empty__Collect__eq,axiom,
    ! [P3: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P3 ) )
      = ( ! [X5: $o] :
            ~ ( P3 @ X5 ) ) ) ).

% empty_Collect_eq
thf(fact_59_UNIV__I,axiom,
    ! [X3: real > a] : ( member_real_a @ X3 @ top_top_set_real_a ) ).

% UNIV_I
thf(fact_60_UNIV__I,axiom,
    ! [X3: $o > real] : ( member_o_real @ X3 @ top_top_set_o_real ) ).

% UNIV_I
thf(fact_61_UNIV__I,axiom,
    ! [X3: nat > real] : ( member_nat_real @ X3 @ top_top_set_nat_real ) ).

% UNIV_I
thf(fact_62_UNIV__I,axiom,
    ! [X3: c > b] : ( member_c_b @ X3 @ top_top_set_c_b ) ).

% UNIV_I
thf(fact_63_UNIV__I,axiom,
    ! [X3: a > b] : ( member_a_b @ X3 @ top_top_set_a_b ) ).

% UNIV_I
thf(fact_64_UNIV__I,axiom,
    ! [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).

% UNIV_I
thf(fact_65_UNIV__I,axiom,
    ! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).

% UNIV_I
thf(fact_66_iso__tuple__UNIV__I,axiom,
    ! [X3: real > a] : ( member_real_a @ X3 @ top_top_set_real_a ) ).

% iso_tuple_UNIV_I
thf(fact_67_iso__tuple__UNIV__I,axiom,
    ! [X3: $o > real] : ( member_o_real @ X3 @ top_top_set_o_real ) ).

% iso_tuple_UNIV_I
thf(fact_68_iso__tuple__UNIV__I,axiom,
    ! [X3: nat > real] : ( member_nat_real @ X3 @ top_top_set_nat_real ) ).

% iso_tuple_UNIV_I
thf(fact_69_iso__tuple__UNIV__I,axiom,
    ! [X3: c > b] : ( member_c_b @ X3 @ top_top_set_c_b ) ).

% iso_tuple_UNIV_I
thf(fact_70_iso__tuple__UNIV__I,axiom,
    ! [X3: a > b] : ( member_a_b @ X3 @ top_top_set_a_b ) ).

% iso_tuple_UNIV_I
thf(fact_71_iso__tuple__UNIV__I,axiom,
    ! [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).

% iso_tuple_UNIV_I
thf(fact_72_iso__tuple__UNIV__I,axiom,
    ! [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).

% iso_tuple_UNIV_I
thf(fact_73_obj__sumE,axiom,
    ! [S2: sum_sum_a_c] :
      ( ! [X: a] :
          ( S2
         != ( sum_Inl_a_c @ X ) )
     => ~ ! [X: c] :
            ( S2
           != ( sum_Inr_c_a @ X ) ) ) ).

% obj_sumE
thf(fact_74_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_75_empty__not__UNIV,axiom,
    bot_bo4854962954004695426nnreal != top_to7994903218803871134nnreal ).

% empty_not_UNIV
thf(fact_76_empty__not__UNIV,axiom,
    bot_bot_set_real != top_top_set_real ).

% empty_not_UNIV
thf(fact_77_empty__not__UNIV,axiom,
    bot_bot_set_o != top_top_set_o ).

% empty_not_UNIV
thf(fact_78_top__set__def,axiom,
    ( top_top_set_real
    = ( collect_real @ top_top_real_o ) ) ).

% top_set_def
thf(fact_79_top__set__def,axiom,
    ( top_top_set_o
    = ( collect_o @ top_top_o_o ) ) ).

% top_set_def
thf(fact_80_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_81_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_82_bot__set__def,axiom,
    ( bot_bo4854962954004695426nnreal
    = ( collec6648975593938027277nnreal @ bot_bo412624608084785539real_o ) ) ).

% bot_set_def
thf(fact_83_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_84_UNIV__witness,axiom,
    ? [X: real > a] : ( member_real_a @ X @ top_top_set_real_a ) ).

% UNIV_witness
thf(fact_85_UNIV__witness,axiom,
    ? [X: $o > real] : ( member_o_real @ X @ top_top_set_o_real ) ).

% UNIV_witness
thf(fact_86_UNIV__witness,axiom,
    ? [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).

% UNIV_witness
thf(fact_87_UNIV__witness,axiom,
    ? [X: c > b] : ( member_c_b @ X @ top_top_set_c_b ) ).

% UNIV_witness
thf(fact_88_UNIV__witness,axiom,
    ? [X: a > b] : ( member_a_b @ X @ top_top_set_a_b ) ).

% UNIV_witness
thf(fact_89_UNIV__witness,axiom,
    ? [X: real] : ( member_real @ X @ top_top_set_real ) ).

% UNIV_witness
thf(fact_90_UNIV__witness,axiom,
    ? [X: $o] : ( member_o @ X @ top_top_set_o ) ).

% UNIV_witness
thf(fact_91_UNIV__eq__I,axiom,
    ! [A4: set_real_a] :
      ( ! [X: real > a] : ( member_real_a @ X @ A4 )
     => ( top_top_set_real_a = A4 ) ) ).

% UNIV_eq_I
thf(fact_92_UNIV__eq__I,axiom,
    ! [A4: set_o_real] :
      ( ! [X: $o > real] : ( member_o_real @ X @ A4 )
     => ( top_top_set_o_real = A4 ) ) ).

% UNIV_eq_I
thf(fact_93_UNIV__eq__I,axiom,
    ! [A4: set_nat_real] :
      ( ! [X: nat > real] : ( member_nat_real @ X @ A4 )
     => ( top_top_set_nat_real = A4 ) ) ).

% UNIV_eq_I
thf(fact_94_UNIV__eq__I,axiom,
    ! [A4: set_c_b] :
      ( ! [X: c > b] : ( member_c_b @ X @ A4 )
     => ( top_top_set_c_b = A4 ) ) ).

% UNIV_eq_I
thf(fact_95_UNIV__eq__I,axiom,
    ! [A4: set_a_b] :
      ( ! [X: a > b] : ( member_a_b @ X @ A4 )
     => ( top_top_set_a_b = A4 ) ) ).

% UNIV_eq_I
thf(fact_96_UNIV__eq__I,axiom,
    ! [A4: set_real] :
      ( ! [X: real] : ( member_real @ X @ A4 )
     => ( top_top_set_real = A4 ) ) ).

% UNIV_eq_I
thf(fact_97_UNIV__eq__I,axiom,
    ! [A4: set_o] :
      ( ! [X: $o] : ( member_o @ X @ A4 )
     => ( top_top_set_o = A4 ) ) ).

% UNIV_eq_I
thf(fact_98_mem__Collect__eq,axiom,
    ! [A: real > a,P3: ( real > a ) > $o] :
      ( ( member_real_a @ A @ ( collect_real_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_99_mem__Collect__eq,axiom,
    ! [A: $o > real,P3: ( $o > real ) > $o] :
      ( ( member_o_real @ A @ ( collect_o_real @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_100_mem__Collect__eq,axiom,
    ! [A: nat > real,P3: ( nat > real ) > $o] :
      ( ( member_nat_real @ A @ ( collect_nat_real @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_101_mem__Collect__eq,axiom,
    ! [A: c > b,P3: ( c > b ) > $o] :
      ( ( member_c_b @ A @ ( collect_c_b @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_102_mem__Collect__eq,axiom,
    ! [A: a > b,P3: ( a > b ) > $o] :
      ( ( member_a_b @ A @ ( collect_a_b @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_103_Collect__mem__eq,axiom,
    ! [A4: set_real_a] :
      ( ( collect_real_a
        @ ^ [X5: real > a] : ( member_real_a @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_104_Collect__mem__eq,axiom,
    ! [A4: set_o_real] :
      ( ( collect_o_real
        @ ^ [X5: $o > real] : ( member_o_real @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_105_Collect__mem__eq,axiom,
    ! [A4: set_nat_real] :
      ( ( collect_nat_real
        @ ^ [X5: nat > real] : ( member_nat_real @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_106_Collect__mem__eq,axiom,
    ! [A4: set_c_b] :
      ( ( collect_c_b
        @ ^ [X5: c > b] : ( member_c_b @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_107_Collect__mem__eq,axiom,
    ! [A4: set_a_b] :
      ( ( collect_a_b
        @ ^ [X5: a > b] : ( member_a_b @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_108_ex__in__conv,axiom,
    ! [A4: set_real_a] :
      ( ( ? [X5: real > a] : ( member_real_a @ X5 @ A4 ) )
      = ( A4 != bot_bot_set_real_a ) ) ).

% ex_in_conv
thf(fact_109_ex__in__conv,axiom,
    ! [A4: set_o_real] :
      ( ( ? [X5: $o > real] : ( member_o_real @ X5 @ A4 ) )
      = ( A4 != bot_bot_set_o_real ) ) ).

% ex_in_conv
thf(fact_110_ex__in__conv,axiom,
    ! [A4: set_nat_real] :
      ( ( ? [X5: nat > real] : ( member_nat_real @ X5 @ A4 ) )
      = ( A4 != bot_bot_set_nat_real ) ) ).

% ex_in_conv
thf(fact_111_ex__in__conv,axiom,
    ! [A4: set_c_b] :
      ( ( ? [X5: c > b] : ( member_c_b @ X5 @ A4 ) )
      = ( A4 != bot_bot_set_c_b ) ) ).

% ex_in_conv
thf(fact_112_ex__in__conv,axiom,
    ! [A4: set_a_b] :
      ( ( ? [X5: a > b] : ( member_a_b @ X5 @ A4 ) )
      = ( A4 != bot_bot_set_a_b ) ) ).

% ex_in_conv
thf(fact_113_ex__in__conv,axiom,
    ! [A4: set_real] :
      ( ( ? [X5: real] : ( member_real @ X5 @ A4 ) )
      = ( A4 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_114_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ? [X5: nat] : ( member_nat @ X5 @ A4 ) )
      = ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_115_ex__in__conv,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( ? [X5: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X5 @ A4 ) )
      = ( A4 != bot_bo4854962954004695426nnreal ) ) ).

% ex_in_conv
thf(fact_116_ex__in__conv,axiom,
    ! [A4: set_o] :
      ( ( ? [X5: $o] : ( member_o @ X5 @ A4 ) )
      = ( A4 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_117_equals0I,axiom,
    ! [A4: set_real_a] :
      ( ! [Y3: real > a] :
          ~ ( member_real_a @ Y3 @ A4 )
     => ( A4 = bot_bot_set_real_a ) ) ).

% equals0I
thf(fact_118_equals0I,axiom,
    ! [A4: set_o_real] :
      ( ! [Y3: $o > real] :
          ~ ( member_o_real @ Y3 @ A4 )
     => ( A4 = bot_bot_set_o_real ) ) ).

% equals0I
thf(fact_119_equals0I,axiom,
    ! [A4: set_nat_real] :
      ( ! [Y3: nat > real] :
          ~ ( member_nat_real @ Y3 @ A4 )
     => ( A4 = bot_bot_set_nat_real ) ) ).

% equals0I
thf(fact_120_equals0I,axiom,
    ! [A4: set_c_b] :
      ( ! [Y3: c > b] :
          ~ ( member_c_b @ Y3 @ A4 )
     => ( A4 = bot_bot_set_c_b ) ) ).

% equals0I
thf(fact_121_equals0I,axiom,
    ! [A4: set_a_b] :
      ( ! [Y3: a > b] :
          ~ ( member_a_b @ Y3 @ A4 )
     => ( A4 = bot_bot_set_a_b ) ) ).

% equals0I
thf(fact_122_equals0I,axiom,
    ! [A4: set_real] :
      ( ! [Y3: real] :
          ~ ( member_real @ Y3 @ A4 )
     => ( A4 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_123_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A4 )
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_124_equals0I,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ! [Y3: extend8495563244428889912nnreal] :
          ~ ( member7908768830364227535nnreal @ Y3 @ A4 )
     => ( A4 = bot_bo4854962954004695426nnreal ) ) ).

% equals0I
thf(fact_125_equals0I,axiom,
    ! [A4: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A4 )
     => ( A4 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_126_equals0D,axiom,
    ! [A4: set_real_a,A: real > a] :
      ( ( A4 = bot_bot_set_real_a )
     => ~ ( member_real_a @ A @ A4 ) ) ).

% equals0D
thf(fact_127_equals0D,axiom,
    ! [A4: set_o_real,A: $o > real] :
      ( ( A4 = bot_bot_set_o_real )
     => ~ ( member_o_real @ A @ A4 ) ) ).

% equals0D
thf(fact_128_equals0D,axiom,
    ! [A4: set_nat_real,A: nat > real] :
      ( ( A4 = bot_bot_set_nat_real )
     => ~ ( member_nat_real @ A @ A4 ) ) ).

% equals0D
thf(fact_129_equals0D,axiom,
    ! [A4: set_c_b,A: c > b] :
      ( ( A4 = bot_bot_set_c_b )
     => ~ ( member_c_b @ A @ A4 ) ) ).

% equals0D
thf(fact_130_equals0D,axiom,
    ! [A4: set_a_b,A: a > b] :
      ( ( A4 = bot_bot_set_a_b )
     => ~ ( member_a_b @ A @ A4 ) ) ).

% equals0D
thf(fact_131_equals0D,axiom,
    ! [A4: set_real,A: real] :
      ( ( A4 = bot_bot_set_real )
     => ~ ( member_real @ A @ A4 ) ) ).

% equals0D
thf(fact_132_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_133_equals0D,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] :
      ( ( A4 = bot_bo4854962954004695426nnreal )
     => ~ ( member7908768830364227535nnreal @ A @ A4 ) ) ).

% equals0D
thf(fact_134_equals0D,axiom,
    ! [A4: set_o,A: $o] :
      ( ( A4 = bot_bot_set_o )
     => ~ ( member_o @ A @ A4 ) ) ).

% equals0D
thf(fact_135_emptyE,axiom,
    ! [A: real > a] :
      ~ ( member_real_a @ A @ bot_bot_set_real_a ) ).

% emptyE
thf(fact_136_emptyE,axiom,
    ! [A: $o > real] :
      ~ ( member_o_real @ A @ bot_bot_set_o_real ) ).

% emptyE
thf(fact_137_emptyE,axiom,
    ! [A: nat > real] :
      ~ ( member_nat_real @ A @ bot_bot_set_nat_real ) ).

% emptyE
thf(fact_138_emptyE,axiom,
    ! [A: c > b] :
      ~ ( member_c_b @ A @ bot_bot_set_c_b ) ).

% emptyE
thf(fact_139_emptyE,axiom,
    ! [A: a > b] :
      ~ ( member_a_b @ A @ bot_bot_set_a_b ) ).

% emptyE
thf(fact_140_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_141_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_142_emptyE,axiom,
    ! [A: extend8495563244428889912nnreal] :
      ~ ( member7908768830364227535nnreal @ A @ bot_bo4854962954004695426nnreal ) ).

% emptyE
thf(fact_143_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_144_not__arg__cong__Inr,axiom,
    ! [X3: c,Y2: c] :
      ( ( X3 != Y2 )
     => ( ( sum_Inr_c_a @ X3 )
       != ( sum_Inr_c_a @ Y2 ) ) ) ).

% not_arg_cong_Inr
thf(fact_145_copair__qbs__Mx__equiv,axiom,
    binary8286901584692334522Mx_a_c = binary6242423198552412156x2_a_c ).

% copair_qbs_Mx_equiv
thf(fact_146_copair__qbs__Mx,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_c] :
      ( ( qbs_Mx_Sum_sum_a_c @ ( binary8555328655094383375bs_a_c @ X2 @ Y ) )
      = ( binary8286901584692334522Mx_a_c @ X2 @ Y ) ) ).

% copair_qbs_Mx
thf(fact_147_eqb__Mx,axiom,
    ( ( qbs_Mx_a @ empty_quasi_borel_a )
    = bot_bot_set_real_a ) ).

% eqb_Mx
thf(fact_148_eqb__Mx,axiom,
    ( ( qbs_Mx_c @ empty_quasi_borel_c )
    = bot_bot_set_real_c ) ).

% eqb_Mx
thf(fact_149_Set_Ois__empty__def,axiom,
    ( is_empty_real
    = ( ^ [A5: set_real] : ( A5 = bot_bot_set_real ) ) ) ).

% Set.is_empty_def
thf(fact_150_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A5: set_nat] : ( A5 = bot_bot_set_nat ) ) ) ).

% Set.is_empty_def
thf(fact_151_Set_Ois__empty__def,axiom,
    ( is_emp182806100662350310nnreal
    = ( ^ [A5: set_Ex3793607809372303086nnreal] : ( A5 = bot_bo4854962954004695426nnreal ) ) ) ).

% Set.is_empty_def
thf(fact_152_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A5: set_o] : ( A5 = bot_bot_set_o ) ) ) ).

% Set.is_empty_def
thf(fact_153_sum__set__simps_I3_J,axiom,
    ! [X3: a] :
      ( ( basic_setr_a_c @ ( sum_Inl_a_c @ X3 ) )
      = bot_bot_set_c ) ).

% sum_set_simps(3)
thf(fact_154_sum__set__simps_I2_J,axiom,
    ! [X3: c] :
      ( ( basic_setl_a_c @ ( sum_Inr_c_a @ X3 ) )
      = bot_bot_set_a ) ).

% sum_set_simps(2)
thf(fact_155_sets__Ball,axiom,
    ! [I: set_real_a,A4: ( real > a ) > set_real,M: ( real > a ) > sigma_measure_real,I2: real > a] :
      ( ! [X: real > a] :
          ( ( member_real_a @ X @ I )
         => ( member_set_real @ ( A4 @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_real_a @ I2 @ I )
       => ( member_set_real @ ( A4 @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_156_sets__Ball,axiom,
    ! [I: set_o_real,A4: ( $o > real ) > set_real,M: ( $o > real ) > sigma_measure_real,I2: $o > real] :
      ( ! [X: $o > real] :
          ( ( member_o_real @ X @ I )
         => ( member_set_real @ ( A4 @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_o_real @ I2 @ I )
       => ( member_set_real @ ( A4 @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_157_sets__Ball,axiom,
    ! [I: set_nat_real,A4: ( nat > real ) > set_real,M: ( nat > real ) > sigma_measure_real,I2: nat > real] :
      ( ! [X: nat > real] :
          ( ( member_nat_real @ X @ I )
         => ( member_set_real @ ( A4 @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_nat_real @ I2 @ I )
       => ( member_set_real @ ( A4 @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_158_sets__Ball,axiom,
    ! [I: set_c_b,A4: ( c > b ) > set_real,M: ( c > b ) > sigma_measure_real,I2: c > b] :
      ( ! [X: c > b] :
          ( ( member_c_b @ X @ I )
         => ( member_set_real @ ( A4 @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_c_b @ I2 @ I )
       => ( member_set_real @ ( A4 @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_159_sets__Ball,axiom,
    ! [I: set_a_b,A4: ( a > b ) > set_real,M: ( a > b ) > sigma_measure_real,I2: a > b] :
      ( ! [X: a > b] :
          ( ( member_a_b @ X @ I )
         => ( member_set_real @ ( A4 @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_a_b @ I2 @ I )
       => ( member_set_real @ ( A4 @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_160_sets__Ball,axiom,
    ! [I: set_real_a,A4: ( real > a ) > set_Ex3793607809372303086nnreal,M: ( real > a ) > sigma_7234349610311085201nnreal,I2: real > a] :
      ( ! [X: real > a] :
          ( ( member_real_a @ X @ I )
         => ( member603777416030116741nnreal @ ( A4 @ X ) @ ( sigma_5465916536984168985nnreal @ ( M @ X ) ) ) )
     => ( ( member_real_a @ I2 @ I )
       => ( member603777416030116741nnreal @ ( A4 @ I2 ) @ ( sigma_5465916536984168985nnreal @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_161_sets__Ball,axiom,
    ! [I: set_o_real,A4: ( $o > real ) > set_Ex3793607809372303086nnreal,M: ( $o > real ) > sigma_7234349610311085201nnreal,I2: $o > real] :
      ( ! [X: $o > real] :
          ( ( member_o_real @ X @ I )
         => ( member603777416030116741nnreal @ ( A4 @ X ) @ ( sigma_5465916536984168985nnreal @ ( M @ X ) ) ) )
     => ( ( member_o_real @ I2 @ I )
       => ( member603777416030116741nnreal @ ( A4 @ I2 ) @ ( sigma_5465916536984168985nnreal @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_162_sets__Ball,axiom,
    ! [I: set_nat_real,A4: ( nat > real ) > set_Ex3793607809372303086nnreal,M: ( nat > real ) > sigma_7234349610311085201nnreal,I2: nat > real] :
      ( ! [X: nat > real] :
          ( ( member_nat_real @ X @ I )
         => ( member603777416030116741nnreal @ ( A4 @ X ) @ ( sigma_5465916536984168985nnreal @ ( M @ X ) ) ) )
     => ( ( member_nat_real @ I2 @ I )
       => ( member603777416030116741nnreal @ ( A4 @ I2 ) @ ( sigma_5465916536984168985nnreal @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_163_sets__Ball,axiom,
    ! [I: set_c_b,A4: ( c > b ) > set_Ex3793607809372303086nnreal,M: ( c > b ) > sigma_7234349610311085201nnreal,I2: c > b] :
      ( ! [X: c > b] :
          ( ( member_c_b @ X @ I )
         => ( member603777416030116741nnreal @ ( A4 @ X ) @ ( sigma_5465916536984168985nnreal @ ( M @ X ) ) ) )
     => ( ( member_c_b @ I2 @ I )
       => ( member603777416030116741nnreal @ ( A4 @ I2 ) @ ( sigma_5465916536984168985nnreal @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_164_sets__Ball,axiom,
    ! [I: set_a_b,A4: ( a > b ) > set_Ex3793607809372303086nnreal,M: ( a > b ) > sigma_7234349610311085201nnreal,I2: a > b] :
      ( ! [X: a > b] :
          ( ( member_a_b @ X @ I )
         => ( member603777416030116741nnreal @ ( A4 @ X ) @ ( sigma_5465916536984168985nnreal @ ( M @ X ) ) ) )
     => ( ( member_a_b @ I2 @ I )
       => ( member603777416030116741nnreal @ ( A4 @ I2 ) @ ( sigma_5465916536984168985nnreal @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_165_Collect__empty__eq__bot,axiom,
    ! [P3: real > $o] :
      ( ( ( collect_real @ P3 )
        = bot_bot_set_real )
      = ( P3 = bot_bot_real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_166_Collect__empty__eq__bot,axiom,
    ! [P3: nat > $o] :
      ( ( ( collect_nat @ P3 )
        = bot_bot_set_nat )
      = ( P3 = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_167_Collect__empty__eq__bot,axiom,
    ! [P3: extend8495563244428889912nnreal > $o] :
      ( ( ( collec6648975593938027277nnreal @ P3 )
        = bot_bo4854962954004695426nnreal )
      = ( P3 = bot_bo412624608084785539real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_168_Collect__empty__eq__bot,axiom,
    ! [P3: $o > $o] :
      ( ( ( collect_o @ P3 )
        = bot_bot_set_o )
      = ( P3 = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_169_top__empty__eq,axiom,
    ( top_top_real_a_o
    = ( ^ [X5: real > a] : ( member_real_a @ X5 @ top_top_set_real_a ) ) ) ).

% top_empty_eq
thf(fact_170_top__empty__eq,axiom,
    ( top_top_o_real_o
    = ( ^ [X5: $o > real] : ( member_o_real @ X5 @ top_top_set_o_real ) ) ) ).

% top_empty_eq
thf(fact_171_top__empty__eq,axiom,
    ( top_top_nat_real_o
    = ( ^ [X5: nat > real] : ( member_nat_real @ X5 @ top_top_set_nat_real ) ) ) ).

% top_empty_eq
thf(fact_172_top__empty__eq,axiom,
    ( top_top_c_b_o
    = ( ^ [X5: c > b] : ( member_c_b @ X5 @ top_top_set_c_b ) ) ) ).

% top_empty_eq
thf(fact_173_top__empty__eq,axiom,
    ( top_top_a_b_o
    = ( ^ [X5: a > b] : ( member_a_b @ X5 @ top_top_set_a_b ) ) ) ).

% top_empty_eq
thf(fact_174_top__empty__eq,axiom,
    ( top_top_real_o
    = ( ^ [X5: real] : ( member_real @ X5 @ top_top_set_real ) ) ) ).

% top_empty_eq
thf(fact_175_top__empty__eq,axiom,
    ( top_top_o_o
    = ( ^ [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ) ) ).

% top_empty_eq
thf(fact_176_setl_Ointros,axiom,
    ! [S2: sum_sum_a_c,X3: a] :
      ( ( S2
        = ( sum_Inl_a_c @ X3 ) )
     => ( member_a @ X3 @ ( basic_setl_a_c @ S2 ) ) ) ).

% setl.intros
thf(fact_177_setl_Osimps,axiom,
    ! [A: a,S2: sum_sum_a_c] :
      ( ( member_a @ A @ ( basic_setl_a_c @ S2 ) )
      = ( ? [X5: a] :
            ( ( A = X5 )
            & ( S2
              = ( sum_Inl_a_c @ X5 ) ) ) ) ) ).

% setl.simps
thf(fact_178_setl_Ocases,axiom,
    ! [A: a,S2: sum_sum_a_c] :
      ( ( member_a @ A @ ( basic_setl_a_c @ S2 ) )
     => ( S2
        = ( sum_Inl_a_c @ A ) ) ) ).

% setl.cases
thf(fact_179_setr_Ocases,axiom,
    ! [A: c,S2: sum_sum_a_c] :
      ( ( member_c @ A @ ( basic_setr_a_c @ S2 ) )
     => ( S2
        = ( sum_Inr_c_a @ A ) ) ) ).

% setr.cases
thf(fact_180_setr_Osimps,axiom,
    ! [A: c,S2: sum_sum_a_c] :
      ( ( member_c @ A @ ( basic_setr_a_c @ S2 ) )
      = ( ? [X5: c] :
            ( ( A = X5 )
            & ( S2
              = ( sum_Inr_c_a @ X5 ) ) ) ) ) ).

% setr.simps
thf(fact_181_setr_Ointros,axiom,
    ! [S2: sum_sum_a_c,X3: c] :
      ( ( S2
        = ( sum_Inr_c_a @ X3 ) )
     => ( member_c @ X3 @ ( basic_setr_a_c @ S2 ) ) ) ).

% setr.intros
thf(fact_182_bot__empty__eq,axiom,
    ( bot_bot_real_a_o
    = ( ^ [X5: real > a] : ( member_real_a @ X5 @ bot_bot_set_real_a ) ) ) ).

% bot_empty_eq
thf(fact_183_bot__empty__eq,axiom,
    ( bot_bot_o_real_o
    = ( ^ [X5: $o > real] : ( member_o_real @ X5 @ bot_bot_set_o_real ) ) ) ).

% bot_empty_eq
thf(fact_184_bot__empty__eq,axiom,
    ( bot_bot_nat_real_o
    = ( ^ [X5: nat > real] : ( member_nat_real @ X5 @ bot_bot_set_nat_real ) ) ) ).

% bot_empty_eq
thf(fact_185_bot__empty__eq,axiom,
    ( bot_bot_c_b_o
    = ( ^ [X5: c > b] : ( member_c_b @ X5 @ bot_bot_set_c_b ) ) ) ).

% bot_empty_eq
thf(fact_186_bot__empty__eq,axiom,
    ( bot_bot_a_b_o
    = ( ^ [X5: a > b] : ( member_a_b @ X5 @ bot_bot_set_a_b ) ) ) ).

% bot_empty_eq
thf(fact_187_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X5: real] : ( member_real @ X5 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_188_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X5: nat] : ( member_nat @ X5 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_189_bot__empty__eq,axiom,
    ( bot_bo412624608084785539real_o
    = ( ^ [X5: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X5 @ bot_bo4854962954004695426nnreal ) ) ) ).

% bot_empty_eq
thf(fact_190_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X5: $o] : ( member_o @ X5 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_191_sum__set__simps_I1_J,axiom,
    ! [X3: a] :
      ( ( basic_setl_a_c @ ( sum_Inl_a_c @ X3 ) )
      = ( insert_a @ X3 @ bot_bot_set_a ) ) ).

% sum_set_simps(1)
thf(fact_192_sum__set__simps_I4_J,axiom,
    ! [X3: c] :
      ( ( basic_setr_a_c @ ( sum_Inr_c_a @ X3 ) )
      = ( insert_c @ X3 @ bot_bot_set_c ) ) ).

% sum_set_simps(4)
thf(fact_193_Inl__qbs__morphism,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_c] : ( member_a_Sum_sum_a_c @ sum_Inl_a_c @ ( qbs_mo7250741323400969261um_a_c @ X2 @ ( binary8555328655094383375bs_a_c @ X2 @ Y ) ) ) ).

% Inl_qbs_morphism
thf(fact_194_Inr__qbs__morphism,axiom,
    ! [Y: quasi_borel_c,X2: quasi_borel_a] : ( member_c_Sum_sum_a_c @ sum_Inr_c_a @ ( qbs_mo5084992033439934511um_a_c @ Y @ ( binary8555328655094383375bs_a_c @ X2 @ Y ) ) ) ).

% Inr_qbs_morphism
thf(fact_195_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_a] :
      ( ( ( qbs_space_a @ X2 )
        = bot_bot_set_a )
      = ( ( qbs_Mx_a @ X2 )
        = bot_bot_set_real_a ) ) ).

% qbs_empty_equiv
thf(fact_196_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_c] :
      ( ( ( qbs_space_c @ X2 )
        = bot_bot_set_c )
      = ( ( qbs_Mx_c @ X2 )
        = bot_bot_set_real_c ) ) ).

% qbs_empty_equiv
thf(fact_197_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_real] :
      ( ( ( qbs_space_real @ X2 )
        = bot_bot_set_real )
      = ( ( qbs_Mx_real @ X2 )
        = bot_bo6767488733719836353l_real ) ) ).

% qbs_empty_equiv
thf(fact_198_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_nat] :
      ( ( ( qbs_space_nat @ X2 )
        = bot_bot_set_nat )
      = ( ( qbs_Mx_nat @ X2 )
        = bot_bot_set_real_nat ) ) ).

% qbs_empty_equiv
thf(fact_199_qbs__empty__equiv,axiom,
    ! [X2: quasi_9015997321629101608nnreal] :
      ( ( ( qbs_sp175953267596557954nnreal @ X2 )
        = bot_bo4854962954004695426nnreal )
      = ( ( qbs_Mx6523938229262583809nnreal @ X2 )
        = bot_bo6037503491064675021nnreal ) ) ).

% qbs_empty_equiv
thf(fact_200_qbs__empty__equiv,axiom,
    ! [X2: quasi_borel_o] :
      ( ( ( qbs_space_o @ X2 )
        = bot_bot_set_o )
      = ( ( qbs_Mx_o @ X2 )
        = bot_bot_set_real_o ) ) ).

% qbs_empty_equiv
thf(fact_201_sum_Opred__mono__strong,axiom,
    ! [P1: ( real > a ) > $o,P22: ( real > a ) > $o,X3: sum_su2571395965866611557real_a,P1a: ( real > a ) > $o,P2a: ( real > a ) > $o] :
      ( ( basic_8747522359897721711real_a @ P1 @ P22 @ X3 )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_1952671662426305125real_a @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: real > a] :
              ( ( member_real_a @ Z2 @ ( basic_2020826412209680235real_a @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_8747522359897721711real_a @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_202_sum_Opred__mono__strong,axiom,
    ! [P1: ( real > a ) > $o,P22: ( $o > real ) > $o,X3: sum_su2067798924538045457o_real,P1a: ( real > a ) > $o,P2a: ( $o > real ) > $o] :
      ( ( basic_2577378067801423003o_real @ P1 @ P22 @ X3 )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_3331954996559478417o_real @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: $o > real] :
              ( ( member_o_real @ Z2 @ ( basic_7007390903827930775o_real @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_2577378067801423003o_real @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_203_sum_Opred__mono__strong,axiom,
    ! [P1: ( real > a ) > $o,P22: ( nat > real ) > $o,X3: sum_su5472343219575513685t_real,P1a: ( real > a ) > $o,P2a: ( nat > real ) > $o] :
      ( ( basic_7720748700303055285t_real @ P1 @ P22 @ X3 )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_4771245887642665279t_real @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: nat > real] :
              ( ( member_nat_real @ Z2 @ ( basic_7028784127217823417t_real @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_7720748700303055285t_real @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_204_sum_Opred__mono__strong,axiom,
    ! [P1: ( real > a ) > $o,P22: ( c > b ) > $o,X3: sum_sum_real_a_c_b,P1a: ( real > a ) > $o,P2a: ( c > b ) > $o] :
      ( ( basic_6994455019795175422_a_c_b @ P1 @ P22 @ X3 )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_2782221633019357064_a_c_b @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: c > b] :
              ( ( member_c_b @ Z2 @ ( basic_1287377912808766722_a_c_b @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_6994455019795175422_a_c_b @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_205_sum_Opred__mono__strong,axiom,
    ! [P1: ( real > a ) > $o,P22: ( a > b ) > $o,X3: sum_sum_real_a_a_b,P1a: ( real > a ) > $o,P2a: ( a > b ) > $o] :
      ( ( basic_3345994146885751680_a_a_b @ P1 @ P22 @ X3 )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_8357132796964709130_a_a_b @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: a > b] :
              ( ( member_a_b @ Z2 @ ( basic_6862289076754118788_a_a_b @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_3345994146885751680_a_a_b @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_206_sum_Opred__mono__strong,axiom,
    ! [P1: ( $o > real ) > $o,P22: ( real > a ) > $o,X3: sum_su7886454506223791033real_a,P1a: ( $o > real ) > $o,P2a: ( real > a ) > $o] :
      ( ( basic_4697917730111463491real_a @ P1 @ P22 @ X3 )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_5452494658869518905real_a @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: real > a] :
              ( ( member_real_a @ Z2 @ ( basic_9127930566137971263real_a @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_4697917730111463491real_a @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_207_sum_Opred__mono__strong,axiom,
    ! [P1: ( $o > real ) > $o,P22: ( $o > real ) > $o,X3: sum_su815935806896055909o_real,P1a: ( $o > real ) > $o,P2a: ( $o > real ) > $o] :
      ( ( basic_4205938908511992687o_real @ P1 @ P22 @ X3 )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_1079876227406846053o_real @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: $o > real] :
              ( ( member_o_real @ Z2 @ ( basic_3048226007810710379o_real @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_4205938908511992687o_real @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_208_sum_Opred__mono__strong,axiom,
    ! [P1: ( $o > real ) > $o,P22: ( nat > real ) > $o,X3: sum_su7765403328973102337t_real,P1a: ( $o > real ) > $o,P2a: ( nat > real ) > $o] :
      ( ( basic_4510141698895607777t_real @ P1 @ P22 @ X3 )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_343682719230182507t_real @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: nat > real] :
              ( ( member_nat_real @ Z2 @ ( basic_8384512648665255653t_real @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_4510141698895607777t_real @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_209_sum_Opred__mono__strong,axiom,
    ! [P1: ( $o > real ) > $o,P22: ( c > b ) > $o,X3: sum_sum_o_real_c_b,P1a: ( $o > real ) > $o,P2a: ( c > b ) > $o] :
      ( ( basic_641583842347427370al_c_b @ P1 @ P22 @ X3 )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_4764176403307764404al_c_b @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: c > b] :
              ( ( member_c_b @ Z2 @ ( basic_900879951930756398al_c_b @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_641583842347427370al_c_b @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_210_sum_Opred__mono__strong,axiom,
    ! [P1: ( $o > real ) > $o,P22: ( a > b ) > $o,X3: sum_sum_o_real_a_b,P1a: ( $o > real ) > $o,P2a: ( a > b ) > $o] :
      ( ( basic_6216495006292779436al_a_b @ P1 @ P22 @ X3 )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_1115715530398340662al_a_b @ X3 ) )
           => ( ( P1 @ Z1 )
             => ( P1a @ Z1 ) ) )
       => ( ! [Z2: a > b] :
              ( ( member_a_b @ Z2 @ ( basic_6475791115876108464al_a_b @ X3 ) )
             => ( ( P22 @ Z2 )
               => ( P2a @ Z2 ) ) )
         => ( basic_6216495006292779436al_a_b @ P1a @ P2a @ X3 ) ) ) ) ).

% sum.pred_mono_strong
thf(fact_211_sum_Opred__cong,axiom,
    ! [X3: sum_su2571395965866611557real_a,Ya: sum_su2571395965866611557real_a,P1: ( real > a ) > $o,P1a: ( real > a ) > $o,P22: ( real > a ) > $o,P2a: ( real > a ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_1952671662426305125real_a @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: real > a] :
              ( ( member_real_a @ Z2 @ ( basic_2020826412209680235real_a @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_8747522359897721711real_a @ P1 @ P22 @ X3 )
            = ( basic_8747522359897721711real_a @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_212_sum_Opred__cong,axiom,
    ! [X3: sum_su2067798924538045457o_real,Ya: sum_su2067798924538045457o_real,P1: ( real > a ) > $o,P1a: ( real > a ) > $o,P22: ( $o > real ) > $o,P2a: ( $o > real ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_3331954996559478417o_real @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: $o > real] :
              ( ( member_o_real @ Z2 @ ( basic_7007390903827930775o_real @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_2577378067801423003o_real @ P1 @ P22 @ X3 )
            = ( basic_2577378067801423003o_real @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_213_sum_Opred__cong,axiom,
    ! [X3: sum_su5472343219575513685t_real,Ya: sum_su5472343219575513685t_real,P1: ( real > a ) > $o,P1a: ( real > a ) > $o,P22: ( nat > real ) > $o,P2a: ( nat > real ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_4771245887642665279t_real @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: nat > real] :
              ( ( member_nat_real @ Z2 @ ( basic_7028784127217823417t_real @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_7720748700303055285t_real @ P1 @ P22 @ X3 )
            = ( basic_7720748700303055285t_real @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_214_sum_Opred__cong,axiom,
    ! [X3: sum_sum_real_a_c_b,Ya: sum_sum_real_a_c_b,P1: ( real > a ) > $o,P1a: ( real > a ) > $o,P22: ( c > b ) > $o,P2a: ( c > b ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_2782221633019357064_a_c_b @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: c > b] :
              ( ( member_c_b @ Z2 @ ( basic_1287377912808766722_a_c_b @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_6994455019795175422_a_c_b @ P1 @ P22 @ X3 )
            = ( basic_6994455019795175422_a_c_b @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_215_sum_Opred__cong,axiom,
    ! [X3: sum_sum_real_a_a_b,Ya: sum_sum_real_a_a_b,P1: ( real > a ) > $o,P1a: ( real > a ) > $o,P22: ( a > b ) > $o,P2a: ( a > b ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: real > a] :
            ( ( member_real_a @ Z1 @ ( basic_8357132796964709130_a_a_b @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: a > b] :
              ( ( member_a_b @ Z2 @ ( basic_6862289076754118788_a_a_b @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_3345994146885751680_a_a_b @ P1 @ P22 @ X3 )
            = ( basic_3345994146885751680_a_a_b @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_216_sum_Opred__cong,axiom,
    ! [X3: sum_su7886454506223791033real_a,Ya: sum_su7886454506223791033real_a,P1: ( $o > real ) > $o,P1a: ( $o > real ) > $o,P22: ( real > a ) > $o,P2a: ( real > a ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_5452494658869518905real_a @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: real > a] :
              ( ( member_real_a @ Z2 @ ( basic_9127930566137971263real_a @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_4697917730111463491real_a @ P1 @ P22 @ X3 )
            = ( basic_4697917730111463491real_a @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_217_sum_Opred__cong,axiom,
    ! [X3: sum_su815935806896055909o_real,Ya: sum_su815935806896055909o_real,P1: ( $o > real ) > $o,P1a: ( $o > real ) > $o,P22: ( $o > real ) > $o,P2a: ( $o > real ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_1079876227406846053o_real @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: $o > real] :
              ( ( member_o_real @ Z2 @ ( basic_3048226007810710379o_real @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_4205938908511992687o_real @ P1 @ P22 @ X3 )
            = ( basic_4205938908511992687o_real @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_218_sum_Opred__cong,axiom,
    ! [X3: sum_su7765403328973102337t_real,Ya: sum_su7765403328973102337t_real,P1: ( $o > real ) > $o,P1a: ( $o > real ) > $o,P22: ( nat > real ) > $o,P2a: ( nat > real ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_343682719230182507t_real @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: nat > real] :
              ( ( member_nat_real @ Z2 @ ( basic_8384512648665255653t_real @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_4510141698895607777t_real @ P1 @ P22 @ X3 )
            = ( basic_4510141698895607777t_real @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_219_sum_Opred__cong,axiom,
    ! [X3: sum_sum_o_real_c_b,Ya: sum_sum_o_real_c_b,P1: ( $o > real ) > $o,P1a: ( $o > real ) > $o,P22: ( c > b ) > $o,P2a: ( c > b ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_4764176403307764404al_c_b @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: c > b] :
              ( ( member_c_b @ Z2 @ ( basic_900879951930756398al_c_b @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_641583842347427370al_c_b @ P1 @ P22 @ X3 )
            = ( basic_641583842347427370al_c_b @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_220_sum_Opred__cong,axiom,
    ! [X3: sum_sum_o_real_a_b,Ya: sum_sum_o_real_a_b,P1: ( $o > real ) > $o,P1a: ( $o > real ) > $o,P22: ( a > b ) > $o,P2a: ( a > b ) > $o] :
      ( ( X3 = Ya )
     => ( ! [Z1: $o > real] :
            ( ( member_o_real @ Z1 @ ( basic_1115715530398340662al_a_b @ Ya ) )
           => ( ( P1 @ Z1 )
              = ( P1a @ Z1 ) ) )
       => ( ! [Z2: a > b] :
              ( ( member_a_b @ Z2 @ ( basic_6475791115876108464al_a_b @ Ya ) )
             => ( ( P22 @ Z2 )
                = ( P2a @ Z2 ) ) )
         => ( ( basic_6216495006292779436al_a_b @ P1 @ P22 @ X3 )
            = ( basic_6216495006292779436al_a_b @ P1a @ P2a @ Ya ) ) ) ) ) ).

% sum.pred_cong
thf(fact_221_insert__absorb2,axiom,
    ! [X3: real,A4: set_real] :
      ( ( insert_real @ X3 @ ( insert_real @ X3 @ A4 ) )
      = ( insert_real @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_222_insert__absorb2,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A4 ) )
      = ( insert_nat @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_223_insert__absorb2,axiom,
    ! [X3: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ X3 @ ( insert7407984058720857448nnreal @ X3 @ A4 ) )
      = ( insert7407984058720857448nnreal @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_224_insert__absorb2,axiom,
    ! [X3: $o,A4: set_o] :
      ( ( insert_o @ X3 @ ( insert_o @ X3 @ A4 ) )
      = ( insert_o @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_225_insert__iff,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A4 ) )
      = ( ( A = B )
        | ( member_real @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_226_insert__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_227_insert__iff,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ A4 ) )
      = ( ( A = B )
        | ( member7908768830364227535nnreal @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_228_insert__iff,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A4 ) )
      = ( ( A = B )
        | ( member_o @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_229_insert__iff,axiom,
    ! [A: real > a,B: real > a,A4: set_real_a] :
      ( ( member_real_a @ A @ ( insert_real_a @ B @ A4 ) )
      = ( ( A = B )
        | ( member_real_a @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_230_insert__iff,axiom,
    ! [A: $o > real,B: $o > real,A4: set_o_real] :
      ( ( member_o_real @ A @ ( insert_o_real @ B @ A4 ) )
      = ( ( A = B )
        | ( member_o_real @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_231_insert__iff,axiom,
    ! [A: nat > real,B: nat > real,A4: set_nat_real] :
      ( ( member_nat_real @ A @ ( insert_nat_real @ B @ A4 ) )
      = ( ( A = B )
        | ( member_nat_real @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_232_insert__iff,axiom,
    ! [A: c > b,B: c > b,A4: set_c_b] :
      ( ( member_c_b @ A @ ( insert_c_b @ B @ A4 ) )
      = ( ( A = B )
        | ( member_c_b @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_233_insert__iff,axiom,
    ! [A: a > b,B: a > b,A4: set_a_b] :
      ( ( member_a_b @ A @ ( insert_a_b @ B @ A4 ) )
      = ( ( A = B )
        | ( member_a_b @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_234_insertCI,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( ~ ( member_real @ A @ B4 )
       => ( A = B ) )
     => ( member_real @ A @ ( insert_real @ B @ B4 ) ) ) ).

% insertCI
thf(fact_235_insertCI,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B4 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_236_insertCI,axiom,
    ! [A: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( ~ ( member7908768830364227535nnreal @ A @ B4 )
       => ( A = B ) )
     => ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ B4 ) ) ) ).

% insertCI
thf(fact_237_insertCI,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( ~ ( member_o @ A @ B4 )
       => ( A = B ) )
     => ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).

% insertCI
thf(fact_238_insertCI,axiom,
    ! [A: real > a,B4: set_real_a,B: real > a] :
      ( ( ~ ( member_real_a @ A @ B4 )
       => ( A = B ) )
     => ( member_real_a @ A @ ( insert_real_a @ B @ B4 ) ) ) ).

% insertCI
thf(fact_239_insertCI,axiom,
    ! [A: $o > real,B4: set_o_real,B: $o > real] :
      ( ( ~ ( member_o_real @ A @ B4 )
       => ( A = B ) )
     => ( member_o_real @ A @ ( insert_o_real @ B @ B4 ) ) ) ).

% insertCI
thf(fact_240_insertCI,axiom,
    ! [A: nat > real,B4: set_nat_real,B: nat > real] :
      ( ( ~ ( member_nat_real @ A @ B4 )
       => ( A = B ) )
     => ( member_nat_real @ A @ ( insert_nat_real @ B @ B4 ) ) ) ).

% insertCI
thf(fact_241_insertCI,axiom,
    ! [A: c > b,B4: set_c_b,B: c > b] :
      ( ( ~ ( member_c_b @ A @ B4 )
       => ( A = B ) )
     => ( member_c_b @ A @ ( insert_c_b @ B @ B4 ) ) ) ).

% insertCI
thf(fact_242_insertCI,axiom,
    ! [A: a > b,B4: set_a_b,B: a > b] :
      ( ( ~ ( member_a_b @ A @ B4 )
       => ( A = B ) )
     => ( member_a_b @ A @ ( insert_a_b @ B @ B4 ) ) ) ).

% insertCI
thf(fact_243_singletonI,axiom,
    ! [A: real > a] : ( member_real_a @ A @ ( insert_real_a @ A @ bot_bot_set_real_a ) ) ).

% singletonI
thf(fact_244_singletonI,axiom,
    ! [A: $o > real] : ( member_o_real @ A @ ( insert_o_real @ A @ bot_bot_set_o_real ) ) ).

% singletonI
thf(fact_245_singletonI,axiom,
    ! [A: nat > real] : ( member_nat_real @ A @ ( insert_nat_real @ A @ bot_bot_set_nat_real ) ) ).

% singletonI
thf(fact_246_singletonI,axiom,
    ! [A: c > b] : ( member_c_b @ A @ ( insert_c_b @ A @ bot_bot_set_c_b ) ) ).

% singletonI
thf(fact_247_singletonI,axiom,
    ! [A: a > b] : ( member_a_b @ A @ ( insert_a_b @ A @ bot_bot_set_a_b ) ) ).

% singletonI
thf(fact_248_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_249_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_250_singletonI,axiom,
    ! [A: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) ).

% singletonI
thf(fact_251_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_252_pred__sum__inject_I2_J,axiom,
    ! [P1: a > $o,P22: c > $o,B: c] :
      ( ( basic_pred_sum_a_c @ P1 @ P22 @ ( sum_Inr_c_a @ B ) )
      = ( P22 @ B ) ) ).

% pred_sum_inject(2)
thf(fact_253_pred__sum__inject_I1_J,axiom,
    ! [P1: a > $o,P22: c > $o,A: a] :
      ( ( basic_pred_sum_a_c @ P1 @ P22 @ ( sum_Inl_a_c @ A ) )
      = ( P1 @ A ) ) ).

% pred_sum_inject(1)
thf(fact_254_eqb__space,axiom,
    ( ( qbs_space_real @ empty_1876425439295802446l_real )
    = bot_bot_set_real ) ).

% eqb_space
thf(fact_255_eqb__space,axiom,
    ( ( qbs_space_nat @ empty_8278123436611590770el_nat )
    = bot_bot_set_nat ) ).

% eqb_space
thf(fact_256_eqb__space,axiom,
    ( ( qbs_sp175953267596557954nnreal @ empty_1788085430566700506nnreal )
    = bot_bo4854962954004695426nnreal ) ).

% eqb_space
thf(fact_257_eqb__space,axiom,
    ( ( qbs_space_o @ empty_quasi_borel_o )
    = bot_bot_set_o ) ).

% eqb_space
thf(fact_258_mk__disjoint__insert,axiom,
    ! [A: real,A4: set_real] :
      ( ( member_real @ A @ A4 )
     => ? [B5: set_real] :
          ( ( A4
            = ( insert_real @ A @ B5 ) )
          & ~ ( member_real @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_259_mk__disjoint__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ? [B5: set_nat] :
          ( ( A4
            = ( insert_nat @ A @ B5 ) )
          & ~ ( member_nat @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_260_mk__disjoint__insert,axiom,
    ! [A: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ A4 )
     => ? [B5: set_Ex3793607809372303086nnreal] :
          ( ( A4
            = ( insert7407984058720857448nnreal @ A @ B5 ) )
          & ~ ( member7908768830364227535nnreal @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_261_mk__disjoint__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ? [B5: set_o] :
          ( ( A4
            = ( insert_o @ A @ B5 ) )
          & ~ ( member_o @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_262_mk__disjoint__insert,axiom,
    ! [A: real > a,A4: set_real_a] :
      ( ( member_real_a @ A @ A4 )
     => ? [B5: set_real_a] :
          ( ( A4
            = ( insert_real_a @ A @ B5 ) )
          & ~ ( member_real_a @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_263_mk__disjoint__insert,axiom,
    ! [A: $o > real,A4: set_o_real] :
      ( ( member_o_real @ A @ A4 )
     => ? [B5: set_o_real] :
          ( ( A4
            = ( insert_o_real @ A @ B5 ) )
          & ~ ( member_o_real @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_264_mk__disjoint__insert,axiom,
    ! [A: nat > real,A4: set_nat_real] :
      ( ( member_nat_real @ A @ A4 )
     => ? [B5: set_nat_real] :
          ( ( A4
            = ( insert_nat_real @ A @ B5 ) )
          & ~ ( member_nat_real @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_265_mk__disjoint__insert,axiom,
    ! [A: c > b,A4: set_c_b] :
      ( ( member_c_b @ A @ A4 )
     => ? [B5: set_c_b] :
          ( ( A4
            = ( insert_c_b @ A @ B5 ) )
          & ~ ( member_c_b @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_266_mk__disjoint__insert,axiom,
    ! [A: a > b,A4: set_a_b] :
      ( ( member_a_b @ A @ A4 )
     => ? [B5: set_a_b] :
          ( ( A4
            = ( insert_a_b @ A @ B5 ) )
          & ~ ( member_a_b @ A @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_267_insert__commute,axiom,
    ! [X3: real,Y2: real,A4: set_real] :
      ( ( insert_real @ X3 @ ( insert_real @ Y2 @ A4 ) )
      = ( insert_real @ Y2 @ ( insert_real @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_268_insert__commute,axiom,
    ! [X3: nat,Y2: nat,A4: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ Y2 @ A4 ) )
      = ( insert_nat @ Y2 @ ( insert_nat @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_269_insert__commute,axiom,
    ! [X3: extend8495563244428889912nnreal,Y2: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ X3 @ ( insert7407984058720857448nnreal @ Y2 @ A4 ) )
      = ( insert7407984058720857448nnreal @ Y2 @ ( insert7407984058720857448nnreal @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_270_insert__commute,axiom,
    ! [X3: $o,Y2: $o,A4: set_o] :
      ( ( insert_o @ X3 @ ( insert_o @ Y2 @ A4 ) )
      = ( insert_o @ Y2 @ ( insert_o @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_271_insert__eq__iff,axiom,
    ! [A: real,A4: set_real,B: real,B4: set_real] :
      ( ~ ( member_real @ A @ A4 )
     => ( ~ ( member_real @ B @ B4 )
       => ( ( ( insert_real @ A @ A4 )
            = ( insert_real @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_real] :
                  ( ( A4
                    = ( insert_real @ B @ C2 ) )
                  & ~ ( member_real @ B @ C2 )
                  & ( B4
                    = ( insert_real @ A @ C2 ) )
                  & ~ ( member_real @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_272_insert__eq__iff,axiom,
    ! [A: nat,A4: set_nat,B: nat,B4: set_nat] :
      ( ~ ( member_nat @ A @ A4 )
     => ( ~ ( member_nat @ B @ B4 )
       => ( ( ( insert_nat @ A @ A4 )
            = ( insert_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_nat] :
                  ( ( A4
                    = ( insert_nat @ B @ C2 ) )
                  & ~ ( member_nat @ B @ C2 )
                  & ( B4
                    = ( insert_nat @ A @ C2 ) )
                  & ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_273_insert__eq__iff,axiom,
    ! [A: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ A @ A4 )
     => ( ~ ( member7908768830364227535nnreal @ B @ B4 )
       => ( ( ( insert7407984058720857448nnreal @ A @ A4 )
            = ( insert7407984058720857448nnreal @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_Ex3793607809372303086nnreal] :
                  ( ( A4
                    = ( insert7407984058720857448nnreal @ B @ C2 ) )
                  & ~ ( member7908768830364227535nnreal @ B @ C2 )
                  & ( B4
                    = ( insert7407984058720857448nnreal @ A @ C2 ) )
                  & ~ ( member7908768830364227535nnreal @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_274_insert__eq__iff,axiom,
    ! [A: $o,A4: set_o,B: $o,B4: set_o] :
      ( ~ ( member_o @ A @ A4 )
     => ( ~ ( member_o @ B @ B4 )
       => ( ( ( insert_o @ A @ A4 )
            = ( insert_o @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A = ~ B )
             => ? [C2: set_o] :
                  ( ( A4
                    = ( insert_o @ B @ C2 ) )
                  & ~ ( member_o @ B @ C2 )
                  & ( B4
                    = ( insert_o @ A @ C2 ) )
                  & ~ ( member_o @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_275_insert__eq__iff,axiom,
    ! [A: real > a,A4: set_real_a,B: real > a,B4: set_real_a] :
      ( ~ ( member_real_a @ A @ A4 )
     => ( ~ ( member_real_a @ B @ B4 )
       => ( ( ( insert_real_a @ A @ A4 )
            = ( insert_real_a @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_real_a] :
                  ( ( A4
                    = ( insert_real_a @ B @ C2 ) )
                  & ~ ( member_real_a @ B @ C2 )
                  & ( B4
                    = ( insert_real_a @ A @ C2 ) )
                  & ~ ( member_real_a @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_276_insert__eq__iff,axiom,
    ! [A: $o > real,A4: set_o_real,B: $o > real,B4: set_o_real] :
      ( ~ ( member_o_real @ A @ A4 )
     => ( ~ ( member_o_real @ B @ B4 )
       => ( ( ( insert_o_real @ A @ A4 )
            = ( insert_o_real @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_o_real] :
                  ( ( A4
                    = ( insert_o_real @ B @ C2 ) )
                  & ~ ( member_o_real @ B @ C2 )
                  & ( B4
                    = ( insert_o_real @ A @ C2 ) )
                  & ~ ( member_o_real @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_277_insert__eq__iff,axiom,
    ! [A: nat > real,A4: set_nat_real,B: nat > real,B4: set_nat_real] :
      ( ~ ( member_nat_real @ A @ A4 )
     => ( ~ ( member_nat_real @ B @ B4 )
       => ( ( ( insert_nat_real @ A @ A4 )
            = ( insert_nat_real @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_nat_real] :
                  ( ( A4
                    = ( insert_nat_real @ B @ C2 ) )
                  & ~ ( member_nat_real @ B @ C2 )
                  & ( B4
                    = ( insert_nat_real @ A @ C2 ) )
                  & ~ ( member_nat_real @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_278_insert__eq__iff,axiom,
    ! [A: c > b,A4: set_c_b,B: c > b,B4: set_c_b] :
      ( ~ ( member_c_b @ A @ A4 )
     => ( ~ ( member_c_b @ B @ B4 )
       => ( ( ( insert_c_b @ A @ A4 )
            = ( insert_c_b @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_c_b] :
                  ( ( A4
                    = ( insert_c_b @ B @ C2 ) )
                  & ~ ( member_c_b @ B @ C2 )
                  & ( B4
                    = ( insert_c_b @ A @ C2 ) )
                  & ~ ( member_c_b @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_279_insert__eq__iff,axiom,
    ! [A: a > b,A4: set_a_b,B: a > b,B4: set_a_b] :
      ( ~ ( member_a_b @ A @ A4 )
     => ( ~ ( member_a_b @ B @ B4 )
       => ( ( ( insert_a_b @ A @ A4 )
            = ( insert_a_b @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C2: set_a_b] :
                  ( ( A4
                    = ( insert_a_b @ B @ C2 ) )
                  & ~ ( member_a_b @ B @ C2 )
                  & ( B4
                    = ( insert_a_b @ A @ C2 ) )
                  & ~ ( member_a_b @ A @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_280_insert__absorb,axiom,
    ! [A: real,A4: set_real] :
      ( ( member_real @ A @ A4 )
     => ( ( insert_real @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_281_insert__absorb,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( insert_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_282_insert__absorb,axiom,
    ! [A: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ A4 )
     => ( ( insert7407984058720857448nnreal @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_283_insert__absorb,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( insert_o @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_284_insert__absorb,axiom,
    ! [A: real > a,A4: set_real_a] :
      ( ( member_real_a @ A @ A4 )
     => ( ( insert_real_a @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_285_insert__absorb,axiom,
    ! [A: $o > real,A4: set_o_real] :
      ( ( member_o_real @ A @ A4 )
     => ( ( insert_o_real @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_286_insert__absorb,axiom,
    ! [A: nat > real,A4: set_nat_real] :
      ( ( member_nat_real @ A @ A4 )
     => ( ( insert_nat_real @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_287_insert__absorb,axiom,
    ! [A: c > b,A4: set_c_b] :
      ( ( member_c_b @ A @ A4 )
     => ( ( insert_c_b @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_288_insert__absorb,axiom,
    ! [A: a > b,A4: set_a_b] :
      ( ( member_a_b @ A @ A4 )
     => ( ( insert_a_b @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_289_insert__ident,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ X3 @ A4 )
     => ( ~ ( member_real @ X3 @ B4 )
       => ( ( ( insert_real @ X3 @ A4 )
            = ( insert_real @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_290_insert__ident,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X3 @ A4 )
     => ( ~ ( member_nat @ X3 @ B4 )
       => ( ( ( insert_nat @ X3 @ A4 )
            = ( insert_nat @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_291_insert__ident,axiom,
    ! [X3: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ X3 @ A4 )
     => ( ~ ( member7908768830364227535nnreal @ X3 @ B4 )
       => ( ( ( insert7407984058720857448nnreal @ X3 @ A4 )
            = ( insert7407984058720857448nnreal @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_292_insert__ident,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ X3 @ A4 )
     => ( ~ ( member_o @ X3 @ B4 )
       => ( ( ( insert_o @ X3 @ A4 )
            = ( insert_o @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_293_insert__ident,axiom,
    ! [X3: real > a,A4: set_real_a,B4: set_real_a] :
      ( ~ ( member_real_a @ X3 @ A4 )
     => ( ~ ( member_real_a @ X3 @ B4 )
       => ( ( ( insert_real_a @ X3 @ A4 )
            = ( insert_real_a @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_294_insert__ident,axiom,
    ! [X3: $o > real,A4: set_o_real,B4: set_o_real] :
      ( ~ ( member_o_real @ X3 @ A4 )
     => ( ~ ( member_o_real @ X3 @ B4 )
       => ( ( ( insert_o_real @ X3 @ A4 )
            = ( insert_o_real @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_295_insert__ident,axiom,
    ! [X3: nat > real,A4: set_nat_real,B4: set_nat_real] :
      ( ~ ( member_nat_real @ X3 @ A4 )
     => ( ~ ( member_nat_real @ X3 @ B4 )
       => ( ( ( insert_nat_real @ X3 @ A4 )
            = ( insert_nat_real @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_296_insert__ident,axiom,
    ! [X3: c > b,A4: set_c_b,B4: set_c_b] :
      ( ~ ( member_c_b @ X3 @ A4 )
     => ( ~ ( member_c_b @ X3 @ B4 )
       => ( ( ( insert_c_b @ X3 @ A4 )
            = ( insert_c_b @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_297_insert__ident,axiom,
    ! [X3: a > b,A4: set_a_b,B4: set_a_b] :
      ( ~ ( member_a_b @ X3 @ A4 )
     => ( ~ ( member_a_b @ X3 @ B4 )
       => ( ( ( insert_a_b @ X3 @ A4 )
            = ( insert_a_b @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_298_Set_Oset__insert,axiom,
    ! [X3: real,A4: set_real] :
      ( ( member_real @ X3 @ A4 )
     => ~ ! [B5: set_real] :
            ( ( A4
              = ( insert_real @ X3 @ B5 ) )
           => ( member_real @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_299_Set_Oset__insert,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ( member_nat @ X3 @ A4 )
     => ~ ! [B5: set_nat] :
            ( ( A4
              = ( insert_nat @ X3 @ B5 ) )
           => ( member_nat @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_300_Set_Oset__insert,axiom,
    ! [X3: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ X3 @ A4 )
     => ~ ! [B5: set_Ex3793607809372303086nnreal] :
            ( ( A4
              = ( insert7407984058720857448nnreal @ X3 @ B5 ) )
           => ( member7908768830364227535nnreal @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_301_Set_Oset__insert,axiom,
    ! [X3: $o,A4: set_o] :
      ( ( member_o @ X3 @ A4 )
     => ~ ! [B5: set_o] :
            ( ( A4
              = ( insert_o @ X3 @ B5 ) )
           => ( member_o @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_302_Set_Oset__insert,axiom,
    ! [X3: real > a,A4: set_real_a] :
      ( ( member_real_a @ X3 @ A4 )
     => ~ ! [B5: set_real_a] :
            ( ( A4
              = ( insert_real_a @ X3 @ B5 ) )
           => ( member_real_a @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_303_Set_Oset__insert,axiom,
    ! [X3: $o > real,A4: set_o_real] :
      ( ( member_o_real @ X3 @ A4 )
     => ~ ! [B5: set_o_real] :
            ( ( A4
              = ( insert_o_real @ X3 @ B5 ) )
           => ( member_o_real @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_304_Set_Oset__insert,axiom,
    ! [X3: nat > real,A4: set_nat_real] :
      ( ( member_nat_real @ X3 @ A4 )
     => ~ ! [B5: set_nat_real] :
            ( ( A4
              = ( insert_nat_real @ X3 @ B5 ) )
           => ( member_nat_real @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_305_Set_Oset__insert,axiom,
    ! [X3: c > b,A4: set_c_b] :
      ( ( member_c_b @ X3 @ A4 )
     => ~ ! [B5: set_c_b] :
            ( ( A4
              = ( insert_c_b @ X3 @ B5 ) )
           => ( member_c_b @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_306_Set_Oset__insert,axiom,
    ! [X3: a > b,A4: set_a_b] :
      ( ( member_a_b @ X3 @ A4 )
     => ~ ! [B5: set_a_b] :
            ( ( A4
              = ( insert_a_b @ X3 @ B5 ) )
           => ( member_a_b @ X3 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_307_insertI2,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( member_real @ A @ B4 )
     => ( member_real @ A @ ( insert_real @ B @ B4 ) ) ) ).

% insertI2
thf(fact_308_insertI2,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( member_nat @ A @ B4 )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_309_insertI2,axiom,
    ! [A: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ A @ B4 )
     => ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ B4 ) ) ) ).

% insertI2
thf(fact_310_insertI2,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( member_o @ A @ B4 )
     => ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).

% insertI2
thf(fact_311_insertI2,axiom,
    ! [A: real > a,B4: set_real_a,B: real > a] :
      ( ( member_real_a @ A @ B4 )
     => ( member_real_a @ A @ ( insert_real_a @ B @ B4 ) ) ) ).

% insertI2
thf(fact_312_insertI2,axiom,
    ! [A: $o > real,B4: set_o_real,B: $o > real] :
      ( ( member_o_real @ A @ B4 )
     => ( member_o_real @ A @ ( insert_o_real @ B @ B4 ) ) ) ).

% insertI2
thf(fact_313_insertI2,axiom,
    ! [A: nat > real,B4: set_nat_real,B: nat > real] :
      ( ( member_nat_real @ A @ B4 )
     => ( member_nat_real @ A @ ( insert_nat_real @ B @ B4 ) ) ) ).

% insertI2
thf(fact_314_insertI2,axiom,
    ! [A: c > b,B4: set_c_b,B: c > b] :
      ( ( member_c_b @ A @ B4 )
     => ( member_c_b @ A @ ( insert_c_b @ B @ B4 ) ) ) ).

% insertI2
thf(fact_315_insertI2,axiom,
    ! [A: a > b,B4: set_a_b,B: a > b] :
      ( ( member_a_b @ A @ B4 )
     => ( member_a_b @ A @ ( insert_a_b @ B @ B4 ) ) ) ).

% insertI2
thf(fact_316_insertI1,axiom,
    ! [A: real,B4: set_real] : ( member_real @ A @ ( insert_real @ A @ B4 ) ) ).

% insertI1
thf(fact_317_insertI1,axiom,
    ! [A: nat,B4: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_318_insertI1,axiom,
    ! [A: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] : ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ A @ B4 ) ) ).

% insertI1
thf(fact_319_insertI1,axiom,
    ! [A: $o,B4: set_o] : ( member_o @ A @ ( insert_o @ A @ B4 ) ) ).

% insertI1
thf(fact_320_insertI1,axiom,
    ! [A: real > a,B4: set_real_a] : ( member_real_a @ A @ ( insert_real_a @ A @ B4 ) ) ).

% insertI1
thf(fact_321_insertI1,axiom,
    ! [A: $o > real,B4: set_o_real] : ( member_o_real @ A @ ( insert_o_real @ A @ B4 ) ) ).

% insertI1
thf(fact_322_insertI1,axiom,
    ! [A: nat > real,B4: set_nat_real] : ( member_nat_real @ A @ ( insert_nat_real @ A @ B4 ) ) ).

% insertI1
thf(fact_323_insertI1,axiom,
    ! [A: c > b,B4: set_c_b] : ( member_c_b @ A @ ( insert_c_b @ A @ B4 ) ) ).

% insertI1
thf(fact_324_insertI1,axiom,
    ! [A: a > b,B4: set_a_b] : ( member_a_b @ A @ ( insert_a_b @ A @ B4 ) ) ).

% insertI1
thf(fact_325_insertE,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A4 ) )
     => ( ( A != B )
       => ( member_real @ A @ A4 ) ) ) ).

% insertE
thf(fact_326_insertE,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_327_insertE,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ ( insert7407984058720857448nnreal @ B @ A4 ) )
     => ( ( A != B )
       => ( member7908768830364227535nnreal @ A @ A4 ) ) ) ).

% insertE
thf(fact_328_insertE,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A4 ) )
     => ( ( A = ~ B )
       => ( member_o @ A @ A4 ) ) ) ).

% insertE
thf(fact_329_insertE,axiom,
    ! [A: real > a,B: real > a,A4: set_real_a] :
      ( ( member_real_a @ A @ ( insert_real_a @ B @ A4 ) )
     => ( ( A != B )
       => ( member_real_a @ A @ A4 ) ) ) ).

% insertE
thf(fact_330_insertE,axiom,
    ! [A: $o > real,B: $o > real,A4: set_o_real] :
      ( ( member_o_real @ A @ ( insert_o_real @ B @ A4 ) )
     => ( ( A != B )
       => ( member_o_real @ A @ A4 ) ) ) ).

% insertE
thf(fact_331_insertE,axiom,
    ! [A: nat > real,B: nat > real,A4: set_nat_real] :
      ( ( member_nat_real @ A @ ( insert_nat_real @ B @ A4 ) )
     => ( ( A != B )
       => ( member_nat_real @ A @ A4 ) ) ) ).

% insertE
thf(fact_332_insertE,axiom,
    ! [A: c > b,B: c > b,A4: set_c_b] :
      ( ( member_c_b @ A @ ( insert_c_b @ B @ A4 ) )
     => ( ( A != B )
       => ( member_c_b @ A @ A4 ) ) ) ).

% insertE
thf(fact_333_insertE,axiom,
    ! [A: a > b,B: a > b,A4: set_a_b] :
      ( ( member_a_b @ A @ ( insert_a_b @ B @ A4 ) )
     => ( ( A != B )
       => ( member_a_b @ A @ A4 ) ) ) ).

% insertE
thf(fact_334_qbs__space__eq__Mx,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X2 )
        = ( qbs_Mx_a @ Y ) )
     => ( ( qbs_space_a @ X2 )
        = ( qbs_space_a @ Y ) ) ) ).

% qbs_space_eq_Mx
thf(fact_335_qbs__space__eq__Mx,axiom,
    ! [X2: quasi_borel_c,Y: quasi_borel_c] :
      ( ( ( qbs_Mx_c @ X2 )
        = ( qbs_Mx_c @ Y ) )
     => ( ( qbs_space_c @ X2 )
        = ( qbs_space_c @ Y ) ) ) ).

% qbs_space_eq_Mx
thf(fact_336_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > a,X2: quasi_borel_real_a,R2: real] :
      ( ( member_real_real_a @ Alpha @ ( qbs_Mx_real_a @ X2 ) )
     => ( member_real_a @ ( Alpha @ R2 ) @ ( qbs_space_real_a @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_337_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > $o > real,X2: quasi_borel_o_real,R2: real] :
      ( ( member_real_o_real @ Alpha @ ( qbs_Mx_o_real @ X2 ) )
     => ( member_o_real @ ( Alpha @ R2 ) @ ( qbs_space_o_real @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_338_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > nat > real,X2: quasi_borel_nat_real,R2: real] :
      ( ( member_real_nat_real @ Alpha @ ( qbs_Mx_nat_real @ X2 ) )
     => ( member_nat_real @ ( Alpha @ R2 ) @ ( qbs_space_nat_real @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_339_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > c > b,X2: quasi_borel_c_b,R2: real] :
      ( ( member_real_c_b @ Alpha @ ( qbs_Mx_c_b @ X2 ) )
     => ( member_c_b @ ( Alpha @ R2 ) @ ( qbs_space_c_b @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_340_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > a > b,X2: quasi_borel_a_b,R2: real] :
      ( ( member_real_a_b @ Alpha @ ( qbs_Mx_a_b @ X2 ) )
     => ( member_a_b @ ( Alpha @ R2 ) @ ( qbs_space_a_b @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_341_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > a,X2: quasi_borel_a,R2: real] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
     => ( member_a @ ( Alpha @ R2 ) @ ( qbs_space_a @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_342_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > c,X2: quasi_borel_c,R2: real] :
      ( ( member_real_c @ Alpha @ ( qbs_Mx_c @ X2 ) )
     => ( member_c @ ( Alpha @ R2 ) @ ( qbs_space_c @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_343_insert__UNIV,axiom,
    ! [X3: nat] :
      ( ( insert_nat @ X3 @ top_top_set_nat )
      = top_top_set_nat ) ).

% insert_UNIV
thf(fact_344_insert__UNIV,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( ( insert7407984058720857448nnreal @ X3 @ top_to7994903218803871134nnreal )
      = top_to7994903218803871134nnreal ) ).

% insert_UNIV
thf(fact_345_insert__UNIV,axiom,
    ! [X3: real] :
      ( ( insert_real @ X3 @ top_top_set_real )
      = top_top_set_real ) ).

% insert_UNIV
thf(fact_346_insert__UNIV,axiom,
    ! [X3: $o] :
      ( ( insert_o @ X3 @ top_top_set_o )
      = top_top_set_o ) ).

% insert_UNIV
thf(fact_347_singletonD,axiom,
    ! [B: real > a,A: real > a] :
      ( ( member_real_a @ B @ ( insert_real_a @ A @ bot_bot_set_real_a ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_348_singletonD,axiom,
    ! [B: $o > real,A: $o > real] :
      ( ( member_o_real @ B @ ( insert_o_real @ A @ bot_bot_set_o_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_349_singletonD,axiom,
    ! [B: nat > real,A: nat > real] :
      ( ( member_nat_real @ B @ ( insert_nat_real @ A @ bot_bot_set_nat_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_350_singletonD,axiom,
    ! [B: c > b,A: c > b] :
      ( ( member_c_b @ B @ ( insert_c_b @ A @ bot_bot_set_c_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_351_singletonD,axiom,
    ! [B: a > b,A: a > b] :
      ( ( member_a_b @ B @ ( insert_a_b @ A @ bot_bot_set_a_b ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_352_singletonD,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_353_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_354_singletonD,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ B @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_355_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_356_singleton__iff,axiom,
    ! [B: real > a,A: real > a] :
      ( ( member_real_a @ B @ ( insert_real_a @ A @ bot_bot_set_real_a ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_357_singleton__iff,axiom,
    ! [B: $o > real,A: $o > real] :
      ( ( member_o_real @ B @ ( insert_o_real @ A @ bot_bot_set_o_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_358_singleton__iff,axiom,
    ! [B: nat > real,A: nat > real] :
      ( ( member_nat_real @ B @ ( insert_nat_real @ A @ bot_bot_set_nat_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_359_singleton__iff,axiom,
    ! [B: c > b,A: c > b] :
      ( ( member_c_b @ B @ ( insert_c_b @ A @ bot_bot_set_c_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_360_singleton__iff,axiom,
    ! [B: a > b,A: a > b] :
      ( ( member_a_b @ B @ ( insert_a_b @ A @ bot_bot_set_a_b ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_361_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_362_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_363_singleton__iff,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ B @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_364_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_365_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_366_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_367_doubleton__eq__iff,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
      ( ( ( insert7407984058720857448nnreal @ A @ ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) )
        = ( insert7407984058720857448nnreal @ C @ ( insert7407984058720857448nnreal @ D @ bot_bo4854962954004695426nnreal ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_368_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_369_insert__not__empty,axiom,
    ! [A: real,A4: set_real] :
      ( ( insert_real @ A @ A4 )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_370_insert__not__empty,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( insert_nat @ A @ A4 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_371_insert__not__empty,axiom,
    ! [A: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ A @ A4 )
     != bot_bo4854962954004695426nnreal ) ).

% insert_not_empty
thf(fact_372_insert__not__empty,axiom,
    ! [A: $o,A4: set_o] :
      ( ( insert_o @ A @ A4 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_373_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( ( insert_real @ A @ bot_bot_set_real )
        = ( insert_real @ B @ bot_bot_set_real ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_374_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_375_singleton__inject,axiom,
    ! [A: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal )
        = ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_376_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_377_qbs__morphismE_I2_J,axiom,
    ! [F: real > a,X2: quasi_borel_real,Y: quasi_borel_a,X3: real] :
      ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X2 @ Y ) )
     => ( ( member_real @ X3 @ ( qbs_space_real @ X2 ) )
       => ( member_a @ ( F @ X3 ) @ ( qbs_space_a @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_378_qbs__morphismE_I2_J,axiom,
    ! [F: $o > real,X2: quasi_borel_o,Y: quasi_borel_real,X3: $o] :
      ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X2 @ Y ) )
     => ( ( member_o @ X3 @ ( qbs_space_o @ X2 ) )
       => ( member_real @ ( F @ X3 ) @ ( qbs_space_real @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_379_qbs__morphismE_I2_J,axiom,
    ! [F: nat > real,X2: quasi_borel_nat,Y: quasi_borel_real,X3: nat] :
      ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X2 @ Y ) )
     => ( ( member_nat @ X3 @ ( qbs_space_nat @ X2 ) )
       => ( member_real @ ( F @ X3 ) @ ( qbs_space_real @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_380_qbs__morphismE_I2_J,axiom,
    ! [F: a > b,X2: quasi_borel_a,Y: quasi_borel_b,X3: a] :
      ( ( member_a_b @ F @ ( qbs_morphism_a_b @ X2 @ Y ) )
     => ( ( member_a @ X3 @ ( qbs_space_a @ X2 ) )
       => ( member_b @ ( F @ X3 ) @ ( qbs_space_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_381_qbs__morphismE_I2_J,axiom,
    ! [F: c > b,X2: quasi_borel_c,Y: quasi_borel_b,X3: c] :
      ( ( member_c_b @ F @ ( qbs_morphism_c_b @ X2 @ Y ) )
     => ( ( member_c @ X3 @ ( qbs_space_c @ X2 ) )
       => ( member_b @ ( F @ X3 ) @ ( qbs_space_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_382_qbs__morphismE_I2_J,axiom,
    ! [F: ( real > a ) > real > a,X2: quasi_borel_real_a,Y: quasi_borel_real_a,X3: real > a] :
      ( ( member_real_a_real_a @ F @ ( qbs_mo6715622035799359544real_a @ X2 @ Y ) )
     => ( ( member_real_a @ X3 @ ( qbs_space_real_a @ X2 ) )
       => ( member_real_a @ ( F @ X3 ) @ ( qbs_space_real_a @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_383_qbs__morphismE_I2_J,axiom,
    ! [F: ( real > a ) > $o > real,X2: quasi_borel_real_a,Y: quasi_borel_o_real,X3: real > a] :
      ( ( member_real_a_o_real @ F @ ( qbs_mo3511297643335615972o_real @ X2 @ Y ) )
     => ( ( member_real_a @ X3 @ ( qbs_space_real_a @ X2 ) )
       => ( member_o_real @ ( F @ X3 ) @ ( qbs_space_o_real @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_384_qbs__morphismE_I2_J,axiom,
    ! [F: ( real > a ) > nat > real,X2: quasi_borel_real_a,Y: quasi_borel_nat_real,X3: real > a] :
      ( ( member1170891962023204286t_real @ F @ ( qbs_mo6662696414006555820t_real @ X2 @ Y ) )
     => ( ( member_real_a @ X3 @ ( qbs_space_real_a @ X2 ) )
       => ( member_nat_real @ ( F @ X3 ) @ ( qbs_space_nat_real @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_385_qbs__morphismE_I2_J,axiom,
    ! [F: ( real > a ) > c > b,X2: quasi_borel_real_a,Y: quasi_borel_c_b,X3: real > a] :
      ( ( member_real_a_c_b @ F @ ( qbs_mo4284245206752562933_a_c_b @ X2 @ Y ) )
     => ( ( member_real_a @ X3 @ ( qbs_space_real_a @ X2 ) )
       => ( member_c_b @ ( F @ X3 ) @ ( qbs_space_c_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_386_qbs__morphismE_I2_J,axiom,
    ! [F: ( real > a ) > a > b,X2: quasi_borel_real_a,Y: quasi_borel_a_b,X3: real > a] :
      ( ( member_real_a_a_b @ F @ ( qbs_mo635784333843139191_a_a_b @ X2 @ Y ) )
     => ( ( member_real_a @ X3 @ ( qbs_space_real_a @ X2 ) )
       => ( member_a_b @ ( F @ X3 ) @ ( qbs_space_a_b @ Y ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_387_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_real,F: real > a,G: real > a,Y: quasi_borel_a] :
      ( ! [X: real] :
          ( ( member_real @ X @ ( qbs_space_real @ X2 ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( member_real_a @ F @ ( qbs_morphism_real_a @ X2 @ Y ) )
       => ( member_real_a @ G @ ( qbs_morphism_real_a @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_388_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_o,F: $o > real,G: $o > real,Y: quasi_borel_real] :
      ( ! [X: $o] :
          ( ( member_o @ X @ ( qbs_space_o @ X2 ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( member_o_real @ F @ ( qbs_morphism_o_real @ X2 @ Y ) )
       => ( member_o_real @ G @ ( qbs_morphism_o_real @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_389_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_nat,F: nat > real,G: nat > real,Y: quasi_borel_real] :
      ( ! [X: nat] :
          ( ( member_nat @ X @ ( qbs_space_nat @ X2 ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X2 @ Y ) )
       => ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_390_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_a,F: a > b,G: a > b,Y: quasi_borel_b] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( qbs_space_a @ X2 ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( member_a_b @ F @ ( qbs_morphism_a_b @ X2 @ Y ) )
       => ( member_a_b @ G @ ( qbs_morphism_a_b @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_391_qbs__morphism__cong,axiom,
    ! [X2: quasi_borel_c,F: c > b,G: c > b,Y: quasi_borel_b] :
      ( ! [X: c] :
          ( ( member_c @ X @ ( qbs_space_c @ X2 ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( member_c_b @ F @ ( qbs_morphism_c_b @ X2 @ Y ) )
       => ( member_c_b @ G @ ( qbs_morphism_c_b @ X2 @ Y ) ) ) ) ).

% qbs_morphism_cong
thf(fact_392_pred__sum_Ointros_I2_J,axiom,
    ! [P22: c > $o,B: c,P1: a > $o] :
      ( ( P22 @ B )
     => ( basic_pred_sum_a_c @ P1 @ P22 @ ( sum_Inr_c_a @ B ) ) ) ).

% pred_sum.intros(2)
thf(fact_393_pred__sum_Ointros_I1_J,axiom,
    ! [P1: a > $o,A: a,P22: c > $o] :
      ( ( P1 @ A )
     => ( basic_pred_sum_a_c @ P1 @ P22 @ ( sum_Inl_a_c @ A ) ) ) ).

% pred_sum.intros(1)
thf(fact_394_sets_Oinsert__in__sets,axiom,
    ! [X3: real,M: sigma_measure_real,A4: set_real] :
      ( ( member_set_real @ ( insert_real @ X3 @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real @ A4 @ ( sigma_sets_real @ M ) )
       => ( member_set_real @ ( insert_real @ X3 @ A4 ) @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_395_sets_Oinsert__in__sets,axiom,
    ! [X3: nat,M: sigma_measure_nat,A4: set_nat] :
      ( ( member_set_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat @ A4 @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat @ ( insert_nat @ X3 @ A4 ) @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_396_sets_Oinsert__in__sets,axiom,
    ! [X3: extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal @ A4 @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X3 @ A4 ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_397_sets_Oinsert__in__sets,axiom,
    ! [X3: $o,M: sigma_measure_o,A4: set_o] :
      ( ( member_set_o @ ( insert_o @ X3 @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o @ A4 @ ( sigma_sets_o @ M ) )
       => ( member_set_o @ ( insert_o @ X3 @ A4 ) @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_398_borel__singleton,axiom,
    ! [A4: set_real,X3: real] :
      ( ( member_set_real @ A4 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( member_set_real @ ( insert_real @ X3 @ A4 ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% borel_singleton
thf(fact_399_borel__singleton,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( member603777416030116741nnreal @ A4 @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
     => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X3 @ A4 ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).

% borel_singleton
thf(fact_400_borel__singleton,axiom,
    ! [A4: set_o,X3: $o] :
      ( ( member_set_o @ A4 @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) )
     => ( member_set_o @ ( insert_o @ X3 @ A4 ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% borel_singleton
thf(fact_401_borel__singleton,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( member_set_nat @ A4 @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) )
     => ( member_set_nat @ ( insert_nat @ X3 @ A4 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% borel_singleton
thf(fact_402_empty__quasi__borel__iff,axiom,
    ! [X2: quasi_borel_real] :
      ( ( ( qbs_space_real @ X2 )
        = bot_bot_set_real )
      = ( X2 = empty_1876425439295802446l_real ) ) ).

% empty_quasi_borel_iff
thf(fact_403_empty__quasi__borel__iff,axiom,
    ! [X2: quasi_borel_nat] :
      ( ( ( qbs_space_nat @ X2 )
        = bot_bot_set_nat )
      = ( X2 = empty_8278123436611590770el_nat ) ) ).

% empty_quasi_borel_iff
thf(fact_404_empty__quasi__borel__iff,axiom,
    ! [X2: quasi_9015997321629101608nnreal] :
      ( ( ( qbs_sp175953267596557954nnreal @ X2 )
        = bot_bo4854962954004695426nnreal )
      = ( X2 = empty_1788085430566700506nnreal ) ) ).

% empty_quasi_borel_iff
thf(fact_405_empty__quasi__borel__iff,axiom,
    ! [X2: quasi_borel_o] :
      ( ( ( qbs_space_o @ X2 )
        = bot_bot_set_o )
      = ( X2 = empty_quasi_borel_o ) ) ).

% empty_quasi_borel_iff
thf(fact_406_setrp_Ocases,axiom,
    ! [S2: sum_sum_a_c,A: c] :
      ( ( basic_setrp_a_c @ S2 @ A )
     => ( S2
        = ( sum_Inr_c_a @ A ) ) ) ).

% setrp.cases
thf(fact_407_setrp_Osimps,axiom,
    ( basic_setrp_a_c
    = ( ^ [S3: sum_sum_a_c,A6: c] :
        ? [X5: c] :
          ( ( A6 = X5 )
          & ( S3
            = ( sum_Inr_c_a @ X5 ) ) ) ) ) ).

% setrp.simps
thf(fact_408_setrp_Ointros,axiom,
    ! [S2: sum_sum_a_c,X3: c] :
      ( ( S2
        = ( sum_Inr_c_a @ X3 ) )
     => ( basic_setrp_a_c @ S2 @ X3 ) ) ).

% setrp.intros
thf(fact_409_setlp_Ointros,axiom,
    ! [S2: sum_sum_a_c,X3: a] :
      ( ( S2
        = ( sum_Inl_a_c @ X3 ) )
     => ( basic_setlp_a_c @ S2 @ X3 ) ) ).

% setlp.intros
thf(fact_410_setlp_Osimps,axiom,
    ( basic_setlp_a_c
    = ( ^ [S3: sum_sum_a_c,A6: a] :
        ? [X5: a] :
          ( ( A6 = X5 )
          & ( S3
            = ( sum_Inl_a_c @ X5 ) ) ) ) ) ).

% setlp.simps
thf(fact_411_setlp_Ocases,axiom,
    ! [S2: sum_sum_a_c,A: a] :
      ( ( basic_setlp_a_c @ S2 @ A )
     => ( S2
        = ( sum_Inl_a_c @ A ) ) ) ).

% setlp.cases
thf(fact_412_pred__sum_Osimps,axiom,
    ( basic_pred_sum_a_c
    = ( ^ [P12: a > $o,P23: c > $o,A6: sum_sum_a_c] :
          ( ? [B6: a] :
              ( ( A6
                = ( sum_Inl_a_c @ B6 ) )
              & ( P12 @ B6 ) )
          | ? [B6: c] :
              ( ( A6
                = ( sum_Inr_c_a @ B6 ) )
              & ( P23 @ B6 ) ) ) ) ) ).

% pred_sum.simps
thf(fact_413_pred__sum_Ocases,axiom,
    ! [P1: a > $o,P22: c > $o,A: sum_sum_a_c] :
      ( ( basic_pred_sum_a_c @ P1 @ P22 @ A )
     => ( ! [A3: a] :
            ( ( A
              = ( sum_Inl_a_c @ A3 ) )
           => ~ ( P1 @ A3 ) )
       => ~ ! [B3: c] :
              ( ( A
                = ( sum_Inr_c_a @ B3 ) )
             => ~ ( P22 @ B3 ) ) ) ) ).

% pred_sum.cases
thf(fact_414_sets__bot,axiom,
    ( ( sigma_sets_real @ bot_bo5982154664989874033e_real )
    = ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ).

% sets_bot
thf(fact_415_sets__bot,axiom,
    ( ( sigma_sets_nat @ bot_bo6718502177978453909re_nat )
    = ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).

% sets_bot
thf(fact_416_sets__bot,axiom,
    ( ( sigma_5465916536984168985nnreal @ bot_bo1740529460517930749nnreal )
    = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) ) ).

% sets_bot
thf(fact_417_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_418_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( top_to7994903218803871134nnreal
     != ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) ).

% perfect_space_class.UNIV_not_singleton
thf(fact_419_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [X3: real] :
      ( top_top_set_real
     != ( insert_real @ X3 @ bot_bot_set_real ) ) ).

% perfect_space_class.UNIV_not_singleton
thf(fact_420_ball__insert,axiom,
    ! [A: real,B4: set_real,P3: real > $o] :
      ( ( ! [X5: real] :
            ( ( member_real @ X5 @ ( insert_real @ A @ B4 ) )
           => ( P3 @ X5 ) ) )
      = ( ( P3 @ A )
        & ! [X5: real] :
            ( ( member_real @ X5 @ B4 )
           => ( P3 @ X5 ) ) ) ) ).

% ball_insert
thf(fact_421_ball__insert,axiom,
    ! [A: nat,B4: set_nat,P3: nat > $o] :
      ( ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( insert_nat @ A @ B4 ) )
           => ( P3 @ X5 ) ) )
      = ( ( P3 @ A )
        & ! [X5: nat] :
            ( ( member_nat @ X5 @ B4 )
           => ( P3 @ X5 ) ) ) ) ).

% ball_insert
thf(fact_422_ball__insert,axiom,
    ! [A: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal,P3: extend8495563244428889912nnreal > $o] :
      ( ( ! [X5: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X5 @ ( insert7407984058720857448nnreal @ A @ B4 ) )
           => ( P3 @ X5 ) ) )
      = ( ( P3 @ A )
        & ! [X5: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X5 @ B4 )
           => ( P3 @ X5 ) ) ) ) ).

% ball_insert
thf(fact_423_ball__insert,axiom,
    ! [A: $o,B4: set_o,P3: $o > $o] :
      ( ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( insert_o @ A @ B4 ) )
           => ( P3 @ X5 ) ) )
      = ( ( P3 @ A )
        & ! [X5: $o] :
            ( ( member_o @ X5 @ B4 )
           => ( P3 @ X5 ) ) ) ) ).

% ball_insert
thf(fact_424_is__singletonI,axiom,
    ! [X3: real] : ( is_singleton_real @ ( insert_real @ X3 @ bot_bot_set_real ) ) ).

% is_singletonI
thf(fact_425_is__singletonI,axiom,
    ! [X3: nat] : ( is_singleton_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_426_is__singletonI,axiom,
    ! [X3: extend8495563244428889912nnreal] : ( is_sin3654761921782142788nnreal @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) ).

% is_singletonI
thf(fact_427_is__singletonI,axiom,
    ! [X3: $o] : ( is_singleton_o @ ( insert_o @ X3 @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_428_sets__eq__bot,axiom,
    ! [M: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) )
      = ( M = bot_bo5982154664989874033e_real ) ) ).

% sets_eq_bot
thf(fact_429_sets__eq__bot,axiom,
    ! [M: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ M )
        = ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) )
      = ( M = bot_bo6718502177978453909re_nat ) ) ).

% sets_eq_bot
thf(fact_430_sets__eq__bot,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) )
      = ( M = bot_bo1740529460517930749nnreal ) ) ).

% sets_eq_bot
thf(fact_431_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_432_sets__eq__bot2,axiom,
    ! [M: sigma_measure_real] :
      ( ( ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real )
        = ( sigma_sets_real @ M ) )
      = ( M = bot_bo5982154664989874033e_real ) ) ).

% sets_eq_bot2
thf(fact_433_sets__eq__bot2,axiom,
    ! [M: sigma_measure_nat] :
      ( ( ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat )
        = ( sigma_sets_nat @ M ) )
      = ( M = bot_bo6718502177978453909re_nat ) ) ).

% sets_eq_bot2
thf(fact_434_sets__eq__bot2,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal )
        = ( sigma_5465916536984168985nnreal @ M ) )
      = ( M = bot_bo1740529460517930749nnreal ) ) ).

% sets_eq_bot2
thf(fact_435_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_436_the__elem__eq,axiom,
    ! [X3: real] :
      ( ( the_elem_real @ ( insert_real @ X3 @ bot_bot_set_real ) )
      = X3 ) ).

% the_elem_eq
thf(fact_437_the__elem__eq,axiom,
    ! [X3: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
      = X3 ) ).

% the_elem_eq
thf(fact_438_the__elem__eq,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( ( the_el3795950934141317635nnreal @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) )
      = X3 ) ).

% the_elem_eq
thf(fact_439_the__elem__eq,axiom,
    ! [X3: $o] :
      ( ( the_elem_o @ ( insert_o @ X3 @ bot_bot_set_o ) )
      = X3 ) ).

% the_elem_eq
thf(fact_440_is__singleton__def,axiom,
    ( is_singleton_real
    = ( ^ [A5: set_real] :
        ? [X5: real] :
          ( A5
          = ( insert_real @ X5 @ bot_bot_set_real ) ) ) ) ).

% is_singleton_def
thf(fact_441_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
        ? [X5: nat] :
          ( A5
          = ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_442_is__singleton__def,axiom,
    ( is_sin3654761921782142788nnreal
    = ( ^ [A5: set_Ex3793607809372303086nnreal] :
        ? [X5: extend8495563244428889912nnreal] :
          ( A5
          = ( insert7407984058720857448nnreal @ X5 @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% is_singleton_def
thf(fact_443_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
        ? [X5: $o] :
          ( A5
          = ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_444_is__singletonE,axiom,
    ! [A4: set_real] :
      ( ( is_singleton_real @ A4 )
     => ~ ! [X: real] :
            ( A4
           != ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% is_singletonE
thf(fact_445_is__singletonE,axiom,
    ! [A4: set_nat] :
      ( ( is_singleton_nat @ A4 )
     => ~ ! [X: nat] :
            ( A4
           != ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_446_is__singletonE,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( is_sin3654761921782142788nnreal @ A4 )
     => ~ ! [X: extend8495563244428889912nnreal] :
            ( A4
           != ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) ) ) ).

% is_singletonE
thf(fact_447_is__singletonE,axiom,
    ! [A4: set_o] :
      ( ( is_singleton_o @ A4 )
     => ~ ! [X: $o] :
            ( A4
           != ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_448_is__singleton__the__elem,axiom,
    ( is_singleton_real
    = ( ^ [A5: set_real] :
          ( A5
          = ( insert_real @ ( the_elem_real @ A5 ) @ bot_bot_set_real ) ) ) ) ).

% is_singleton_the_elem
thf(fact_449_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
          ( A5
          = ( insert_nat @ ( the_elem_nat @ A5 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_450_is__singleton__the__elem,axiom,
    ( is_sin3654761921782142788nnreal
    = ( ^ [A5: set_Ex3793607809372303086nnreal] :
          ( A5
          = ( insert7407984058720857448nnreal @ ( the_el3795950934141317635nnreal @ A5 ) @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% is_singleton_the_elem
thf(fact_451_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
          ( A5
          = ( insert_o @ ( the_elem_o @ A5 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_452_is__singletonI_H,axiom,
    ! [A4: set_real_a] :
      ( ( A4 != bot_bot_set_real_a )
     => ( ! [X: real > a,Y3: real > a] :
            ( ( member_real_a @ X @ A4 )
           => ( ( member_real_a @ Y3 @ A4 )
             => ( X = Y3 ) ) )
       => ( is_singleton_real_a @ A4 ) ) ) ).

% is_singletonI'
thf(fact_453_is__singletonI_H,axiom,
    ! [A4: set_o_real] :
      ( ( A4 != bot_bot_set_o_real )
     => ( ! [X: $o > real,Y3: $o > real] :
            ( ( member_o_real @ X @ A4 )
           => ( ( member_o_real @ Y3 @ A4 )
             => ( X = Y3 ) ) )
       => ( is_singleton_o_real @ A4 ) ) ) ).

% is_singletonI'
thf(fact_454_is__singletonI_H,axiom,
    ! [A4: set_nat_real] :
      ( ( A4 != bot_bot_set_nat_real )
     => ( ! [X: nat > real,Y3: nat > real] :
            ( ( member_nat_real @ X @ A4 )
           => ( ( member_nat_real @ Y3 @ A4 )
             => ( X = Y3 ) ) )
       => ( is_sin3816863272566379559t_real @ A4 ) ) ) ).

% is_singletonI'
thf(fact_455_is__singletonI_H,axiom,
    ! [A4: set_c_b] :
      ( ( A4 != bot_bot_set_c_b )
     => ( ! [X: c > b,Y3: c > b] :
            ( ( member_c_b @ X @ A4 )
           => ( ( member_c_b @ Y3 @ A4 )
             => ( X = Y3 ) ) )
       => ( is_singleton_c_b @ A4 ) ) ) ).

% is_singletonI'
thf(fact_456_is__singletonI_H,axiom,
    ! [A4: set_a_b] :
      ( ( A4 != bot_bot_set_a_b )
     => ( ! [X: a > b,Y3: a > b] :
            ( ( member_a_b @ X @ A4 )
           => ( ( member_a_b @ Y3 @ A4 )
             => ( X = Y3 ) ) )
       => ( is_singleton_a_b @ A4 ) ) ) ).

% is_singletonI'
thf(fact_457_is__singletonI_H,axiom,
    ! [A4: set_real] :
      ( ( A4 != bot_bot_set_real )
     => ( ! [X: real,Y3: real] :
            ( ( member_real @ X @ A4 )
           => ( ( member_real @ Y3 @ A4 )
             => ( X = Y3 ) ) )
       => ( is_singleton_real @ A4 ) ) ) ).

% is_singletonI'
thf(fact_458_is__singletonI_H,axiom,
    ! [A4: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X: nat,Y3: nat] :
            ( ( member_nat @ X @ A4 )
           => ( ( member_nat @ Y3 @ A4 )
             => ( X = Y3 ) ) )
       => ( is_singleton_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_459_is__singletonI_H,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( A4 != bot_bo4854962954004695426nnreal )
     => ( ! [X: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X @ A4 )
           => ( ( member7908768830364227535nnreal @ Y3 @ A4 )
             => ( X = Y3 ) ) )
       => ( is_sin3654761921782142788nnreal @ A4 ) ) ) ).

% is_singletonI'
thf(fact_460_is__singletonI_H,axiom,
    ! [A4: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X: $o,Y3: $o] :
            ( ( member_o @ X @ A4 )
           => ( ( member_o @ Y3 @ A4 )
             => ( X = Y3 ) ) )
       => ( is_singleton_o @ A4 ) ) ) ).

% is_singletonI'
thf(fact_461_space__empty__iff,axiom,
    ! [N: sigma_measure_real] :
      ( ( ( sigma_space_real @ N )
        = bot_bot_set_real )
      = ( ( sigma_sets_real @ N )
        = ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ) ).

% space_empty_iff
thf(fact_462_space__empty__iff,axiom,
    ! [N: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ N )
        = bot_bot_set_nat )
      = ( ( sigma_sets_nat @ N )
        = ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ) ).

% space_empty_iff
thf(fact_463_space__empty__iff,axiom,
    ! [N: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ N )
        = bot_bo4854962954004695426nnreal )
      = ( ( sigma_5465916536984168985nnreal @ N )
        = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) ) ) ).

% space_empty_iff
thf(fact_464_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_465_totally__bounded__empty,axiom,
    topolo4708875952704042879d_real @ bot_bot_set_real ).

% totally_bounded_empty
thf(fact_466_Pow__empty,axiom,
    ( ( pow_real @ bot_bot_set_real )
    = ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ).

% Pow_empty
thf(fact_467_Pow__empty,axiom,
    ( ( pow_nat @ bot_bot_set_nat )
    = ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).

% Pow_empty
thf(fact_468_Pow__empty,axiom,
    ( ( pow_Ex5372160365422184283nnreal @ bot_bo4854962954004695426nnreal )
    = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) ) ).

% Pow_empty
thf(fact_469_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_470_Pow__singleton__iff,axiom,
    ! [X2: set_real,Y: set_real] :
      ( ( ( pow_real @ X2 )
        = ( insert_set_real @ Y @ bot_bot_set_set_real ) )
      = ( ( X2 = bot_bot_set_real )
        & ( Y = bot_bot_set_real ) ) ) ).

% Pow_singleton_iff
thf(fact_471_Pow__singleton__iff,axiom,
    ! [X2: set_nat,Y: set_nat] :
      ( ( ( pow_nat @ X2 )
        = ( insert_set_nat @ Y @ bot_bot_set_set_nat ) )
      = ( ( X2 = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% Pow_singleton_iff
thf(fact_472_Pow__singleton__iff,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,Y: set_Ex3793607809372303086nnreal] :
      ( ( ( pow_Ex5372160365422184283nnreal @ X2 )
        = ( insert1343806209672318238nnreal @ Y @ bot_bo2988155216863113784nnreal ) )
      = ( ( X2 = bot_bo4854962954004695426nnreal )
        & ( Y = bot_bo4854962954004695426nnreal ) ) ) ).

% Pow_singleton_iff
thf(fact_473_Pow__singleton__iff,axiom,
    ! [X2: set_o,Y: set_o] :
      ( ( ( pow_o @ X2 )
        = ( insert_set_o @ Y @ bot_bot_set_set_o ) )
      = ( ( X2 = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% Pow_singleton_iff
thf(fact_474_sigma__sets__single,axiom,
    ! [A4: set_real] :
      ( ( sigma_7195353284648819924s_real @ A4 @ ( insert_set_real @ A4 @ bot_bot_set_set_real ) )
      = ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ A4 @ bot_bot_set_set_real ) ) ) ).

% sigma_sets_single
thf(fact_475_sigma__sets__single,axiom,
    ! [A4: set_nat] :
      ( ( sigma_sigma_sets_nat @ A4 @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) )
      = ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ).

% sigma_sets_single
thf(fact_476_sigma__sets__single,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( sigma_7808855514367478112nnreal @ A4 @ ( insert1343806209672318238nnreal @ A4 @ bot_bo2988155216863113784nnreal ) )
      = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ A4 @ bot_bo2988155216863113784nnreal ) ) ) ).

% sigma_sets_single
thf(fact_477_sigma__sets__single,axiom,
    ! [A4: set_o] :
      ( ( sigma_sigma_sets_o @ A4 @ ( insert_set_o @ A4 @ bot_bot_set_set_o ) )
      = ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ A4 @ bot_bot_set_set_o ) ) ) ).

% sigma_sets_single
thf(fact_478_sets__sup__measure_H,axiom,
    ! [B4: sigma_measure_real,A4: sigma_measure_real] :
      ( ( ( sigma_sets_real @ B4 )
        = ( sigma_sets_real @ A4 ) )
     => ( ( sigma_sets_real @ ( measur2147279183506585690e_real @ A4 @ B4 ) )
        = ( sigma_sets_real @ A4 ) ) ) ).

% sets_sup_measure'
thf(fact_479_sets__sup__measure_H,axiom,
    ! [B4: sigma_7234349610311085201nnreal,A4: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ B4 )
        = ( sigma_5465916536984168985nnreal @ A4 ) )
     => ( ( sigma_5465916536984168985nnreal @ ( measur4473656680840910822nnreal @ A4 @ B4 ) )
        = ( sigma_5465916536984168985nnreal @ A4 ) ) ) ).

% sets_sup_measure'
thf(fact_480_sets__sup__measure_H,axiom,
    ! [B4: sigma_measure_o,A4: sigma_measure_o] :
      ( ( ( sigma_sets_o @ B4 )
        = ( sigma_sets_o @ A4 ) )
     => ( ( sigma_sets_o @ ( measur4529518739368704874sure_o @ A4 @ B4 ) )
        = ( sigma_sets_o @ A4 ) ) ) ).

% sets_sup_measure'
thf(fact_481_sets__sup__measure_H,axiom,
    ! [B4: sigma_measure_nat,A4: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ B4 )
        = ( sigma_sets_nat @ A4 ) )
     => ( ( sigma_sets_nat @ ( measur876423496291765374re_nat @ A4 @ B4 ) )
        = ( sigma_sets_nat @ A4 ) ) ) ).

% sets_sup_measure'
thf(fact_482_sigma__algebra__trivial,axiom,
    ! [Omega: set_real] : ( sigma_1481383337440427903a_real @ Omega @ ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ Omega @ bot_bot_set_set_real ) ) ) ).

% sigma_algebra_trivial
thf(fact_483_sigma__algebra__trivial,axiom,
    ! [Omega: set_nat] : ( sigma_8817008012692346403ra_nat @ Omega @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ Omega @ bot_bot_set_set_nat ) ) ) ).

% sigma_algebra_trivial
thf(fact_484_sigma__algebra__trivial,axiom,
    ! [Omega: set_Ex3793607809372303086nnreal] : ( sigma_2413694886200424843nnreal @ Omega @ ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ Omega @ bot_bo2988155216863113784nnreal ) ) ) ).

% sigma_algebra_trivial
thf(fact_485_sigma__algebra__trivial,axiom,
    ! [Omega: set_o] : ( sigma_3687534776968752773ebra_o @ Omega @ ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ Omega @ bot_bot_set_set_o ) ) ) ).

% sigma_algebra_trivial
thf(fact_486_Set_Oball__empty,axiom,
    ! [P3: real > $o,X6: real] :
      ( ( member_real @ X6 @ bot_bot_set_real )
     => ( P3 @ X6 ) ) ).

% Set.ball_empty
thf(fact_487_Set_Oball__empty,axiom,
    ! [P3: nat > $o,X6: nat] :
      ( ( member_nat @ X6 @ bot_bot_set_nat )
     => ( P3 @ X6 ) ) ).

% Set.ball_empty
thf(fact_488_Set_Oball__empty,axiom,
    ! [P3: extend8495563244428889912nnreal > $o,X6: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ X6 @ bot_bo4854962954004695426nnreal )
     => ( P3 @ X6 ) ) ).

% Set.ball_empty
thf(fact_489_Set_Oball__empty,axiom,
    ! [P3: $o > $o,X6: $o] :
      ( ( member_o @ X6 @ bot_bot_set_o )
     => ( P3 @ X6 ) ) ).

% Set.ball_empty
thf(fact_490_sets_Otop,axiom,
    ! [M: sigma_measure_real] : ( member_set_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).

% sets.top
thf(fact_491_sets_Otop,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( member603777416030116741nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.top
thf(fact_492_sets_Otop,axiom,
    ! [M: sigma_measure_o] : ( member_set_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) ).

% sets.top
thf(fact_493_sets_Otop,axiom,
    ! [M: sigma_measure_nat] : ( member_set_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) ).

% sets.top
thf(fact_494_Pow__UNIV,axiom,
    ( ( pow_real @ top_top_set_real )
    = top_top_set_set_real ) ).

% Pow_UNIV
thf(fact_495_Pow__UNIV,axiom,
    ( ( pow_o @ top_top_set_o )
    = top_top_set_set_o ) ).

% Pow_UNIV
thf(fact_496_UNIV__Plus__UNIV,axiom,
    ( ( sum_Plus_real_real @ top_top_set_real @ top_top_set_real )
    = top_to8895904057622651549l_real ) ).

% UNIV_Plus_UNIV
thf(fact_497_UNIV__Plus__UNIV,axiom,
    ( ( sum_Plus_real_o @ top_top_set_real @ top_top_set_o )
    = top_to798167405261542675real_o ) ).

% UNIV_Plus_UNIV
thf(fact_498_UNIV__Plus__UNIV,axiom,
    ( ( sum_Plus_o_real @ top_top_set_o @ top_top_set_real )
    = top_to2434899875267808933o_real ) ).

% UNIV_Plus_UNIV
thf(fact_499_UNIV__Plus__UNIV,axiom,
    ( ( sum_Plus_o_o @ top_top_set_o @ top_top_set_o )
    = top_to1686961084667892491um_o_o ) ).

% UNIV_Plus_UNIV
thf(fact_500_Plus__eq__empty__conv,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( ( sum_Plus_real_real @ A4 @ B4 )
        = bot_bo3924869615403164801l_real )
      = ( ( A4 = bot_bot_set_real )
        & ( B4 = bot_bot_set_real ) ) ) ).

% Plus_eq_empty_conv
thf(fact_501_Plus__eq__empty__conv,axiom,
    ! [A4: set_real,B4: set_nat] :
      ( ( ( sum_Plus_real_nat @ A4 @ B4 )
        = bot_bo1032908899468160293al_nat )
      = ( ( A4 = bot_bot_set_real )
        & ( B4 = bot_bot_set_nat ) ) ) ).

% Plus_eq_empty_conv
thf(fact_502_Plus__eq__empty__conv,axiom,
    ! [A4: set_real,B4: set_Ex3793607809372303086nnreal] :
      ( ( ( sum_Pl8903377536230004441nnreal @ A4 @ B4 )
        = bot_bo1433428352774195341nnreal )
      = ( ( A4 = bot_bot_set_real )
        & ( B4 = bot_bo4854962954004695426nnreal ) ) ) ).

% Plus_eq_empty_conv
thf(fact_503_Plus__eq__empty__conv,axiom,
    ! [A4: set_real,B4: set_o] :
      ( ( ( sum_Plus_real_o @ A4 @ B4 )
        = bot_bo3440880874437255215real_o )
      = ( ( A4 = bot_bot_set_real )
        & ( B4 = bot_bot_set_o ) ) ) ).

% Plus_eq_empty_conv
thf(fact_504_Plus__eq__empty__conv,axiom,
    ! [A4: set_nat,B4: set_real] :
      ( ( ( sum_Plus_nat_real @ A4 @ B4 )
        = bot_bo7066286797129332901t_real )
      = ( ( A4 = bot_bot_set_nat )
        & ( B4 = bot_bot_set_real ) ) ) ).

% Plus_eq_empty_conv
thf(fact_505_Plus__eq__empty__conv,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( sum_Plus_nat_nat @ A4 @ B4 )
        = bot_bo8389910388014404425at_nat )
      = ( ( A4 = bot_bot_set_nat )
        & ( B4 = bot_bot_set_nat ) ) ) ).

% Plus_eq_empty_conv
thf(fact_506_Plus__eq__empty__conv,axiom,
    ! [A4: set_nat,B4: set_Ex3793607809372303086nnreal] :
      ( ( ( sum_Pl3187918063032462845nnreal @ A4 @ B4 )
        = bot_bo6227762595354431153nnreal )
      = ( ( A4 = bot_bot_set_nat )
        & ( B4 = bot_bo4854962954004695426nnreal ) ) ) ).

% Plus_eq_empty_conv
thf(fact_507_Plus__eq__empty__conv,axiom,
    ! [A4: set_nat,B4: set_o] :
      ( ( ( sum_Plus_nat_o @ A4 @ B4 )
        = bot_bo2414119734386878731_nat_o )
      = ( ( A4 = bot_bot_set_nat )
        & ( B4 = bot_bot_set_o ) ) ) ).

% Plus_eq_empty_conv
thf(fact_508_Plus__eq__empty__conv,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,B4: set_real] :
      ( ( ( sum_Pl6935631266779460185l_real @ A4 @ B4 )
        = bot_bo1763472996691639309l_real )
      = ( ( A4 = bot_bo4854962954004695426nnreal )
        & ( B4 = bot_bot_set_real ) ) ) ).

% Plus_eq_empty_conv
thf(fact_509_Plus__eq__empty__conv,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,B4: set_nat] :
      ( ( ( sum_Pl7961618503699322365al_nat @ A4 @ B4 )
        = bot_bo386220672133221041al_nat )
      = ( ( A4 = bot_bo4854962954004695426nnreal )
        & ( B4 = bot_bot_set_nat ) ) ) ).

% Plus_eq_empty_conv
thf(fact_510_InrI,axiom,
    ! [B: c,B4: set_c,A4: set_a] :
      ( ( member_c @ B @ B4 )
     => ( member_Sum_sum_a_c @ ( sum_Inr_c_a @ B ) @ ( sum_Plus_a_c @ A4 @ B4 ) ) ) ).

% InrI
thf(fact_511_InlI,axiom,
    ! [A: a,A4: set_a,B4: set_c] :
      ( ( member_a @ A @ A4 )
     => ( member_Sum_sum_a_c @ ( sum_Inl_a_c @ A ) @ ( sum_Plus_a_c @ A4 @ B4 ) ) ) ).

% InlI
thf(fact_512_space__borel,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% space_borel
thf(fact_513_space__borel,axiom,
    ( ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal )
    = top_to7994903218803871134nnreal ) ).

% space_borel
thf(fact_514_space__borel,axiom,
    ( ( sigma_space_o @ borel_5500255247093592246orel_o )
    = top_top_set_o ) ).

% space_borel
thf(fact_515_space__borel,axiom,
    ( ( sigma_space_nat @ borel_8449730974584783410el_nat )
    = top_top_set_nat ) ).

% space_borel
thf(fact_516_space__bot,axiom,
    ( ( sigma_space_real @ bot_bo5982154664989874033e_real )
    = bot_bot_set_real ) ).

% space_bot
thf(fact_517_space__bot,axiom,
    ( ( sigma_space_nat @ bot_bo6718502177978453909re_nat )
    = bot_bot_set_nat ) ).

% space_bot
thf(fact_518_space__bot,axiom,
    ( ( sigma_3147302497200244656nnreal @ bot_bo1740529460517930749nnreal )
    = bot_bo4854962954004695426nnreal ) ).

% space_bot
thf(fact_519_space__bot,axiom,
    ( ( sigma_space_o @ bot_bo5758314138661044393sure_o )
    = bot_bot_set_o ) ).

% space_bot
thf(fact_520_space__sup__measure_H,axiom,
    ! [B4: sigma_measure_real,A4: sigma_measure_real] :
      ( ( ( sigma_sets_real @ B4 )
        = ( sigma_sets_real @ A4 ) )
     => ( ( sigma_space_real @ ( measur2147279183506585690e_real @ A4 @ B4 ) )
        = ( sigma_space_real @ A4 ) ) ) ).

% space_sup_measure'
thf(fact_521_space__sup__measure_H,axiom,
    ! [B4: sigma_7234349610311085201nnreal,A4: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ B4 )
        = ( sigma_5465916536984168985nnreal @ A4 ) )
     => ( ( sigma_3147302497200244656nnreal @ ( measur4473656680840910822nnreal @ A4 @ B4 ) )
        = ( sigma_3147302497200244656nnreal @ A4 ) ) ) ).

% space_sup_measure'
thf(fact_522_space__sup__measure_H,axiom,
    ! [B4: sigma_measure_o,A4: sigma_measure_o] :
      ( ( ( sigma_sets_o @ B4 )
        = ( sigma_sets_o @ A4 ) )
     => ( ( sigma_space_o @ ( measur4529518739368704874sure_o @ A4 @ B4 ) )
        = ( sigma_space_o @ A4 ) ) ) ).

% space_sup_measure'
thf(fact_523_space__sup__measure_H,axiom,
    ! [B4: sigma_measure_nat,A4: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ B4 )
        = ( sigma_sets_nat @ A4 ) )
     => ( ( sigma_space_nat @ ( measur876423496291765374re_nat @ A4 @ B4 ) )
        = ( sigma_space_nat @ A4 ) ) ) ).

% space_sup_measure'
thf(fact_524_sets_Osigma__algebra__axioms,axiom,
    ! [M: sigma_measure_real] : ( sigma_1481383337440427903a_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).

% sets.sigma_algebra_axioms
thf(fact_525_sets_Osigma__algebra__axioms,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( sigma_2413694886200424843nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.sigma_algebra_axioms
thf(fact_526_sets_Osigma__algebra__axioms,axiom,
    ! [M: sigma_measure_o] : ( sigma_3687534776968752773ebra_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) ).

% sets.sigma_algebra_axioms
thf(fact_527_sets_Osigma__algebra__axioms,axiom,
    ! [M: sigma_measure_nat] : ( sigma_8817008012692346403ra_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) ).

% sets.sigma_algebra_axioms
thf(fact_528_Ball__def,axiom,
    ( ball_real_a
    = ( ^ [A5: set_real_a,P2: ( real > a ) > $o] :
        ! [X5: real > a] :
          ( ( member_real_a @ X5 @ A5 )
         => ( P2 @ X5 ) ) ) ) ).

% Ball_def
thf(fact_529_Ball__def,axiom,
    ( ball_o_real
    = ( ^ [A5: set_o_real,P2: ( $o > real ) > $o] :
        ! [X5: $o > real] :
          ( ( member_o_real @ X5 @ A5 )
         => ( P2 @ X5 ) ) ) ) ).

% Ball_def
thf(fact_530_Ball__def,axiom,
    ( ball_nat_real
    = ( ^ [A5: set_nat_real,P2: ( nat > real ) > $o] :
        ! [X5: nat > real] :
          ( ( member_nat_real @ X5 @ A5 )
         => ( P2 @ X5 ) ) ) ) ).

% Ball_def
thf(fact_531_Ball__def,axiom,
    ( ball_c_b
    = ( ^ [A5: set_c_b,P2: ( c > b ) > $o] :
        ! [X5: c > b] :
          ( ( member_c_b @ X5 @ A5 )
         => ( P2 @ X5 ) ) ) ) ).

% Ball_def
thf(fact_532_Ball__def,axiom,
    ( ball_a_b
    = ( ^ [A5: set_a_b,P2: ( a > b ) > $o] :
        ! [X5: a > b] :
          ( ( member_a_b @ X5 @ A5 )
         => ( P2 @ X5 ) ) ) ) ).

% Ball_def
thf(fact_533_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_measure_real] :
      ( ( sigma_7195353284648819924s_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) )
      = ( sigma_sets_real @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_534_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( sigma_7808855514367478112nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) )
      = ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_535_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_measure_o] :
      ( ( sigma_sigma_sets_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) )
      = ( sigma_sets_o @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_536_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_measure_nat] :
      ( ( sigma_sigma_sets_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) )
      = ( sigma_sets_nat @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_537_sigma__sets_OEmpty,axiom,
    ! [Sp: set_real,A4: set_set_real] : ( member_set_real @ bot_bot_set_real @ ( sigma_7195353284648819924s_real @ Sp @ A4 ) ) ).

% sigma_sets.Empty
thf(fact_538_sigma__sets_OEmpty,axiom,
    ! [Sp: set_nat,A4: set_set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( sigma_sigma_sets_nat @ Sp @ A4 ) ) ).

% sigma_sets.Empty
thf(fact_539_sigma__sets_OEmpty,axiom,
    ! [Sp: set_Ex3793607809372303086nnreal,A4: set_se4580700918925141924nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( sigma_7808855514367478112nnreal @ Sp @ A4 ) ) ).

% sigma_sets.Empty
thf(fact_540_sigma__sets_OEmpty,axiom,
    ! [Sp: set_o,A4: set_set_o] : ( member_set_o @ bot_bot_set_o @ ( sigma_sigma_sets_o @ Sp @ A4 ) ) ).

% sigma_sets.Empty
thf(fact_541_Pow__bottom,axiom,
    ! [B4: set_real] : ( member_set_real @ bot_bot_set_real @ ( pow_real @ B4 ) ) ).

% Pow_bottom
thf(fact_542_Pow__bottom,axiom,
    ! [B4: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B4 ) ) ).

% Pow_bottom
thf(fact_543_Pow__bottom,axiom,
    ! [B4: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( pow_Ex5372160365422184283nnreal @ B4 ) ) ).

% Pow_bottom
thf(fact_544_Pow__bottom,axiom,
    ! [B4: set_o] : ( member_set_o @ bot_bot_set_o @ ( pow_o @ B4 ) ) ).

% Pow_bottom
thf(fact_545_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_real,M2: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M2 ) )
     => ( ( sigma_space_real @ M )
        = ( sigma_space_real @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_546_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( sigma_5465916536984168985nnreal @ M2 ) )
     => ( ( sigma_3147302497200244656nnreal @ M )
        = ( sigma_3147302497200244656nnreal @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_547_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_o,M2: sigma_measure_o] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ M2 ) )
     => ( ( sigma_space_o @ M )
        = ( sigma_space_o @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_548_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_nat,M2: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ M2 ) )
     => ( ( sigma_space_nat @ M )
        = ( sigma_space_nat @ M2 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_549_PlusE,axiom,
    ! [U: sum_sum_a_c,A4: set_a,B4: set_c] :
      ( ( member_Sum_sum_a_c @ U @ ( sum_Plus_a_c @ A4 @ B4 ) )
     => ( ! [X: a] :
            ( ( member_a @ X @ A4 )
           => ( U
             != ( sum_Inl_a_c @ X ) ) )
       => ~ ! [Y3: c] :
              ( ( member_c @ Y3 @ B4 )
             => ( U
               != ( sum_Inr_c_a @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_550_PlusE,axiom,
    ! [U: sum_su2571395965866611557real_a,A4: set_real_a,B4: set_real_a] :
      ( ( member6990135608157631996real_a @ U @ ( sum_Pl5077908370370176563real_a @ A4 @ B4 ) )
     => ( ! [X: real > a] :
            ( ( member_real_a @ X @ A4 )
           => ( U
             != ( sum_In8182254581923447214real_a @ X ) ) )
       => ~ ! [Y3: real > a] :
              ( ( member_real_a @ Y3 @ B4 )
             => ( U
               != ( sum_In8250409331706822324real_a @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_551_PlusE,axiom,
    ! [U: sum_su2067798924538045457o_real,A4: set_real_a,B4: set_o_real] :
      ( ( member8606992278910886440o_real @ U @ ( sum_Pl2066794120620407135o_real @ A4 @ B4 ) )
     => ( ! [X: real > a] :
            ( ( member_real_a @ X @ A4 )
           => ( U
             != ( sum_In9033506258822570586o_real @ X ) ) )
       => ~ ! [Y3: $o > real] :
              ( ( member_o_real @ Y3 @ B4 )
             => ( U
               != ( sum_In5606109791546287624real_a @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_552_PlusE,axiom,
    ! [U: sum_su5472343219575513685t_real,A4: set_real_a,B4: set_nat_real] :
      ( ( member1761293007767394302t_real @ U @ ( sum_Pl6872154006543717873t_real @ A4 @ B4 ) )
     => ( ! [X: real > a] :
            ( ( member_real_a @ X @ A4 )
           => ( U
             != ( sum_In5343589694070989238t_real @ X ) ) )
       => ~ ! [Y3: nat > real] :
              ( ( member_nat_real @ Y3 @ B4 )
             => ( U
               != ( sum_In8909204494147139768real_a @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_553_PlusE,axiom,
    ! [U: sum_sum_real_a_c_b,A4: set_real_a,B4: set_c_b] :
      ( ( member5294325084747020743_a_c_b @ U @ ( sum_Plus_real_a_c_b @ A4 @ B4 ) )
     => ( ! [X: real > a] :
            ( ( member_real_a @ X @ A4 )
           => ( U
             != ( sum_Inl_real_a_c_b @ X ) ) )
       => ~ ! [Y3: c > b] :
              ( ( member_c_b @ Y3 @ B4 )
             => ( U
               != ( sum_Inr_c_b_real_a @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_554_PlusE,axiom,
    ! [U: sum_sum_real_a_a_b,A4: set_real_a,B4: set_a_b] :
      ( ( member6565623085956994121_a_a_b @ U @ ( sum_Plus_real_a_a_b @ A4 @ B4 ) )
     => ( ! [X: real > a] :
            ( ( member_real_a @ X @ A4 )
           => ( U
             != ( sum_Inl_real_a_a_b @ X ) ) )
       => ~ ! [Y3: a > b] :
              ( ( member_a_b @ Y3 @ B4 )
             => ( U
               != ( sum_Inr_a_b_real_a @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_555_PlusE,axiom,
    ! [U: sum_su7886454506223791033real_a,A4: set_o_real,B4: set_real_a] :
      ( ( member5202275823741856208real_a @ U @ ( sum_Pl4187333782930447623real_a @ A4 @ B4 ) )
     => ( ! [X: $o > real] :
            ( ( member_o_real @ X @ A4 )
           => ( U
             != ( sum_In1930673884277835266real_a @ X ) ) )
       => ~ ! [Y3: real > a] :
              ( ( member_real_a @ Y3 @ B4 )
             => ( U
               != ( sum_In3485570129236247136o_real @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_556_PlusE,axiom,
    ! [U: sum_su815935806896055909o_real,A4: set_o_real,B4: set_o_real] :
      ( ( member4866819706902675452o_real @ U @ ( sum_Pl8533773182915009075o_real @ A4 @ B4 ) )
     => ( ! [X: $o > real] :
            ( ( member_o_real @ X @ A4 )
           => ( U
             != ( sum_In8123802879950721198o_real @ X ) ) )
       => ~ ! [Y3: $o > real] :
              ( ( member_o_real @ Y3 @ B4 )
             => ( U
               != ( sum_In868780623499809716o_real @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_557_PlusE,axiom,
    ! [U: sum_su7765403328973102337t_real,A4: set_o_real,B4: set_nat_real] :
      ( ( member90129604467644202t_real @ U @ ( sum_Pl5478791264714117661t_real @ A4 @ B4 ) )
     => ( ! [X: $o > real] :
            ( ( member_o_real @ X @ A4 )
           => ( U
             != ( sum_In4218321019262890082t_real @ X ) ) )
       => ~ ! [Y3: nat > real] :
              ( ( member_nat_real @ Y3 @ B4 )
             => ( U
               != ( sum_In7724691659553554020o_real @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_558_PlusE,axiom,
    ! [U: sum_sum_o_real_c_b,A4: set_o_real,B4: set_c_b] :
      ( ( member5560441923647006963al_c_b @ U @ ( sum_Plus_o_real_c_b @ A4 @ B4 ) )
     => ( ! [X: $o > real] :
            ( ( member_o_real @ X @ A4 )
           => ( U
             != ( sum_Inl_o_real_c_b @ X ) ) )
       => ~ ! [Y3: c > b] :
              ( ( member_c_b @ Y3 @ B4 )
             => ( U
               != ( sum_Inr_c_b_o_real @ Y3 ) ) ) ) ) ).

% PlusE
thf(fact_559_space__empty__eq__bot,axiom,
    ! [A: sigma_measure_real] :
      ( ( ( sigma_space_real @ A )
        = bot_bot_set_real )
      = ( A = bot_bo5982154664989874033e_real ) ) ).

% space_empty_eq_bot
thf(fact_560_space__empty__eq__bot,axiom,
    ! [A: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ A )
        = bot_bot_set_nat )
      = ( A = bot_bo6718502177978453909re_nat ) ) ).

% space_empty_eq_bot
thf(fact_561_space__empty__eq__bot,axiom,
    ! [A: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ A )
        = bot_bo4854962954004695426nnreal )
      = ( A = bot_bo1740529460517930749nnreal ) ) ).

% space_empty_eq_bot
thf(fact_562_space__empty__eq__bot,axiom,
    ! [A: sigma_measure_o] :
      ( ( ( sigma_space_o @ A )
        = bot_bot_set_o )
      = ( A = bot_bo5758314138661044393sure_o ) ) ).

% space_empty_eq_bot
thf(fact_563_sigma__sets__empty__eq,axiom,
    ! [A4: set_real] :
      ( ( sigma_7195353284648819924s_real @ A4 @ bot_bot_set_set_real )
      = ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ A4 @ bot_bot_set_set_real ) ) ) ).

% sigma_sets_empty_eq
thf(fact_564_sigma__sets__empty__eq,axiom,
    ! [A4: set_nat] :
      ( ( sigma_sigma_sets_nat @ A4 @ bot_bot_set_set_nat )
      = ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ).

% sigma_sets_empty_eq
thf(fact_565_sigma__sets__empty__eq,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( sigma_7808855514367478112nnreal @ A4 @ bot_bo2988155216863113784nnreal )
      = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ A4 @ bot_bo2988155216863113784nnreal ) ) ) ).

% sigma_sets_empty_eq
thf(fact_566_sigma__sets__empty__eq,axiom,
    ! [A4: set_o] :
      ( ( sigma_sigma_sets_o @ A4 @ bot_bot_set_set_o )
      = ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ A4 @ bot_bot_set_set_o ) ) ) ).

% sigma_sets_empty_eq
thf(fact_567_real_Osingleton__sets,axiom,
    ! [X3: real] :
      ( ( member_real @ X3 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( member_set_real @ ( insert_real @ X3 @ bot_bot_set_real ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% real.singleton_sets
thf(fact_568_real_Ospace__UNIV,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% real.space_UNIV
thf(fact_569_copair__qbs__closed2,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_c] : ( qbs_cl8237352598534793137um_a_c @ ( sum_Plus_a_c @ ( qbs_space_a @ X2 ) @ ( qbs_space_c @ Y ) ) @ ( binary8286901584692334522Mx_a_c @ X2 @ Y ) ) ).

% copair_qbs_closed2
thf(fact_570_space__Sup__measure_H,axiom,
    ! [M: set_Si6059263944882162789e_real,A4: sigma_measure_real] :
      ( ! [M3: sigma_measure_real] :
          ( ( member4553183543495551918e_real @ M3 @ M )
         => ( ( sigma_sets_real @ M3 )
            = ( sigma_sets_real @ A4 ) ) )
     => ( ( M != bot_bo5686449298802467025e_real )
       => ( ( sigma_space_real @ ( measur8657758558638653562e_real @ M ) )
          = ( sigma_space_real @ A4 ) ) ) ) ).

% space_Sup_measure'
thf(fact_571_space__Sup__measure_H,axiom,
    ! [M: set_Si97717610131227249nnreal,A4: sigma_7234349610311085201nnreal] :
      ( ! [M3: sigma_7234349610311085201nnreal] :
          ( ( member6261374078160781754nnreal @ M3 @ M )
         => ( ( sigma_5465916536984168985nnreal @ M3 )
            = ( sigma_5465916536984168985nnreal @ A4 ) ) )
     => ( ( M != bot_bo8227844048696536285nnreal )
       => ( ( sigma_3147302497200244656nnreal @ ( measur1651139276328235014nnreal @ M ) )
          = ( sigma_3147302497200244656nnreal @ A4 ) ) ) ) ).

% space_Sup_measure'
thf(fact_572_space__Sup__measure_H,axiom,
    ! [M: set_Sigma_measure_o,A4: sigma_measure_o] :
      ( ! [M3: sigma_measure_o] :
          ( ( member1844656263901471916sure_o @ M3 @ M )
         => ( ( sigma_sets_o @ M3 )
            = ( sigma_sets_o @ A4 ) ) )
     => ( ( M != bot_bo7838039659004643295sure_o )
       => ( ( sigma_space_o @ ( measur1214336222341667658sure_o @ M ) )
          = ( sigma_space_o @ A4 ) ) ) ) ).

% space_Sup_measure'
thf(fact_573_space__Sup__measure_H,axiom,
    ! [M: set_Si3048223896905877257re_nat,A4: sigma_measure_nat] :
      ( ! [M3: sigma_measure_nat] :
          ( ( member4416920341759242834re_nat @ M3 @ M )
         => ( ( sigma_sets_nat @ M3 )
            = ( sigma_sets_nat @ A4 ) ) )
     => ( ( M != bot_bo8872222457363190133re_nat )
       => ( ( sigma_space_nat @ ( measur3575099672463795358re_nat @ M ) )
          = ( sigma_space_nat @ A4 ) ) ) ) ).

% space_Sup_measure'
thf(fact_574_borel__sigma__sets__subset,axiom,
    ! [A4: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A4 @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ top_top_set_real @ A4 ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% borel_sigma_sets_subset
thf(fact_575_borel__sigma__sets__subset,axiom,
    ! [A4: set_se4580700918925141924nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A4 @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
     => ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ top_to7994903218803871134nnreal @ A4 ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).

% borel_sigma_sets_subset
thf(fact_576_borel__sigma__sets__subset,axiom,
    ! [A4: set_set_o] :
      ( ( ord_le4374716579403074808_set_o @ A4 @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) )
     => ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ top_top_set_o @ A4 ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% borel_sigma_sets_subset
thf(fact_577_borel__sigma__sets__subset,axiom,
    ! [A4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) )
     => ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ top_top_set_nat @ A4 ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% borel_sigma_sets_subset
thf(fact_578_sets__eq__countable,axiom,
    ! [A4: set_real_a,M: sigma_measure_real_a] :
      ( ( counta6639396083684174020real_a @ A4 )
     => ( ( ( sigma_space_real_a @ M )
          = A4 )
       => ( ! [X: real > a] :
              ( ( member_real_a @ X @ A4 )
             => ( member_set_real_a @ ( insert_real_a @ X @ bot_bot_set_real_a ) @ ( sigma_sets_real_a @ M ) ) )
         => ( ( sigma_sets_real_a @ M )
            = ( pow_real_a @ A4 ) ) ) ) ) ).

% sets_eq_countable
thf(fact_579_sets__eq__countable,axiom,
    ! [A4: set_o_real,M: sigma_measure_o_real] :
      ( ( counta8783200249485735024o_real @ A4 )
     => ( ( ( sigma_space_o_real @ M )
          = A4 )
       => ( ! [X: $o > real] :
              ( ( member_o_real @ X @ A4 )
             => ( member_set_o_real @ ( insert_o_real @ X @ bot_bot_set_o_real ) @ ( sigma_sets_o_real @ M ) ) )
         => ( ( sigma_sets_o_real @ M )
            = ( pow_o_real @ A4 ) ) ) ) ) ).

% sets_eq_countable
thf(fact_580_sets__eq__countable,axiom,
    ! [A4: set_nat_real,M: sigma_3396294578489551860t_real] :
      ( ( counta2162411829015494944t_real @ A4 )
     => ( ( ( sigma_space_nat_real @ M )
          = A4 )
       => ( ! [X: nat > real] :
              ( ( member_nat_real @ X @ A4 )
             => ( member_set_nat_real @ ( insert_nat_real @ X @ bot_bot_set_nat_real ) @ ( sigma_sets_nat_real @ M ) ) )
         => ( ( sigma_sets_nat_real @ M )
            = ( pow_nat_real @ A4 ) ) ) ) ) ).

% sets_eq_countable
thf(fact_581_sets__eq__countable,axiom,
    ! [A4: set_c_b,M: sigma_measure_c_b] :
      ( ( counta2657777928882154345le_c_b @ A4 )
     => ( ( ( sigma_space_c_b @ M )
          = A4 )
       => ( ! [X: c > b] :
              ( ( member_c_b @ X @ A4 )
             => ( member_set_c_b @ ( insert_c_b @ X @ bot_bot_set_c_b ) @ ( sigma_sets_c_b @ M ) ) )
         => ( ( sigma_sets_c_b @ M )
            = ( pow_c_b @ A4 ) ) ) ) ) ).

% sets_eq_countable
thf(fact_582_sets__eq__countable,axiom,
    ! [A4: set_a_b,M: sigma_measure_a_b] :
      ( ( counta8232689092827506411le_a_b @ A4 )
     => ( ( ( sigma_space_a_b @ M )
          = A4 )
       => ( ! [X: a > b] :
              ( ( member_a_b @ X @ A4 )
             => ( member_set_a_b @ ( insert_a_b @ X @ bot_bot_set_a_b ) @ ( sigma_sets_a_b @ M ) ) )
         => ( ( sigma_sets_a_b @ M )
            = ( pow_a_b @ A4 ) ) ) ) ) ).

% sets_eq_countable
thf(fact_583_sets__eq__countable,axiom,
    ! [A4: set_real,M: sigma_measure_real] :
      ( ( counta7319604579010473777e_real @ A4 )
     => ( ( ( sigma_space_real @ M )
          = A4 )
       => ( ! [X: real] :
              ( ( member_real @ X @ A4 )
             => ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
         => ( ( sigma_sets_real @ M )
            = ( pow_real @ A4 ) ) ) ) ) ).

% sets_eq_countable
thf(fact_584_sets__eq__countable,axiom,
    ! [A4: set_nat,M: sigma_measure_nat] :
      ( ( counta1168086296615599829le_nat @ A4 )
     => ( ( ( sigma_space_nat @ M )
          = A4 )
       => ( ! [X: nat] :
              ( ( member_nat @ X @ A4 )
             => ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) ) )
         => ( ( sigma_sets_nat @ M )
            = ( pow_nat @ A4 ) ) ) ) ) ).

% sets_eq_countable
thf(fact_585_sets__eq__countable,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( counta8439243037236335165nnreal @ A4 )
     => ( ( ( sigma_3147302497200244656nnreal @ M )
          = A4 )
       => ( ! [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ A4 )
             => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
         => ( ( sigma_5465916536984168985nnreal @ M )
            = ( pow_Ex5372160365422184283nnreal @ A4 ) ) ) ) ) ).

% sets_eq_countable
thf(fact_586_sets__eq__countable,axiom,
    ! [A4: set_o,M: sigma_measure_o] :
      ( ( counta5976203206615340371able_o @ A4 )
     => ( ( ( sigma_space_o @ M )
          = A4 )
       => ( ! [X: $o] :
              ( ( member_o @ X @ A4 )
             => ( member_set_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) ) )
         => ( ( sigma_sets_o @ M )
            = ( pow_o @ A4 ) ) ) ) ) ).

% sets_eq_countable
thf(fact_587_sigma__sets__le__sets__iff,axiom,
    ! [X3: sigma_measure_real,A7: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ ( sigma_space_real @ X3 ) @ A7 ) @ ( sigma_sets_real @ X3 ) )
      = ( ord_le3558479182127378552t_real @ A7 @ ( sigma_sets_real @ X3 ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_588_sigma__sets__le__sets__iff,axiom,
    ! [X3: sigma_7234349610311085201nnreal,A7: set_se4580700918925141924nnreal] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ ( sigma_3147302497200244656nnreal @ X3 ) @ A7 ) @ ( sigma_5465916536984168985nnreal @ X3 ) )
      = ( ord_le3366939622266546180nnreal @ A7 @ ( sigma_5465916536984168985nnreal @ X3 ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_589_sigma__sets__le__sets__iff,axiom,
    ! [X3: sigma_measure_o,A7: set_set_o] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ ( sigma_space_o @ X3 ) @ A7 ) @ ( sigma_sets_o @ X3 ) )
      = ( ord_le4374716579403074808_set_o @ A7 @ ( sigma_sets_o @ X3 ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_590_sigma__sets__le__sets__iff,axiom,
    ! [X3: sigma_measure_nat,A7: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ ( sigma_space_nat @ X3 ) @ A7 ) @ ( sigma_sets_nat @ X3 ) )
      = ( ord_le6893508408891458716et_nat @ A7 @ ( sigma_sets_nat @ X3 ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_591_sets_Osigma__sets__subset,axiom,
    ! [A: set_set_real,M: sigma_measure_real] :
      ( ( ord_le3558479182127378552t_real @ A @ ( sigma_sets_real @ M ) )
     => ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ ( sigma_space_real @ M ) @ A ) @ ( sigma_sets_real @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_592_sets_Osigma__sets__subset,axiom,
    ! [A: set_se4580700918925141924nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ A ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_593_sets_Osigma__sets__subset,axiom,
    ! [A: set_set_o,M: sigma_measure_o] :
      ( ( ord_le4374716579403074808_set_o @ A @ ( sigma_sets_o @ M ) )
     => ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ ( sigma_space_o @ M ) @ A ) @ ( sigma_sets_o @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_594_sets_Osigma__sets__subset,axiom,
    ! [A: set_set_nat,M: sigma_measure_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ ( sigma_sets_nat @ M ) )
     => ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ ( sigma_space_nat @ M ) @ A ) @ ( sigma_sets_nat @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_595_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_596_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_597_subsetI,axiom,
    ! [A4: set_real_a,B4: set_real_a] :
      ( ! [X: real > a] :
          ( ( member_real_a @ X @ A4 )
         => ( member_real_a @ X @ B4 ) )
     => ( ord_le5743406823621094409real_a @ A4 @ B4 ) ) ).

% subsetI
thf(fact_598_subsetI,axiom,
    ! [A4: set_o_real,B4: set_o_real] :
      ( ! [X: $o > real] :
          ( ( member_o_real @ X @ A4 )
         => ( member_o_real @ X @ B4 ) )
     => ( ord_le3251842697534426805o_real @ A4 @ B4 ) ) ).

% subsetI
thf(fact_599_subsetI,axiom,
    ! [A4: set_nat_real,B4: set_nat_real] :
      ( ! [X: nat > real] :
          ( ( member_nat_real @ X @ A4 )
         => ( member_nat_real @ X @ B4 ) )
     => ( ord_le2908806416726583473t_real @ A4 @ B4 ) ) ).

% subsetI
thf(fact_600_subsetI,axiom,
    ! [A4: set_c_b,B4: set_c_b] :
      ( ! [X: c > b] :
          ( ( member_c_b @ X @ A4 )
         => ( member_c_b @ X @ B4 ) )
     => ( ord_less_eq_set_c_b @ A4 @ B4 ) ) ).

% subsetI
thf(fact_601_subsetI,axiom,
    ! [A4: set_a_b,B4: set_a_b] :
      ( ! [X: a > b] :
          ( ( member_a_b @ X @ A4 )
         => ( member_a_b @ X @ B4 ) )
     => ( ord_less_eq_set_a_b @ A4 @ B4 ) ) ).

% subsetI
thf(fact_602_empty__subsetI,axiom,
    ! [A4: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A4 ) ).

% empty_subsetI
thf(fact_603_empty__subsetI,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).

% empty_subsetI
thf(fact_604_empty__subsetI,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ bot_bo4854962954004695426nnreal @ A4 ) ).

% empty_subsetI
thf(fact_605_empty__subsetI,axiom,
    ! [A4: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A4 ) ).

% empty_subsetI
thf(fact_606_subset__empty,axiom,
    ! [A4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ bot_bot_set_real )
      = ( A4 = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_607_subset__empty,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
      = ( A4 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_608_subset__empty,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A4 @ bot_bo4854962954004695426nnreal )
      = ( A4 = bot_bo4854962954004695426nnreal ) ) ).

% subset_empty
thf(fact_609_subset__empty,axiom,
    ! [A4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ bot_bot_set_o )
      = ( A4 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_610_insert__subset,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X3 @ A4 ) @ B4 )
      = ( ( member_real @ X3 @ B4 )
        & ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_611_insert__subset,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A4 ) @ B4 )
      = ( ( member_nat @ X3 @ B4 )
        & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_612_insert__subset,axiom,
    ! [X3: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ ( insert7407984058720857448nnreal @ X3 @ A4 ) @ B4 )
      = ( ( member7908768830364227535nnreal @ X3 @ B4 )
        & ( ord_le6787938422905777998nnreal @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_613_insert__subset,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
      = ( ( member_o @ X3 @ B4 )
        & ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_614_insert__subset,axiom,
    ! [X3: real > a,A4: set_real_a,B4: set_real_a] :
      ( ( ord_le5743406823621094409real_a @ ( insert_real_a @ X3 @ A4 ) @ B4 )
      = ( ( member_real_a @ X3 @ B4 )
        & ( ord_le5743406823621094409real_a @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_615_insert__subset,axiom,
    ! [X3: $o > real,A4: set_o_real,B4: set_o_real] :
      ( ( ord_le3251842697534426805o_real @ ( insert_o_real @ X3 @ A4 ) @ B4 )
      = ( ( member_o_real @ X3 @ B4 )
        & ( ord_le3251842697534426805o_real @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_616_insert__subset,axiom,
    ! [X3: nat > real,A4: set_nat_real,B4: set_nat_real] :
      ( ( ord_le2908806416726583473t_real @ ( insert_nat_real @ X3 @ A4 ) @ B4 )
      = ( ( member_nat_real @ X3 @ B4 )
        & ( ord_le2908806416726583473t_real @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_617_insert__subset,axiom,
    ! [X3: c > b,A4: set_c_b,B4: set_c_b] :
      ( ( ord_less_eq_set_c_b @ ( insert_c_b @ X3 @ A4 ) @ B4 )
      = ( ( member_c_b @ X3 @ B4 )
        & ( ord_less_eq_set_c_b @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_618_insert__subset,axiom,
    ! [X3: a > b,A4: set_a_b,B4: set_a_b] :
      ( ( ord_less_eq_set_a_b @ ( insert_a_b @ X3 @ A4 ) @ B4 )
      = ( ( member_a_b @ X3 @ B4 )
        & ( ord_less_eq_set_a_b @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_619_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A4: set_real] :
      ( ( ( insert_real @ B @ bot_bot_set_real )
        = ( insert_real @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A4 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_620_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A4: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_621_singleton__insert__inj__eq,axiom,
    ! [B: extend8495563244428889912nnreal,A: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal )
        = ( insert7407984058720857448nnreal @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_le6787938422905777998nnreal @ A4 @ ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_622_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A4: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_623_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A4: set_real,B: real] :
      ( ( ( insert_real @ A @ A4 )
        = ( insert_real @ B @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A4 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_624_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A4: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A4 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_625_singleton__insert__inj__eq_H,axiom,
    ! [A: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( insert7407984058720857448nnreal @ A @ A4 )
        = ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) )
      = ( ( A = B )
        & ( ord_le6787938422905777998nnreal @ A4 @ ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_626_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A4: set_o,B: $o] :
      ( ( ( insert_o @ A @ A4 )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_627_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_628_le__cases3,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y2 )
       => ~ ( ord_less_eq_nat @ Y2 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y2 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y2 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y2 )
             => ~ ( ord_less_eq_nat @ Y2 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y2 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y2 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_629_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [X5: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_630_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_631_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_632_order__antisym,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ X3 )
       => ( X3 = Y2 ) ) ) ).

% order_antisym
thf(fact_633_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_634_order__trans,axiom,
    ! [X3: nat,Y2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ( ord_less_eq_nat @ Y2 @ Z )
       => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_635_linorder__wlog,axiom,
    ! [P3: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P3 @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P3 @ B3 @ A3 )
           => ( P3 @ A3 @ B3 ) )
       => ( P3 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_636_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_637_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_638_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_639_in__mono,axiom,
    ! [A4: set_real_a,B4: set_real_a,X3: real > a] :
      ( ( ord_le5743406823621094409real_a @ A4 @ B4 )
     => ( ( member_real_a @ X3 @ A4 )
       => ( member_real_a @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_640_in__mono,axiom,
    ! [A4: set_o_real,B4: set_o_real,X3: $o > real] :
      ( ( ord_le3251842697534426805o_real @ A4 @ B4 )
     => ( ( member_o_real @ X3 @ A4 )
       => ( member_o_real @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_641_in__mono,axiom,
    ! [A4: set_nat_real,B4: set_nat_real,X3: nat > real] :
      ( ( ord_le2908806416726583473t_real @ A4 @ B4 )
     => ( ( member_nat_real @ X3 @ A4 )
       => ( member_nat_real @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_642_in__mono,axiom,
    ! [A4: set_c_b,B4: set_c_b,X3: c > b] :
      ( ( ord_less_eq_set_c_b @ A4 @ B4 )
     => ( ( member_c_b @ X3 @ A4 )
       => ( member_c_b @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_643_in__mono,axiom,
    ! [A4: set_a_b,B4: set_a_b,X3: a > b] :
      ( ( ord_less_eq_set_a_b @ A4 @ B4 )
     => ( ( member_a_b @ X3 @ A4 )
       => ( member_a_b @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_644_subsetD,axiom,
    ! [A4: set_real_a,B4: set_real_a,C: real > a] :
      ( ( ord_le5743406823621094409real_a @ A4 @ B4 )
     => ( ( member_real_a @ C @ A4 )
       => ( member_real_a @ C @ B4 ) ) ) ).

% subsetD
thf(fact_645_subsetD,axiom,
    ! [A4: set_o_real,B4: set_o_real,C: $o > real] :
      ( ( ord_le3251842697534426805o_real @ A4 @ B4 )
     => ( ( member_o_real @ C @ A4 )
       => ( member_o_real @ C @ B4 ) ) ) ).

% subsetD
thf(fact_646_subsetD,axiom,
    ! [A4: set_nat_real,B4: set_nat_real,C: nat > real] :
      ( ( ord_le2908806416726583473t_real @ A4 @ B4 )
     => ( ( member_nat_real @ C @ A4 )
       => ( member_nat_real @ C @ B4 ) ) ) ).

% subsetD
thf(fact_647_subsetD,axiom,
    ! [A4: set_c_b,B4: set_c_b,C: c > b] :
      ( ( ord_less_eq_set_c_b @ A4 @ B4 )
     => ( ( member_c_b @ C @ A4 )
       => ( member_c_b @ C @ B4 ) ) ) ).

% subsetD
thf(fact_648_subsetD,axiom,
    ! [A4: set_a_b,B4: set_a_b,C: a > b] :
      ( ( ord_less_eq_set_a_b @ A4 @ B4 )
     => ( ( member_a_b @ C @ A4 )
       => ( member_a_b @ C @ B4 ) ) ) ).

% subsetD
thf(fact_649_subset__eq,axiom,
    ( ord_le5743406823621094409real_a
    = ( ^ [A5: set_real_a,B7: set_real_a] :
        ! [X5: real > a] :
          ( ( member_real_a @ X5 @ A5 )
         => ( member_real_a @ X5 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_650_subset__eq,axiom,
    ( ord_le3251842697534426805o_real
    = ( ^ [A5: set_o_real,B7: set_o_real] :
        ! [X5: $o > real] :
          ( ( member_o_real @ X5 @ A5 )
         => ( member_o_real @ X5 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_651_subset__eq,axiom,
    ( ord_le2908806416726583473t_real
    = ( ^ [A5: set_nat_real,B7: set_nat_real] :
        ! [X5: nat > real] :
          ( ( member_nat_real @ X5 @ A5 )
         => ( member_nat_real @ X5 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_652_subset__eq,axiom,
    ( ord_less_eq_set_c_b
    = ( ^ [A5: set_c_b,B7: set_c_b] :
        ! [X5: c > b] :
          ( ( member_c_b @ X5 @ A5 )
         => ( member_c_b @ X5 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_653_subset__eq,axiom,
    ( ord_less_eq_set_a_b
    = ( ^ [A5: set_a_b,B7: set_a_b] :
        ! [X5: a > b] :
          ( ( member_a_b @ X5 @ A5 )
         => ( member_a_b @ X5 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_654_subset__iff,axiom,
    ( ord_le5743406823621094409real_a
    = ( ^ [A5: set_real_a,B7: set_real_a] :
        ! [T: real > a] :
          ( ( member_real_a @ T @ A5 )
         => ( member_real_a @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_655_subset__iff,axiom,
    ( ord_le3251842697534426805o_real
    = ( ^ [A5: set_o_real,B7: set_o_real] :
        ! [T: $o > real] :
          ( ( member_o_real @ T @ A5 )
         => ( member_o_real @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_656_subset__iff,axiom,
    ( ord_le2908806416726583473t_real
    = ( ^ [A5: set_nat_real,B7: set_nat_real] :
        ! [T: nat > real] :
          ( ( member_nat_real @ T @ A5 )
         => ( member_nat_real @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_657_subset__iff,axiom,
    ( ord_less_eq_set_c_b
    = ( ^ [A5: set_c_b,B7: set_c_b] :
        ! [T: c > b] :
          ( ( member_c_b @ T @ A5 )
         => ( member_c_b @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_658_subset__iff,axiom,
    ( ord_less_eq_set_a_b
    = ( ^ [A5: set_a_b,B7: set_a_b] :
        ! [T: a > b] :
          ( ( member_a_b @ T @ A5 )
         => ( member_a_b @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_659_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_660_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
          & ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_661_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_662_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_663_order__eq__refl,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( X3 = Y2 )
     => ( ord_less_eq_nat @ X3 @ Y2 ) ) ).

% order_eq_refl
thf(fact_664_linorder__linear,axiom,
    ! [X3: nat,Y2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y2 )
      | ( ord_less_eq_nat @ Y2 @ X3 ) ) ).

% linorder_linear
thf(fact_665_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_666_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_667_linorder__le__cases,axiom,
    ! [X3: nat,Y2: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y2 )
     => ( ord_less_eq_nat @ Y2 @ X3 ) ) ).

% linorder_le_cases
thf(fact_668_order__antisym__conv,axiom,
    ! [Y2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y2 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y2 )
        = ( X3 = Y2 ) ) ) ).

% order_antisym_conv
thf(fact_669_sets__le__imp__space__le,axiom,
    ! [A4: sigma_measure_real,B4: sigma_measure_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A4 ) @ ( sigma_sets_real @ B4 ) )
     => ( ord_less_eq_set_real @ ( sigma_space_real @ A4 ) @ ( sigma_space_real @ B4 ) ) ) ).

% sets_le_imp_space_le
thf(fact_670_sets__le__imp__space__le,axiom,
    ! [A4: sigma_7234349610311085201nnreal,B4: sigma_7234349610311085201nnreal] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ A4 ) @ ( sigma_5465916536984168985nnreal @ B4 ) )
     => ( ord_le6787938422905777998nnreal @ ( sigma_3147302497200244656nnreal @ A4 ) @ ( sigma_3147302497200244656nnreal @ B4 ) ) ) ).

% sets_le_imp_space_le
thf(fact_671_sets__le__imp__space__le,axiom,
    ! [A4: sigma_measure_o,B4: sigma_measure_o] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ A4 ) @ ( sigma_sets_o @ B4 ) )
     => ( ord_less_eq_set_o @ ( sigma_space_o @ A4 ) @ ( sigma_space_o @ B4 ) ) ) ).

% sets_le_imp_space_le
thf(fact_672_sets__le__imp__space__le,axiom,
    ! [A4: sigma_measure_nat,B4: sigma_measure_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ A4 ) @ ( sigma_sets_nat @ B4 ) )
     => ( ord_less_eq_set_nat @ ( sigma_space_nat @ A4 ) @ ( sigma_space_nat @ B4 ) ) ) ).

% sets_le_imp_space_le
thf(fact_673_top_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ top_top_set_real @ A )
     => ( A = top_top_set_real ) ) ).

% top.extremum_uniqueI
thf(fact_674_top_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ top_top_set_o @ A )
     => ( A = top_top_set_o ) ) ).

% top.extremum_uniqueI
thf(fact_675_top_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ top_top_set_real @ A )
      = ( A = top_top_set_real ) ) ).

% top.extremum_unique
thf(fact_676_top_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ top_top_set_o @ A )
      = ( A = top_top_set_o ) ) ).

% top.extremum_unique
thf(fact_677_top__greatest,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).

% top_greatest
thf(fact_678_top__greatest,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).

% top_greatest
thf(fact_679_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_680_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_681_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ bot_bo4854962954004695426nnreal )
     => ( A = bot_bo4854962954004695426nnreal ) ) ).

% bot.extremum_uniqueI
thf(fact_682_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_683_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_684_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_685_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_686_bot_Oextremum__unique,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ bot_bo4854962954004695426nnreal )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% bot.extremum_unique
thf(fact_687_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_688_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_689_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_690_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_691_bot_Oextremum,axiom,
    ! [A: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ bot_bo4854962954004695426nnreal @ A ) ).

% bot.extremum
thf(fact_692_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_693_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_694_subset__UNIV,axiom,
    ! [A4: set_real] : ( ord_less_eq_set_real @ A4 @ top_top_set_real ) ).

% subset_UNIV
thf(fact_695_subset__UNIV,axiom,
    ! [A4: set_o] : ( ord_less_eq_set_o @ A4 @ top_top_set_o ) ).

% subset_UNIV
thf(fact_696_Set_Oinsert__mono,axiom,
    ! [C3: set_real,D2: set_real,A: real] :
      ( ( ord_less_eq_set_real @ C3 @ D2 )
     => ( ord_less_eq_set_real @ ( insert_real @ A @ C3 ) @ ( insert_real @ A @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_697_Set_Oinsert__mono,axiom,
    ! [C3: set_nat,D2: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C3 @ D2 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C3 ) @ ( insert_nat @ A @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_698_Set_Oinsert__mono,axiom,
    ! [C3: set_Ex3793607809372303086nnreal,D2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ C3 @ D2 )
     => ( ord_le6787938422905777998nnreal @ ( insert7407984058720857448nnreal @ A @ C3 ) @ ( insert7407984058720857448nnreal @ A @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_699_Set_Oinsert__mono,axiom,
    ! [C3: set_o,D2: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ C3 @ D2 )
     => ( ord_less_eq_set_o @ ( insert_o @ A @ C3 ) @ ( insert_o @ A @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_700_subset__insert,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ X3 @ A4 )
     => ( ( ord_less_eq_set_real @ A4 @ ( insert_real @ X3 @ B4 ) )
        = ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_701_subset__insert,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X3 @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X3 @ B4 ) )
        = ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_702_subset__insert,axiom,
    ! [X3: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ X3 @ A4 )
     => ( ( ord_le6787938422905777998nnreal @ A4 @ ( insert7407984058720857448nnreal @ X3 @ B4 ) )
        = ( ord_le6787938422905777998nnreal @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_703_subset__insert,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ X3 @ A4 )
     => ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
        = ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_704_subset__insert,axiom,
    ! [X3: real > a,A4: set_real_a,B4: set_real_a] :
      ( ~ ( member_real_a @ X3 @ A4 )
     => ( ( ord_le5743406823621094409real_a @ A4 @ ( insert_real_a @ X3 @ B4 ) )
        = ( ord_le5743406823621094409real_a @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_705_subset__insert,axiom,
    ! [X3: $o > real,A4: set_o_real,B4: set_o_real] :
      ( ~ ( member_o_real @ X3 @ A4 )
     => ( ( ord_le3251842697534426805o_real @ A4 @ ( insert_o_real @ X3 @ B4 ) )
        = ( ord_le3251842697534426805o_real @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_706_subset__insert,axiom,
    ! [X3: nat > real,A4: set_nat_real,B4: set_nat_real] :
      ( ~ ( member_nat_real @ X3 @ A4 )
     => ( ( ord_le2908806416726583473t_real @ A4 @ ( insert_nat_real @ X3 @ B4 ) )
        = ( ord_le2908806416726583473t_real @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_707_subset__insert,axiom,
    ! [X3: c > b,A4: set_c_b,B4: set_c_b] :
      ( ~ ( member_c_b @ X3 @ A4 )
     => ( ( ord_less_eq_set_c_b @ A4 @ ( insert_c_b @ X3 @ B4 ) )
        = ( ord_less_eq_set_c_b @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_708_subset__insert,axiom,
    ! [X3: a > b,A4: set_a_b,B4: set_a_b] :
      ( ~ ( member_a_b @ X3 @ A4 )
     => ( ( ord_less_eq_set_a_b @ A4 @ ( insert_a_b @ X3 @ B4 ) )
        = ( ord_less_eq_set_a_b @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_709_subset__insertI,axiom,
    ! [B4: set_real,A: real] : ( ord_less_eq_set_real @ B4 @ ( insert_real @ A @ B4 ) ) ).

% subset_insertI
thf(fact_710_subset__insertI,axiom,
    ! [B4: set_nat,A: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A @ B4 ) ) ).

% subset_insertI
thf(fact_711_subset__insertI,axiom,
    ! [B4: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] : ( ord_le6787938422905777998nnreal @ B4 @ ( insert7407984058720857448nnreal @ A @ B4 ) ) ).

% subset_insertI
thf(fact_712_subset__insertI,axiom,
    ! [B4: set_o,A: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A @ B4 ) ) ).

% subset_insertI
thf(fact_713_subset__insertI2,axiom,
    ! [A4: set_real,B4: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A4 @ B4 )
     => ( ord_less_eq_set_real @ A4 @ ( insert_real @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_714_subset__insertI2,axiom,
    ! [A4: set_nat,B4: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_715_subset__insertI2,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A4 @ B4 )
     => ( ord_le6787938422905777998nnreal @ A4 @ ( insert7407984058720857448nnreal @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_716_subset__insertI2,axiom,
    ! [A4: set_o,B4: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A4 @ B4 )
     => ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_717_subset__singleton__iff,axiom,
    ! [X2: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X2 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( ( X2 = bot_bot_set_real )
        | ( X2
          = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_718_subset__singleton__iff,axiom,
    ! [X2: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X2 = bot_bot_set_nat )
        | ( X2
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_719_subset__singleton__iff,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ X2 @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) )
      = ( ( X2 = bot_bo4854962954004695426nnreal )
        | ( X2
          = ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% subset_singleton_iff
thf(fact_720_subset__singleton__iff,axiom,
    ! [X2: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ X2 @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( ( X2 = bot_bot_set_o )
        | ( X2
          = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_721_subset__singletonD,axiom,
    ! [A4: set_real,X3: real] :
      ( ( ord_less_eq_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) )
     => ( ( A4 = bot_bot_set_real )
        | ( A4
          = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_722_subset__singletonD,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
     => ( ( A4 = bot_bot_set_nat )
        | ( A4
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_723_subset__singletonD,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A4 @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) )
     => ( ( A4 = bot_bo4854962954004695426nnreal )
        | ( A4
          = ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% subset_singletonD
thf(fact_724_subset__singletonD,axiom,
    ! [A4: set_o,X3: $o] :
      ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) )
     => ( ( A4 = bot_bot_set_o )
        | ( A4
          = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_725_sets_Osets__into__space,axiom,
    ! [X3: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ X3 @ ( sigma_sets_real @ M ) )
     => ( ord_less_eq_set_real @ X3 @ ( sigma_space_real @ M ) ) ) ).

% sets.sets_into_space
thf(fact_726_sets_Osets__into__space,axiom,
    ! [X3: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( member603777416030116741nnreal @ X3 @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ord_le6787938422905777998nnreal @ X3 @ ( sigma_3147302497200244656nnreal @ M ) ) ) ).

% sets.sets_into_space
thf(fact_727_sets_Osets__into__space,axiom,
    ! [X3: set_o,M: sigma_measure_o] :
      ( ( member_set_o @ X3 @ ( sigma_sets_o @ M ) )
     => ( ord_less_eq_set_o @ X3 @ ( sigma_space_o @ M ) ) ) ).

% sets.sets_into_space
thf(fact_728_sets_Osets__into__space,axiom,
    ! [X3: set_nat,M: sigma_measure_nat] :
      ( ( member_set_nat @ X3 @ ( sigma_sets_nat @ M ) )
     => ( ord_less_eq_set_nat @ X3 @ ( sigma_space_nat @ M ) ) ) ).

% sets.sets_into_space
thf(fact_729_sets_Osigma__sets__subset_H,axiom,
    ! [A: set_set_real,M: sigma_measure_real,Omega2: set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real @ Omega2 @ ( sigma_sets_real @ M ) )
       => ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ Omega2 @ A ) @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_730_sets_Osigma__sets__subset_H,axiom,
    ! [A: set_se4580700918925141924nnreal,M: sigma_7234349610311085201nnreal,Omega2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal @ Omega2 @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ Omega2 @ A ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_731_sets_Osigma__sets__subset_H,axiom,
    ! [A: set_set_o,M: sigma_measure_o,Omega2: set_o] :
      ( ( ord_le4374716579403074808_set_o @ A @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o @ Omega2 @ ( sigma_sets_o @ M ) )
       => ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ Omega2 @ A ) @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_732_sets_Osigma__sets__subset_H,axiom,
    ! [A: set_set_nat,M: sigma_measure_nat,Omega2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat @ Omega2 @ ( sigma_sets_nat @ M ) )
       => ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ Omega2 @ A ) @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_733_sets__Sup__measure_H,axiom,
    ! [M: set_Si6059263944882162789e_real,A4: sigma_measure_real] :
      ( ! [M3: sigma_measure_real] :
          ( ( member4553183543495551918e_real @ M3 @ M )
         => ( ( sigma_sets_real @ M3 )
            = ( sigma_sets_real @ A4 ) ) )
     => ( ( M != bot_bo5686449298802467025e_real )
       => ( ( sigma_sets_real @ ( measur8657758558638653562e_real @ M ) )
          = ( sigma_sets_real @ A4 ) ) ) ) ).

% sets_Sup_measure'
thf(fact_734_sets__Sup__measure_H,axiom,
    ! [M: set_Si97717610131227249nnreal,A4: sigma_7234349610311085201nnreal] :
      ( ! [M3: sigma_7234349610311085201nnreal] :
          ( ( member6261374078160781754nnreal @ M3 @ M )
         => ( ( sigma_5465916536984168985nnreal @ M3 )
            = ( sigma_5465916536984168985nnreal @ A4 ) ) )
     => ( ( M != bot_bo8227844048696536285nnreal )
       => ( ( sigma_5465916536984168985nnreal @ ( measur1651139276328235014nnreal @ M ) )
          = ( sigma_5465916536984168985nnreal @ A4 ) ) ) ) ).

% sets_Sup_measure'
thf(fact_735_sets__Sup__measure_H,axiom,
    ! [M: set_Sigma_measure_o,A4: sigma_measure_o] :
      ( ! [M3: sigma_measure_o] :
          ( ( member1844656263901471916sure_o @ M3 @ M )
         => ( ( sigma_sets_o @ M3 )
            = ( sigma_sets_o @ A4 ) ) )
     => ( ( M != bot_bo7838039659004643295sure_o )
       => ( ( sigma_sets_o @ ( measur1214336222341667658sure_o @ M ) )
          = ( sigma_sets_o @ A4 ) ) ) ) ).

% sets_Sup_measure'
thf(fact_736_sets__Sup__measure_H,axiom,
    ! [M: set_Si3048223896905877257re_nat,A4: sigma_measure_nat] :
      ( ! [M3: sigma_measure_nat] :
          ( ( member4416920341759242834re_nat @ M3 @ M )
         => ( ( sigma_sets_nat @ M3 )
            = ( sigma_sets_nat @ A4 ) ) )
     => ( ( M != bot_bo8872222457363190133re_nat )
       => ( ( sigma_sets_nat @ ( measur3575099672463795358re_nat @ M ) )
          = ( sigma_sets_nat @ A4 ) ) ) ) ).

% sets_Sup_measure'
thf(fact_737_sets_Ocountable,axiom,
    ! [A4: set_real_a,M: sigma_measure_real_a] :
      ( ! [A3: real > a] :
          ( ( member_real_a @ A3 @ A4 )
         => ( member_set_real_a @ ( insert_real_a @ A3 @ bot_bot_set_real_a ) @ ( sigma_sets_real_a @ M ) ) )
     => ( ( counta6639396083684174020real_a @ A4 )
       => ( member_set_real_a @ A4 @ ( sigma_sets_real_a @ M ) ) ) ) ).

% sets.countable
thf(fact_738_sets_Ocountable,axiom,
    ! [A4: set_o_real,M: sigma_measure_o_real] :
      ( ! [A3: $o > real] :
          ( ( member_o_real @ A3 @ A4 )
         => ( member_set_o_real @ ( insert_o_real @ A3 @ bot_bot_set_o_real ) @ ( sigma_sets_o_real @ M ) ) )
     => ( ( counta8783200249485735024o_real @ A4 )
       => ( member_set_o_real @ A4 @ ( sigma_sets_o_real @ M ) ) ) ) ).

% sets.countable
thf(fact_739_sets_Ocountable,axiom,
    ! [A4: set_nat_real,M: sigma_3396294578489551860t_real] :
      ( ! [A3: nat > real] :
          ( ( member_nat_real @ A3 @ A4 )
         => ( member_set_nat_real @ ( insert_nat_real @ A3 @ bot_bot_set_nat_real ) @ ( sigma_sets_nat_real @ M ) ) )
     => ( ( counta2162411829015494944t_real @ A4 )
       => ( member_set_nat_real @ A4 @ ( sigma_sets_nat_real @ M ) ) ) ) ).

% sets.countable
thf(fact_740_sets_Ocountable,axiom,
    ! [A4: set_c_b,M: sigma_measure_c_b] :
      ( ! [A3: c > b] :
          ( ( member_c_b @ A3 @ A4 )
         => ( member_set_c_b @ ( insert_c_b @ A3 @ bot_bot_set_c_b ) @ ( sigma_sets_c_b @ M ) ) )
     => ( ( counta2657777928882154345le_c_b @ A4 )
       => ( member_set_c_b @ A4 @ ( sigma_sets_c_b @ M ) ) ) ) ).

% sets.countable
thf(fact_741_sets_Ocountable,axiom,
    ! [A4: set_a_b,M: sigma_measure_a_b] :
      ( ! [A3: a > b] :
          ( ( member_a_b @ A3 @ A4 )
         => ( member_set_a_b @ ( insert_a_b @ A3 @ bot_bot_set_a_b ) @ ( sigma_sets_a_b @ M ) ) )
     => ( ( counta8232689092827506411le_a_b @ A4 )
       => ( member_set_a_b @ A4 @ ( sigma_sets_a_b @ M ) ) ) ) ).

% sets.countable
thf(fact_742_sets_Ocountable,axiom,
    ! [A4: set_real,M: sigma_measure_real] :
      ( ! [A3: real] :
          ( ( member_real @ A3 @ A4 )
         => ( member_set_real @ ( insert_real @ A3 @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
     => ( ( counta7319604579010473777e_real @ A4 )
       => ( member_set_real @ A4 @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.countable
thf(fact_743_sets_Ocountable,axiom,
    ! [A4: set_nat,M: sigma_measure_nat] :
      ( ! [A3: nat] :
          ( ( member_nat @ A3 @ A4 )
         => ( member_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) ) )
     => ( ( counta1168086296615599829le_nat @ A4 )
       => ( member_set_nat @ A4 @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.countable
thf(fact_744_sets_Ocountable,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ! [A3: extend8495563244428889912nnreal] :
          ( ( member7908768830364227535nnreal @ A3 @ A4 )
         => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ A3 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
     => ( ( counta8439243037236335165nnreal @ A4 )
       => ( member603777416030116741nnreal @ A4 @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.countable
thf(fact_745_sets_Ocountable,axiom,
    ! [A4: set_o,M: sigma_measure_o] :
      ( ! [A3: $o] :
          ( ( member_o @ A3 @ A4 )
         => ( member_set_o @ ( insert_o @ A3 @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) ) )
     => ( ( counta5976203206615340371able_o @ A4 )
       => ( member_set_o @ A4 @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.countable
thf(fact_746_sigma__algebra_Ocountable,axiom,
    ! [Omega: set_real_a,M: set_set_real_a,A4: set_real_a] :
      ( ( sigma_6829682388519410934real_a @ Omega @ M )
     => ( ! [A3: real > a] :
            ( ( member_real_a @ A3 @ A4 )
           => ( member_set_real_a @ ( insert_real_a @ A3 @ bot_bot_set_real_a ) @ M ) )
       => ( ( counta6639396083684174020real_a @ A4 )
         => ( member_set_real_a @ A4 @ M ) ) ) ) ).

% sigma_algebra.countable
thf(fact_747_sigma__algebra_Ocountable,axiom,
    ! [Omega: set_o_real,M: set_set_o_real,A4: set_o_real] :
      ( ( sigma_1431946479552111010o_real @ Omega @ M )
     => ( ! [A3: $o > real] :
            ( ( member_o_real @ A3 @ A4 )
           => ( member_set_o_real @ ( insert_o_real @ A3 @ bot_bot_set_o_real ) @ M ) )
       => ( ( counta8783200249485735024o_real @ A4 )
         => ( member_set_o_real @ A4 @ M ) ) ) ) ).

% sigma_algebra.countable
thf(fact_748_sigma__algebra_Ocountable,axiom,
    ! [Omega: set_nat_real,M: set_set_nat_real,A4: set_nat_real] :
      ( ( sigma_5696473160036025454t_real @ Omega @ M )
     => ( ! [A3: nat > real] :
            ( ( member_nat_real @ A3 @ A4 )
           => ( member_set_nat_real @ ( insert_nat_real @ A3 @ bot_bot_set_nat_real ) @ M ) )
       => ( ( counta2162411829015494944t_real @ A4 )
         => ( member_set_nat_real @ A4 @ M ) ) ) ) ).

% sigma_algebra.countable
thf(fact_749_sigma__algebra_Ocountable,axiom,
    ! [Omega: set_c_b,M: set_set_c_b,A4: set_c_b] :
      ( ( sigma_3665481007338186423ra_c_b @ Omega @ M )
     => ( ! [A3: c > b] :
            ( ( member_c_b @ A3 @ A4 )
           => ( member_set_c_b @ ( insert_c_b @ A3 @ bot_bot_set_c_b ) @ M ) )
       => ( ( counta2657777928882154345le_c_b @ A4 )
         => ( member_set_c_b @ A4 @ M ) ) ) ) ).

% sigma_algebra.countable
thf(fact_750_sigma__algebra_Ocountable,axiom,
    ! [Omega: set_a_b,M: set_set_a_b,A4: set_a_b] :
      ( ( sigma_17020134428762681ra_a_b @ Omega @ M )
     => ( ! [A3: a > b] :
            ( ( member_a_b @ A3 @ A4 )
           => ( member_set_a_b @ ( insert_a_b @ A3 @ bot_bot_set_a_b ) @ M ) )
       => ( ( counta8232689092827506411le_a_b @ A4 )
         => ( member_set_a_b @ A4 @ M ) ) ) ) ).

% sigma_algebra.countable
thf(fact_751_sigma__algebra_Ocountable,axiom,
    ! [Omega: set_real,M: set_set_real,A4: set_real] :
      ( ( sigma_1481383337440427903a_real @ Omega @ M )
     => ( ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ( member_set_real @ ( insert_real @ A3 @ bot_bot_set_real ) @ M ) )
       => ( ( counta7319604579010473777e_real @ A4 )
         => ( member_set_real @ A4 @ M ) ) ) ) ).

% sigma_algebra.countable
thf(fact_752_sigma__algebra_Ocountable,axiom,
    ! [Omega: set_nat,M: set_set_nat,A4: set_nat] :
      ( ( sigma_8817008012692346403ra_nat @ Omega @ M )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ( member_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) @ M ) )
       => ( ( counta1168086296615599829le_nat @ A4 )
         => ( member_set_nat @ A4 @ M ) ) ) ) ).

% sigma_algebra.countable
thf(fact_753_sigma__algebra_Ocountable,axiom,
    ! [Omega: set_Ex3793607809372303086nnreal,M: set_se4580700918925141924nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( sigma_2413694886200424843nnreal @ Omega @ M )
     => ( ! [A3: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ A3 @ A4 )
           => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ A3 @ bot_bo4854962954004695426nnreal ) @ M ) )
       => ( ( counta8439243037236335165nnreal @ A4 )
         => ( member603777416030116741nnreal @ A4 @ M ) ) ) ) ).

% sigma_algebra.countable
thf(fact_754_sigma__algebra_Ocountable,axiom,
    ! [Omega: set_o,M: set_set_o,A4: set_o] :
      ( ( sigma_3687534776968752773ebra_o @ Omega @ M )
     => ( ! [A3: $o] :
            ( ( member_o @ A3 @ A4 )
           => ( member_set_o @ ( insert_o @ A3 @ bot_bot_set_o ) @ M ) )
       => ( ( counta5976203206615340371able_o @ A4 )
         => ( member_set_o @ A4 @ M ) ) ) ) ).

% sigma_algebra.countable
thf(fact_755_sets_Ospace__closed,axiom,
    ! [M: sigma_measure_real] : ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( pow_real @ ( sigma_space_real @ M ) ) ) ).

% sets.space_closed
thf(fact_756_sets_Ospace__closed,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( pow_Ex5372160365422184283nnreal @ ( sigma_3147302497200244656nnreal @ M ) ) ) ).

% sets.space_closed
thf(fact_757_sets_Ospace__closed,axiom,
    ! [M: sigma_measure_o] : ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ M ) @ ( pow_o @ ( sigma_space_o @ M ) ) ) ).

% sets.space_closed
thf(fact_758_sets_Ospace__closed,axiom,
    ! [M: sigma_measure_nat] : ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ M ) @ ( pow_nat @ ( sigma_space_nat @ M ) ) ) ).

% sets.space_closed
thf(fact_759_countable__insert__eq,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ( counta1168086296615599829le_nat @ ( insert_nat @ X3 @ A4 ) )
      = ( counta1168086296615599829le_nat @ A4 ) ) ).

% countable_insert_eq
thf(fact_760_countable__insert__eq,axiom,
    ! [X3: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( counta8439243037236335165nnreal @ ( insert7407984058720857448nnreal @ X3 @ A4 ) )
      = ( counta8439243037236335165nnreal @ A4 ) ) ).

% countable_insert_eq
thf(fact_761_countable__insert__eq,axiom,
    ! [X3: $o,A4: set_o] :
      ( ( counta5976203206615340371able_o @ ( insert_o @ X3 @ A4 ) )
      = ( counta5976203206615340371able_o @ A4 ) ) ).

% countable_insert_eq
thf(fact_762_countable__insert__eq,axiom,
    ! [X3: real,A4: set_real] :
      ( ( counta7319604579010473777e_real @ ( insert_real @ X3 @ A4 ) )
      = ( counta7319604579010473777e_real @ A4 ) ) ).

% countable_insert_eq
thf(fact_763_countable__insert,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( counta1168086296615599829le_nat @ A4 )
     => ( counta1168086296615599829le_nat @ ( insert_nat @ A @ A4 ) ) ) ).

% countable_insert
thf(fact_764_countable__insert,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal] :
      ( ( counta8439243037236335165nnreal @ A4 )
     => ( counta8439243037236335165nnreal @ ( insert7407984058720857448nnreal @ A @ A4 ) ) ) ).

% countable_insert
thf(fact_765_countable__insert,axiom,
    ! [A4: set_o,A: $o] :
      ( ( counta5976203206615340371able_o @ A4 )
     => ( counta5976203206615340371able_o @ ( insert_o @ A @ A4 ) ) ) ).

% countable_insert
thf(fact_766_countable__insert,axiom,
    ! [A4: set_real,A: real] :
      ( ( counta7319604579010473777e_real @ A4 )
     => ( counta7319604579010473777e_real @ ( insert_real @ A @ A4 ) ) ) ).

% countable_insert
thf(fact_767_countable__empty,axiom,
    counta7319604579010473777e_real @ bot_bot_set_real ).

% countable_empty
thf(fact_768_countable__empty,axiom,
    counta1168086296615599829le_nat @ bot_bot_set_nat ).

% countable_empty
thf(fact_769_countable__empty,axiom,
    counta8439243037236335165nnreal @ bot_bo4854962954004695426nnreal ).

% countable_empty
thf(fact_770_countable__empty,axiom,
    counta5976203206615340371able_o @ bot_bot_set_o ).

% countable_empty
thf(fact_771_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_measure_real,C3: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ C3 )
     => ( ( sigma_227922725797042522i_real @ ( sigma_space_real @ M ) @ C3 )
       => ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) @ C3 ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_772_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_7234349610311085201nnreal,C3: set_se4580700918925141924nnreal] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ C3 )
     => ( ( sigma_114563780369365222nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ C3 )
       => ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) ) @ C3 ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_773_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_measure_o,C3: set_set_o] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ M ) @ C3 )
     => ( ( sigma_closed_cdi_o @ ( sigma_space_o @ M ) @ C3 )
       => ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) @ C3 ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_774_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_measure_nat,C3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ M ) @ C3 )
     => ( ( sigma_closed_cdi_nat @ ( sigma_space_nat @ M ) @ C3 )
       => ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) @ C3 ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_775_insert__subsetI,axiom,
    ! [X3: real,A4: set_real,X2: set_real] :
      ( ( member_real @ X3 @ A4 )
     => ( ( ord_less_eq_set_real @ X2 @ A4 )
       => ( ord_less_eq_set_real @ ( insert_real @ X3 @ X2 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_776_insert__subsetI,axiom,
    ! [X3: nat,A4: set_nat,X2: set_nat] :
      ( ( member_nat @ X3 @ A4 )
     => ( ( ord_less_eq_set_nat @ X2 @ A4 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ X2 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_777_insert__subsetI,axiom,
    ! [X3: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal,X2: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ X3 @ A4 )
     => ( ( ord_le6787938422905777998nnreal @ X2 @ A4 )
       => ( ord_le6787938422905777998nnreal @ ( insert7407984058720857448nnreal @ X3 @ X2 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_778_insert__subsetI,axiom,
    ! [X3: $o,A4: set_o,X2: set_o] :
      ( ( member_o @ X3 @ A4 )
     => ( ( ord_less_eq_set_o @ X2 @ A4 )
       => ( ord_less_eq_set_o @ ( insert_o @ X3 @ X2 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_779_insert__subsetI,axiom,
    ! [X3: real > a,A4: set_real_a,X2: set_real_a] :
      ( ( member_real_a @ X3 @ A4 )
     => ( ( ord_le5743406823621094409real_a @ X2 @ A4 )
       => ( ord_le5743406823621094409real_a @ ( insert_real_a @ X3 @ X2 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_780_insert__subsetI,axiom,
    ! [X3: $o > real,A4: set_o_real,X2: set_o_real] :
      ( ( member_o_real @ X3 @ A4 )
     => ( ( ord_le3251842697534426805o_real @ X2 @ A4 )
       => ( ord_le3251842697534426805o_real @ ( insert_o_real @ X3 @ X2 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_781_insert__subsetI,axiom,
    ! [X3: nat > real,A4: set_nat_real,X2: set_nat_real] :
      ( ( member_nat_real @ X3 @ A4 )
     => ( ( ord_le2908806416726583473t_real @ X2 @ A4 )
       => ( ord_le2908806416726583473t_real @ ( insert_nat_real @ X3 @ X2 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_782_insert__subsetI,axiom,
    ! [X3: c > b,A4: set_c_b,X2: set_c_b] :
      ( ( member_c_b @ X3 @ A4 )
     => ( ( ord_less_eq_set_c_b @ X2 @ A4 )
       => ( ord_less_eq_set_c_b @ ( insert_c_b @ X3 @ X2 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_783_insert__subsetI,axiom,
    ! [X3: a > b,A4: set_a_b,X2: set_a_b] :
      ( ( member_a_b @ X3 @ A4 )
     => ( ( ord_less_eq_set_a_b @ X2 @ A4 )
       => ( ord_less_eq_set_a_b @ ( insert_a_b @ X3 @ X2 ) @ A4 ) ) ) ).

% insert_subsetI
thf(fact_784_subset__emptyI,axiom,
    ! [A4: set_real_a] :
      ( ! [X: real > a] :
          ~ ( member_real_a @ X @ A4 )
     => ( ord_le5743406823621094409real_a @ A4 @ bot_bot_set_real_a ) ) ).

% subset_emptyI
thf(fact_785_subset__emptyI,axiom,
    ! [A4: set_o_real] :
      ( ! [X: $o > real] :
          ~ ( member_o_real @ X @ A4 )
     => ( ord_le3251842697534426805o_real @ A4 @ bot_bot_set_o_real ) ) ).

% subset_emptyI
thf(fact_786_subset__emptyI,axiom,
    ! [A4: set_nat_real] :
      ( ! [X: nat > real] :
          ~ ( member_nat_real @ X @ A4 )
     => ( ord_le2908806416726583473t_real @ A4 @ bot_bot_set_nat_real ) ) ).

% subset_emptyI
thf(fact_787_subset__emptyI,axiom,
    ! [A4: set_c_b] :
      ( ! [X: c > b] :
          ~ ( member_c_b @ X @ A4 )
     => ( ord_less_eq_set_c_b @ A4 @ bot_bot_set_c_b ) ) ).

% subset_emptyI
thf(fact_788_subset__emptyI,axiom,
    ! [A4: set_a_b] :
      ( ! [X: a > b] :
          ~ ( member_a_b @ X @ A4 )
     => ( ord_less_eq_set_a_b @ A4 @ bot_bot_set_a_b ) ) ).

% subset_emptyI
thf(fact_789_subset__emptyI,axiom,
    ! [A4: set_real] :
      ( ! [X: real] :
          ~ ( member_real @ X @ A4 )
     => ( ord_less_eq_set_real @ A4 @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_790_subset__emptyI,axiom,
    ! [A4: set_nat] :
      ( ! [X: nat] :
          ~ ( member_nat @ X @ A4 )
     => ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_791_subset__emptyI,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ! [X: extend8495563244428889912nnreal] :
          ~ ( member7908768830364227535nnreal @ X @ A4 )
     => ( ord_le6787938422905777998nnreal @ A4 @ bot_bo4854962954004695426nnreal ) ) ).

% subset_emptyI
thf(fact_792_subset__emptyI,axiom,
    ! [A4: set_o] :
      ( ! [X: $o] :
          ~ ( member_o @ X @ A4 )
     => ( ord_less_eq_set_o @ A4 @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_793_sets__measure__of__conv,axiom,
    ! [A4: set_set_real,Omega: set_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( ( ord_le3558479182127378552t_real @ A4 @ ( pow_real @ Omega ) )
       => ( ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega @ A4 @ Mu ) )
          = ( sigma_7195353284648819924s_real @ Omega @ A4 ) ) )
      & ( ~ ( ord_le3558479182127378552t_real @ A4 @ ( pow_real @ Omega ) )
       => ( ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega @ A4 @ Mu ) )
          = ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ Omega @ bot_bot_set_set_real ) ) ) ) ) ).

% sets_measure_of_conv
thf(fact_794_sets__measure__of__conv,axiom,
    ! [A4: set_set_nat,Omega: set_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( ( ord_le6893508408891458716et_nat @ A4 @ ( pow_nat @ Omega ) )
       => ( ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega @ A4 @ Mu ) )
          = ( sigma_sigma_sets_nat @ Omega @ A4 ) ) )
      & ( ~ ( ord_le6893508408891458716et_nat @ A4 @ ( pow_nat @ Omega ) )
       => ( ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega @ A4 @ Mu ) )
          = ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ Omega @ bot_bot_set_set_nat ) ) ) ) ) ).

% sets_measure_of_conv
thf(fact_795_sets__measure__of__conv,axiom,
    ! [A4: set_se4580700918925141924nnreal,Omega: set_Ex3793607809372303086nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( ( ord_le3366939622266546180nnreal @ A4 @ ( pow_Ex5372160365422184283nnreal @ Omega ) )
       => ( ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ A4 @ Mu ) )
          = ( sigma_7808855514367478112nnreal @ Omega @ A4 ) ) )
      & ( ~ ( ord_le3366939622266546180nnreal @ A4 @ ( pow_Ex5372160365422184283nnreal @ Omega ) )
       => ( ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ A4 @ Mu ) )
          = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ Omega @ bot_bo2988155216863113784nnreal ) ) ) ) ) ).

% sets_measure_of_conv
thf(fact_796_sets__measure__of__conv,axiom,
    ! [A4: set_set_o,Omega: set_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( ( ord_le4374716579403074808_set_o @ A4 @ ( pow_o @ Omega ) )
       => ( ( sigma_sets_o @ ( sigma_measure_of_o @ Omega @ A4 @ Mu ) )
          = ( sigma_sigma_sets_o @ Omega @ A4 ) ) )
      & ( ~ ( ord_le4374716579403074808_set_o @ A4 @ ( pow_o @ Omega ) )
       => ( ( sigma_sets_o @ ( sigma_measure_of_o @ Omega @ A4 @ Mu ) )
          = ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ Omega @ bot_bot_set_set_o ) ) ) ) ) ).

% sets_measure_of_conv
thf(fact_797_sigma__algebra__single__set,axiom,
    ! [X2: set_real,S4: set_real] :
      ( ( ord_less_eq_set_real @ X2 @ S4 )
     => ( sigma_1481383337440427903a_real @ S4 @ ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ X2 @ ( insert_set_real @ ( minus_minus_set_real @ S4 @ X2 ) @ ( insert_set_real @ S4 @ bot_bot_set_set_real ) ) ) ) ) ) ).

% sigma_algebra_single_set
thf(fact_798_sigma__algebra__single__set,axiom,
    ! [X2: set_nat,S4: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ S4 )
     => ( sigma_8817008012692346403ra_nat @ S4 @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X2 @ ( insert_set_nat @ ( minus_minus_set_nat @ S4 @ X2 ) @ ( insert_set_nat @ S4 @ bot_bot_set_set_nat ) ) ) ) ) ) ).

% sigma_algebra_single_set
thf(fact_799_sigma__algebra__single__set,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,S4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ X2 @ S4 )
     => ( sigma_2413694886200424843nnreal @ S4 @ ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ X2 @ ( insert1343806209672318238nnreal @ ( minus_104578273773384135nnreal @ S4 @ X2 ) @ ( insert1343806209672318238nnreal @ S4 @ bot_bo2988155216863113784nnreal ) ) ) ) ) ) ).

% sigma_algebra_single_set
thf(fact_800_sigma__algebra__single__set,axiom,
    ! [X2: set_o,S4: set_o] :
      ( ( ord_less_eq_set_o @ X2 @ S4 )
     => ( sigma_3687534776968752773ebra_o @ S4 @ ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ X2 @ ( insert_set_o @ ( minus_minus_set_o @ S4 @ X2 ) @ ( insert_set_o @ S4 @ bot_bot_set_set_o ) ) ) ) ) ) ).

% sigma_algebra_single_set
thf(fact_801_Diff__iff,axiom,
    ! [C: real > a,A4: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ ( minus_6532636778494125008real_a @ A4 @ B4 ) )
      = ( ( member_real_a @ C @ A4 )
        & ~ ( member_real_a @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_802_Diff__iff,axiom,
    ! [C: $o > real,A4: set_o_real,B4: set_o_real] :
      ( ( member_o_real @ C @ ( minus_2870878895999678972o_real @ A4 @ B4 ) )
      = ( ( member_o_real @ C @ A4 )
        & ~ ( member_o_real @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_803_Diff__iff,axiom,
    ! [C: nat > real,A4: set_nat_real,B4: set_nat_real] :
      ( ( member_nat_real @ C @ ( minus_3492551254948764970t_real @ A4 @ B4 ) )
      = ( ( member_nat_real @ C @ A4 )
        & ~ ( member_nat_real @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_804_Diff__iff,axiom,
    ! [C: c > b,A4: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ ( minus_minus_set_c_b @ A4 @ B4 ) )
      = ( ( member_c_b @ C @ A4 )
        & ~ ( member_c_b @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_805_Diff__iff,axiom,
    ! [C: a > b,A4: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ ( minus_minus_set_a_b @ A4 @ B4 ) )
      = ( ( member_a_b @ C @ A4 )
        & ~ ( member_a_b @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_806_DiffI,axiom,
    ! [C: real > a,A4: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ A4 )
     => ( ~ ( member_real_a @ C @ B4 )
       => ( member_real_a @ C @ ( minus_6532636778494125008real_a @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_807_DiffI,axiom,
    ! [C: $o > real,A4: set_o_real,B4: set_o_real] :
      ( ( member_o_real @ C @ A4 )
     => ( ~ ( member_o_real @ C @ B4 )
       => ( member_o_real @ C @ ( minus_2870878895999678972o_real @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_808_DiffI,axiom,
    ! [C: nat > real,A4: set_nat_real,B4: set_nat_real] :
      ( ( member_nat_real @ C @ A4 )
     => ( ~ ( member_nat_real @ C @ B4 )
       => ( member_nat_real @ C @ ( minus_3492551254948764970t_real @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_809_DiffI,axiom,
    ! [C: c > b,A4: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ A4 )
     => ( ~ ( member_c_b @ C @ B4 )
       => ( member_c_b @ C @ ( minus_minus_set_c_b @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_810_DiffI,axiom,
    ! [C: a > b,A4: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ A4 )
     => ( ~ ( member_a_b @ C @ B4 )
       => ( member_a_b @ C @ ( minus_minus_set_a_b @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_811_Diff__empty,axiom,
    ! [A4: set_real] :
      ( ( minus_minus_set_real @ A4 @ bot_bot_set_real )
      = A4 ) ).

% Diff_empty
thf(fact_812_Diff__empty,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
      = A4 ) ).

% Diff_empty
thf(fact_813_Diff__empty,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( minus_104578273773384135nnreal @ A4 @ bot_bo4854962954004695426nnreal )
      = A4 ) ).

% Diff_empty
thf(fact_814_Diff__empty,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ A4 @ bot_bot_set_o )
      = A4 ) ).

% Diff_empty
thf(fact_815_empty__Diff,axiom,
    ! [A4: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A4 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_816_empty__Diff,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_817_empty__Diff,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( minus_104578273773384135nnreal @ bot_bo4854962954004695426nnreal @ A4 )
      = bot_bo4854962954004695426nnreal ) ).

% empty_Diff
thf(fact_818_empty__Diff,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A4 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_819_Diff__cancel,axiom,
    ! [A4: set_real] :
      ( ( minus_minus_set_real @ A4 @ A4 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_820_Diff__cancel,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ A4 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_821_Diff__cancel,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( minus_104578273773384135nnreal @ A4 @ A4 )
      = bot_bo4854962954004695426nnreal ) ).

% Diff_cancel
thf(fact_822_Diff__cancel,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ A4 @ A4 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_823_insert__Diff1,axiom,
    ! [X3: real,B4: set_real,A4: set_real] :
      ( ( member_real @ X3 @ B4 )
     => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A4 ) @ B4 )
        = ( minus_minus_set_real @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_824_insert__Diff1,axiom,
    ! [X3: nat,B4: set_nat,A4: set_nat] :
      ( ( member_nat @ X3 @ B4 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A4 ) @ B4 )
        = ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_825_insert__Diff1,axiom,
    ! [X3: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ X3 @ B4 )
     => ( ( minus_104578273773384135nnreal @ ( insert7407984058720857448nnreal @ X3 @ A4 ) @ B4 )
        = ( minus_104578273773384135nnreal @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_826_insert__Diff1,axiom,
    ! [X3: $o,B4: set_o,A4: set_o] :
      ( ( member_o @ X3 @ B4 )
     => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
        = ( minus_minus_set_o @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_827_insert__Diff1,axiom,
    ! [X3: real > a,B4: set_real_a,A4: set_real_a] :
      ( ( member_real_a @ X3 @ B4 )
     => ( ( minus_6532636778494125008real_a @ ( insert_real_a @ X3 @ A4 ) @ B4 )
        = ( minus_6532636778494125008real_a @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_828_insert__Diff1,axiom,
    ! [X3: $o > real,B4: set_o_real,A4: set_o_real] :
      ( ( member_o_real @ X3 @ B4 )
     => ( ( minus_2870878895999678972o_real @ ( insert_o_real @ X3 @ A4 ) @ B4 )
        = ( minus_2870878895999678972o_real @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_829_insert__Diff1,axiom,
    ! [X3: nat > real,B4: set_nat_real,A4: set_nat_real] :
      ( ( member_nat_real @ X3 @ B4 )
     => ( ( minus_3492551254948764970t_real @ ( insert_nat_real @ X3 @ A4 ) @ B4 )
        = ( minus_3492551254948764970t_real @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_830_insert__Diff1,axiom,
    ! [X3: c > b,B4: set_c_b,A4: set_c_b] :
      ( ( member_c_b @ X3 @ B4 )
     => ( ( minus_minus_set_c_b @ ( insert_c_b @ X3 @ A4 ) @ B4 )
        = ( minus_minus_set_c_b @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_831_insert__Diff1,axiom,
    ! [X3: a > b,B4: set_a_b,A4: set_a_b] :
      ( ( member_a_b @ X3 @ B4 )
     => ( ( minus_minus_set_a_b @ ( insert_a_b @ X3 @ A4 ) @ B4 )
        = ( minus_minus_set_a_b @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_832_Diff__insert0,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ X3 @ A4 )
     => ( ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ B4 ) )
        = ( minus_minus_set_real @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_833_Diff__insert0,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X3 @ A4 )
     => ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ B4 ) )
        = ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_834_Diff__insert0,axiom,
    ! [X3: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ X3 @ A4 )
     => ( ( minus_104578273773384135nnreal @ A4 @ ( insert7407984058720857448nnreal @ X3 @ B4 ) )
        = ( minus_104578273773384135nnreal @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_835_Diff__insert0,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ X3 @ A4 )
     => ( ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
        = ( minus_minus_set_o @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_836_Diff__insert0,axiom,
    ! [X3: real > a,A4: set_real_a,B4: set_real_a] :
      ( ~ ( member_real_a @ X3 @ A4 )
     => ( ( minus_6532636778494125008real_a @ A4 @ ( insert_real_a @ X3 @ B4 ) )
        = ( minus_6532636778494125008real_a @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_837_Diff__insert0,axiom,
    ! [X3: $o > real,A4: set_o_real,B4: set_o_real] :
      ( ~ ( member_o_real @ X3 @ A4 )
     => ( ( minus_2870878895999678972o_real @ A4 @ ( insert_o_real @ X3 @ B4 ) )
        = ( minus_2870878895999678972o_real @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_838_Diff__insert0,axiom,
    ! [X3: nat > real,A4: set_nat_real,B4: set_nat_real] :
      ( ~ ( member_nat_real @ X3 @ A4 )
     => ( ( minus_3492551254948764970t_real @ A4 @ ( insert_nat_real @ X3 @ B4 ) )
        = ( minus_3492551254948764970t_real @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_839_Diff__insert0,axiom,
    ! [X3: c > b,A4: set_c_b,B4: set_c_b] :
      ( ~ ( member_c_b @ X3 @ A4 )
     => ( ( minus_minus_set_c_b @ A4 @ ( insert_c_b @ X3 @ B4 ) )
        = ( minus_minus_set_c_b @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_840_Diff__insert0,axiom,
    ! [X3: a > b,A4: set_a_b,B4: set_a_b] :
      ( ~ ( member_a_b @ X3 @ A4 )
     => ( ( minus_minus_set_a_b @ A4 @ ( insert_a_b @ X3 @ B4 ) )
        = ( minus_minus_set_a_b @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_841_sets_ODiff,axiom,
    ! [A: set_real,M: sigma_measure_real,B: set_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ M ) )
       => ( member_set_real @ ( minus_minus_set_real @ A @ B ) @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.Diff
thf(fact_842_sets_ODiff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal @ B @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( member603777416030116741nnreal @ ( minus_104578273773384135nnreal @ A @ B ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.Diff
thf(fact_843_sets_ODiff,axiom,
    ! [A: set_o,M: sigma_measure_o,B: set_o] :
      ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o @ B @ ( sigma_sets_o @ M ) )
       => ( member_set_o @ ( minus_minus_set_o @ A @ B ) @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.Diff
thf(fact_844_sets_ODiff,axiom,
    ! [A: set_nat,M: sigma_measure_nat,B: set_nat] :
      ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat @ B @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat @ ( minus_minus_set_nat @ A @ B ) @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.Diff
thf(fact_845_Diff__UNIV,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ top_top_set_nat )
      = bot_bot_set_nat ) ).

% Diff_UNIV
thf(fact_846_Diff__UNIV,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( minus_104578273773384135nnreal @ A4 @ top_to7994903218803871134nnreal )
      = bot_bo4854962954004695426nnreal ) ).

% Diff_UNIV
thf(fact_847_Diff__UNIV,axiom,
    ! [A4: set_real] :
      ( ( minus_minus_set_real @ A4 @ top_top_set_real )
      = bot_bot_set_real ) ).

% Diff_UNIV
thf(fact_848_Diff__UNIV,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ A4 @ top_top_set_o )
      = bot_bot_set_o ) ).

% Diff_UNIV
thf(fact_849_Diff__eq__empty__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( ( minus_minus_set_real @ A4 @ B4 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_850_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( minus_minus_set_nat @ A4 @ B4 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_851_Diff__eq__empty__iff,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( ( minus_104578273773384135nnreal @ A4 @ B4 )
        = bot_bo4854962954004695426nnreal )
      = ( ord_le6787938422905777998nnreal @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_852_Diff__eq__empty__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ( minus_minus_set_o @ A4 @ B4 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_853_insert__Diff__single,axiom,
    ! [A: real,A4: set_real] :
      ( ( insert_real @ A @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
      = ( insert_real @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_854_insert__Diff__single,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_855_insert__Diff__single,axiom,
    ! [A: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ A @ ( minus_104578273773384135nnreal @ A4 @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) )
      = ( insert7407984058720857448nnreal @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_856_insert__Diff__single,axiom,
    ! [A: $o,A4: set_o] :
      ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
      = ( insert_o @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_857_sets_Ocompl__sets,axiom,
    ! [A: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
     => ( member_set_real @ ( minus_minus_set_real @ ( sigma_space_real @ M ) @ A ) @ ( sigma_sets_real @ M ) ) ) ).

% sets.compl_sets
thf(fact_858_sets_Ocompl__sets,axiom,
    ! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( member603777416030116741nnreal @ ( minus_104578273773384135nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ A ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).

% sets.compl_sets
thf(fact_859_sets_Ocompl__sets,axiom,
    ! [A: set_o,M: sigma_measure_o] :
      ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
     => ( member_set_o @ ( minus_minus_set_o @ ( sigma_space_o @ M ) @ A ) @ ( sigma_sets_o @ M ) ) ) ).

% sets.compl_sets
thf(fact_860_sets_Ocompl__sets,axiom,
    ! [A: set_nat,M: sigma_measure_nat] :
      ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
     => ( member_set_nat @ ( minus_minus_set_nat @ ( sigma_space_nat @ M ) @ A ) @ ( sigma_sets_nat @ M ) ) ) ).

% sets.compl_sets
thf(fact_861_sets_Osets__measure__of__eq,axiom,
    ! [M: sigma_measure_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) @ Mu ) )
      = ( sigma_sets_real @ M ) ) ).

% sets.sets_measure_of_eq
thf(fact_862_sets_Osets__measure__of__eq,axiom,
    ! [M: sigma_7234349610311085201nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) @ Mu ) )
      = ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.sets_measure_of_eq
thf(fact_863_sets_Osets__measure__of__eq,axiom,
    ! [M: sigma_measure_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( sigma_sets_o @ ( sigma_measure_of_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) @ Mu ) )
      = ( sigma_sets_o @ M ) ) ).

% sets.sets_measure_of_eq
thf(fact_864_sets_Osets__measure__of__eq,axiom,
    ! [M: sigma_measure_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( sigma_sets_nat @ ( sigma_measure_of_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) @ Mu ) )
      = ( sigma_sets_nat @ M ) ) ).

% sets.sets_measure_of_eq
thf(fact_865_sets_Ospace__measure__of__eq,axiom,
    ! [M: sigma_measure_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( sigma_space_real @ ( sigma_2693083824694760531f_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) @ Mu ) )
      = ( sigma_space_real @ M ) ) ).

% sets.space_measure_of_eq
thf(fact_866_sets_Ospace__measure__of__eq,axiom,
    ! [M: sigma_7234349610311085201nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( sigma_3147302497200244656nnreal @ ( sigma_8167827323036178527nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) @ Mu ) )
      = ( sigma_3147302497200244656nnreal @ M ) ) ).

% sets.space_measure_of_eq
thf(fact_867_sets_Ospace__measure__of__eq,axiom,
    ! [M: sigma_measure_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( sigma_space_o @ ( sigma_measure_of_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) @ Mu ) )
      = ( sigma_space_o @ M ) ) ).

% sets.space_measure_of_eq
thf(fact_868_sets_Ospace__measure__of__eq,axiom,
    ! [M: sigma_measure_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( sigma_space_nat @ ( sigma_measure_of_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) @ Mu ) )
      = ( sigma_space_nat @ M ) ) ).

% sets.space_measure_of_eq
thf(fact_869_countable__Diff__eq,axiom,
    ! [A4: set_real,X3: real] :
      ( ( counta7319604579010473777e_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) )
      = ( counta7319604579010473777e_real @ A4 ) ) ).

% countable_Diff_eq
thf(fact_870_countable__Diff__eq,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( counta1168086296615599829le_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
      = ( counta1168086296615599829le_nat @ A4 ) ) ).

% countable_Diff_eq
thf(fact_871_countable__Diff__eq,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal] :
      ( ( counta8439243037236335165nnreal @ ( minus_104578273773384135nnreal @ A4 @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) )
      = ( counta8439243037236335165nnreal @ A4 ) ) ).

% countable_Diff_eq
thf(fact_872_countable__Diff__eq,axiom,
    ! [A4: set_o,X3: $o] :
      ( ( counta5976203206615340371able_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) )
      = ( counta5976203206615340371able_o @ A4 ) ) ).

% countable_Diff_eq
thf(fact_873_in__measure__of,axiom,
    ! [M: set_set_real,Omega: set_real,A4: set_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( ord_le3558479182127378552t_real @ M @ ( pow_real @ Omega ) )
     => ( ( member_set_real @ A4 @ M )
       => ( member_set_real @ A4 @ ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega @ M @ Mu ) ) ) ) ) ).

% in_measure_of
thf(fact_874_in__measure__of,axiom,
    ! [M: set_se4580700918925141924nnreal,Omega: set_Ex3793607809372303086nnreal,A4: set_Ex3793607809372303086nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( ord_le3366939622266546180nnreal @ M @ ( pow_Ex5372160365422184283nnreal @ Omega ) )
     => ( ( member603777416030116741nnreal @ A4 @ M )
       => ( member603777416030116741nnreal @ A4 @ ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ M @ Mu ) ) ) ) ) ).

% in_measure_of
thf(fact_875_in__measure__of,axiom,
    ! [M: set_set_o,Omega: set_o,A4: set_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( ord_le4374716579403074808_set_o @ M @ ( pow_o @ Omega ) )
     => ( ( member_set_o @ A4 @ M )
       => ( member_set_o @ A4 @ ( sigma_sets_o @ ( sigma_measure_of_o @ Omega @ M @ Mu ) ) ) ) ) ).

% in_measure_of
thf(fact_876_in__measure__of,axiom,
    ! [M: set_set_nat,Omega: set_nat,A4: set_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( ord_le6893508408891458716et_nat @ M @ ( pow_nat @ Omega ) )
     => ( ( member_set_nat @ A4 @ M )
       => ( member_set_nat @ A4 @ ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega @ M @ Mu ) ) ) ) ) ).

% in_measure_of
thf(fact_877_space__measure__of,axiom,
    ! [A4: set_set_real,Omega: set_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( ord_le3558479182127378552t_real @ A4 @ ( pow_real @ Omega ) )
     => ( ( sigma_space_real @ ( sigma_2693083824694760531f_real @ Omega @ A4 @ Mu ) )
        = Omega ) ) ).

% space_measure_of
thf(fact_878_space__measure__of,axiom,
    ! [A4: set_se4580700918925141924nnreal,Omega: set_Ex3793607809372303086nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A4 @ ( pow_Ex5372160365422184283nnreal @ Omega ) )
     => ( ( sigma_3147302497200244656nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ A4 @ Mu ) )
        = Omega ) ) ).

% space_measure_of
thf(fact_879_space__measure__of,axiom,
    ! [A4: set_set_o,Omega: set_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( ord_le4374716579403074808_set_o @ A4 @ ( pow_o @ Omega ) )
     => ( ( sigma_space_o @ ( sigma_measure_of_o @ Omega @ A4 @ Mu ) )
        = Omega ) ) ).

% space_measure_of
thf(fact_880_space__measure__of,axiom,
    ! [A4: set_set_nat,Omega: set_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ ( pow_nat @ Omega ) )
     => ( ( sigma_space_nat @ ( sigma_measure_of_nat @ Omega @ A4 @ Mu ) )
        = Omega ) ) ).

% space_measure_of
thf(fact_881_sets__measure__of,axiom,
    ! [A4: set_set_real,Omega: set_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( ord_le3558479182127378552t_real @ A4 @ ( pow_real @ Omega ) )
     => ( ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega @ A4 @ Mu ) )
        = ( sigma_7195353284648819924s_real @ Omega @ A4 ) ) ) ).

% sets_measure_of
thf(fact_882_sets__measure__of,axiom,
    ! [A4: set_se4580700918925141924nnreal,Omega: set_Ex3793607809372303086nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A4 @ ( pow_Ex5372160365422184283nnreal @ Omega ) )
     => ( ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ A4 @ Mu ) )
        = ( sigma_7808855514367478112nnreal @ Omega @ A4 ) ) ) ).

% sets_measure_of
thf(fact_883_sets__measure__of,axiom,
    ! [A4: set_set_o,Omega: set_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( ord_le4374716579403074808_set_o @ A4 @ ( pow_o @ Omega ) )
     => ( ( sigma_sets_o @ ( sigma_measure_of_o @ Omega @ A4 @ Mu ) )
        = ( sigma_sigma_sets_o @ Omega @ A4 ) ) ) ).

% sets_measure_of
thf(fact_884_sets__measure__of,axiom,
    ! [A4: set_set_nat,Omega: set_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ ( pow_nat @ Omega ) )
     => ( ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega @ A4 @ Mu ) )
        = ( sigma_sigma_sets_nat @ Omega @ A4 ) ) ) ).

% sets_measure_of
thf(fact_885_DiffD2,axiom,
    ! [C: real > a,A4: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ ( minus_6532636778494125008real_a @ A4 @ B4 ) )
     => ~ ( member_real_a @ C @ B4 ) ) ).

% DiffD2
thf(fact_886_DiffD2,axiom,
    ! [C: $o > real,A4: set_o_real,B4: set_o_real] :
      ( ( member_o_real @ C @ ( minus_2870878895999678972o_real @ A4 @ B4 ) )
     => ~ ( member_o_real @ C @ B4 ) ) ).

% DiffD2
thf(fact_887_DiffD2,axiom,
    ! [C: nat > real,A4: set_nat_real,B4: set_nat_real] :
      ( ( member_nat_real @ C @ ( minus_3492551254948764970t_real @ A4 @ B4 ) )
     => ~ ( member_nat_real @ C @ B4 ) ) ).

% DiffD2
thf(fact_888_DiffD2,axiom,
    ! [C: c > b,A4: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ ( minus_minus_set_c_b @ A4 @ B4 ) )
     => ~ ( member_c_b @ C @ B4 ) ) ).

% DiffD2
thf(fact_889_DiffD2,axiom,
    ! [C: a > b,A4: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ ( minus_minus_set_a_b @ A4 @ B4 ) )
     => ~ ( member_a_b @ C @ B4 ) ) ).

% DiffD2
thf(fact_890_DiffD1,axiom,
    ! [C: real > a,A4: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ ( minus_6532636778494125008real_a @ A4 @ B4 ) )
     => ( member_real_a @ C @ A4 ) ) ).

% DiffD1
thf(fact_891_DiffD1,axiom,
    ! [C: $o > real,A4: set_o_real,B4: set_o_real] :
      ( ( member_o_real @ C @ ( minus_2870878895999678972o_real @ A4 @ B4 ) )
     => ( member_o_real @ C @ A4 ) ) ).

% DiffD1
thf(fact_892_DiffD1,axiom,
    ! [C: nat > real,A4: set_nat_real,B4: set_nat_real] :
      ( ( member_nat_real @ C @ ( minus_3492551254948764970t_real @ A4 @ B4 ) )
     => ( member_nat_real @ C @ A4 ) ) ).

% DiffD1
thf(fact_893_DiffD1,axiom,
    ! [C: c > b,A4: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ ( minus_minus_set_c_b @ A4 @ B4 ) )
     => ( member_c_b @ C @ A4 ) ) ).

% DiffD1
thf(fact_894_DiffD1,axiom,
    ! [C: a > b,A4: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ ( minus_minus_set_a_b @ A4 @ B4 ) )
     => ( member_a_b @ C @ A4 ) ) ).

% DiffD1
thf(fact_895_DiffE,axiom,
    ! [C: real > a,A4: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ ( minus_6532636778494125008real_a @ A4 @ B4 ) )
     => ~ ( ( member_real_a @ C @ A4 )
         => ( member_real_a @ C @ B4 ) ) ) ).

% DiffE
thf(fact_896_DiffE,axiom,
    ! [C: $o > real,A4: set_o_real,B4: set_o_real] :
      ( ( member_o_real @ C @ ( minus_2870878895999678972o_real @ A4 @ B4 ) )
     => ~ ( ( member_o_real @ C @ A4 )
         => ( member_o_real @ C @ B4 ) ) ) ).

% DiffE
thf(fact_897_DiffE,axiom,
    ! [C: nat > real,A4: set_nat_real,B4: set_nat_real] :
      ( ( member_nat_real @ C @ ( minus_3492551254948764970t_real @ A4 @ B4 ) )
     => ~ ( ( member_nat_real @ C @ A4 )
         => ( member_nat_real @ C @ B4 ) ) ) ).

% DiffE
thf(fact_898_DiffE,axiom,
    ! [C: c > b,A4: set_c_b,B4: set_c_b] :
      ( ( member_c_b @ C @ ( minus_minus_set_c_b @ A4 @ B4 ) )
     => ~ ( ( member_c_b @ C @ A4 )
         => ( member_c_b @ C @ B4 ) ) ) ).

% DiffE
thf(fact_899_DiffE,axiom,
    ! [C: a > b,A4: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ ( minus_minus_set_a_b @ A4 @ B4 ) )
     => ~ ( ( member_a_b @ C @ A4 )
         => ( member_a_b @ C @ B4 ) ) ) ).

% DiffE
thf(fact_900_space__in__measure__of,axiom,
    ! [Omega: set_real,M: set_set_real,Mu: set_real > extend8495563244428889912nnreal] : ( member_set_real @ Omega @ ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega @ M @ Mu ) ) ) ).

% space_in_measure_of
thf(fact_901_space__in__measure__of,axiom,
    ! [Omega: set_Ex3793607809372303086nnreal,M: set_se4580700918925141924nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] : ( member603777416030116741nnreal @ Omega @ ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ M @ Mu ) ) ) ).

% space_in_measure_of
thf(fact_902_space__in__measure__of,axiom,
    ! [Omega: set_o,M: set_set_o,Mu: set_o > extend8495563244428889912nnreal] : ( member_set_o @ Omega @ ( sigma_sets_o @ ( sigma_measure_of_o @ Omega @ M @ Mu ) ) ) ).

% space_in_measure_of
thf(fact_903_space__in__measure__of,axiom,
    ! [Omega: set_nat,M: set_set_nat,Mu: set_nat > extend8495563244428889912nnreal] : ( member_set_nat @ Omega @ ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega @ M @ Mu ) ) ) ).

% space_in_measure_of
thf(fact_904_space__measure__of__conv,axiom,
    ! [Omega: set_real,A4: set_set_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( sigma_space_real @ ( sigma_2693083824694760531f_real @ Omega @ A4 @ Mu ) )
      = Omega ) ).

% space_measure_of_conv
thf(fact_905_space__measure__of__conv,axiom,
    ! [Omega: set_Ex3793607809372303086nnreal,A4: set_se4580700918925141924nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( sigma_3147302497200244656nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ A4 @ Mu ) )
      = Omega ) ).

% space_measure_of_conv
thf(fact_906_space__measure__of__conv,axiom,
    ! [Omega: set_o,A4: set_set_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( sigma_space_o @ ( sigma_measure_of_o @ Omega @ A4 @ Mu ) )
      = Omega ) ).

% space_measure_of_conv
thf(fact_907_space__measure__of__conv,axiom,
    ! [Omega: set_nat,A4: set_set_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( sigma_space_nat @ ( sigma_measure_of_nat @ Omega @ A4 @ Mu ) )
      = Omega ) ).

% space_measure_of_conv
thf(fact_908_less__eq__quasi__borel_Ointros_I2_J,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_space_a @ X2 )
        = ( qbs_space_a @ Y ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ Y ) @ ( qbs_Mx_a @ X2 ) )
       => ( ord_le1843388692487544644orel_a @ X2 @ Y ) ) ) ).

% less_eq_quasi_borel.intros(2)
thf(fact_909_less__eq__quasi__borel_Ointros_I2_J,axiom,
    ! [X2: quasi_borel_c,Y: quasi_borel_c] :
      ( ( ( qbs_space_c @ X2 )
        = ( qbs_space_c @ Y ) )
     => ( ( ord_le5885474903713786379real_c @ ( qbs_Mx_c @ Y ) @ ( qbs_Mx_c @ X2 ) )
       => ( ord_le1843388701094002246orel_c @ X2 @ Y ) ) ) ).

% less_eq_quasi_borel.intros(2)
thf(fact_910_insert__Diff__if,axiom,
    ! [X3: real,B4: set_real,A4: set_real] :
      ( ( ( member_real @ X3 @ B4 )
       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A4 ) @ B4 )
          = ( minus_minus_set_real @ A4 @ B4 ) ) )
      & ( ~ ( member_real @ X3 @ B4 )
       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A4 ) @ B4 )
          = ( insert_real @ X3 @ ( minus_minus_set_real @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_911_insert__Diff__if,axiom,
    ! [X3: nat,B4: set_nat,A4: set_nat] :
      ( ( ( member_nat @ X3 @ B4 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A4 ) @ B4 )
          = ( minus_minus_set_nat @ A4 @ B4 ) ) )
      & ( ~ ( member_nat @ X3 @ B4 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A4 ) @ B4 )
          = ( insert_nat @ X3 @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_912_insert__Diff__if,axiom,
    ! [X3: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( ( member7908768830364227535nnreal @ X3 @ B4 )
       => ( ( minus_104578273773384135nnreal @ ( insert7407984058720857448nnreal @ X3 @ A4 ) @ B4 )
          = ( minus_104578273773384135nnreal @ A4 @ B4 ) ) )
      & ( ~ ( member7908768830364227535nnreal @ X3 @ B4 )
       => ( ( minus_104578273773384135nnreal @ ( insert7407984058720857448nnreal @ X3 @ A4 ) @ B4 )
          = ( insert7407984058720857448nnreal @ X3 @ ( minus_104578273773384135nnreal @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_913_insert__Diff__if,axiom,
    ! [X3: $o,B4: set_o,A4: set_o] :
      ( ( ( member_o @ X3 @ B4 )
       => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
          = ( minus_minus_set_o @ A4 @ B4 ) ) )
      & ( ~ ( member_o @ X3 @ B4 )
       => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
          = ( insert_o @ X3 @ ( minus_minus_set_o @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_914_insert__Diff__if,axiom,
    ! [X3: real > a,B4: set_real_a,A4: set_real_a] :
      ( ( ( member_real_a @ X3 @ B4 )
       => ( ( minus_6532636778494125008real_a @ ( insert_real_a @ X3 @ A4 ) @ B4 )
          = ( minus_6532636778494125008real_a @ A4 @ B4 ) ) )
      & ( ~ ( member_real_a @ X3 @ B4 )
       => ( ( minus_6532636778494125008real_a @ ( insert_real_a @ X3 @ A4 ) @ B4 )
          = ( insert_real_a @ X3 @ ( minus_6532636778494125008real_a @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_915_insert__Diff__if,axiom,
    ! [X3: $o > real,B4: set_o_real,A4: set_o_real] :
      ( ( ( member_o_real @ X3 @ B4 )
       => ( ( minus_2870878895999678972o_real @ ( insert_o_real @ X3 @ A4 ) @ B4 )
          = ( minus_2870878895999678972o_real @ A4 @ B4 ) ) )
      & ( ~ ( member_o_real @ X3 @ B4 )
       => ( ( minus_2870878895999678972o_real @ ( insert_o_real @ X3 @ A4 ) @ B4 )
          = ( insert_o_real @ X3 @ ( minus_2870878895999678972o_real @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_916_insert__Diff__if,axiom,
    ! [X3: nat > real,B4: set_nat_real,A4: set_nat_real] :
      ( ( ( member_nat_real @ X3 @ B4 )
       => ( ( minus_3492551254948764970t_real @ ( insert_nat_real @ X3 @ A4 ) @ B4 )
          = ( minus_3492551254948764970t_real @ A4 @ B4 ) ) )
      & ( ~ ( member_nat_real @ X3 @ B4 )
       => ( ( minus_3492551254948764970t_real @ ( insert_nat_real @ X3 @ A4 ) @ B4 )
          = ( insert_nat_real @ X3 @ ( minus_3492551254948764970t_real @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_917_insert__Diff__if,axiom,
    ! [X3: c > b,B4: set_c_b,A4: set_c_b] :
      ( ( ( member_c_b @ X3 @ B4 )
       => ( ( minus_minus_set_c_b @ ( insert_c_b @ X3 @ A4 ) @ B4 )
          = ( minus_minus_set_c_b @ A4 @ B4 ) ) )
      & ( ~ ( member_c_b @ X3 @ B4 )
       => ( ( minus_minus_set_c_b @ ( insert_c_b @ X3 @ A4 ) @ B4 )
          = ( insert_c_b @ X3 @ ( minus_minus_set_c_b @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_918_insert__Diff__if,axiom,
    ! [X3: a > b,B4: set_a_b,A4: set_a_b] :
      ( ( ( member_a_b @ X3 @ B4 )
       => ( ( minus_minus_set_a_b @ ( insert_a_b @ X3 @ A4 ) @ B4 )
          = ( minus_minus_set_a_b @ A4 @ B4 ) ) )
      & ( ~ ( member_a_b @ X3 @ B4 )
       => ( ( minus_minus_set_a_b @ ( insert_a_b @ X3 @ A4 ) @ B4 )
          = ( insert_a_b @ X3 @ ( minus_minus_set_a_b @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_919_sets__eq__iff__bounded,axiom,
    ! [A4: sigma_measure_real,B4: sigma_measure_real,C3: sigma_measure_real] :
      ( ( ord_le487379304121309861e_real @ A4 @ B4 )
     => ( ( ord_le487379304121309861e_real @ B4 @ C3 )
       => ( ( ( sigma_sets_real @ A4 )
            = ( sigma_sets_real @ C3 ) )
         => ( ( sigma_sets_real @ B4 )
            = ( sigma_sets_real @ A4 ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_920_sets__eq__iff__bounded,axiom,
    ! [A4: sigma_7234349610311085201nnreal,B4: sigma_7234349610311085201nnreal,C3: sigma_7234349610311085201nnreal] :
      ( ( ord_le1854472233513649201nnreal @ A4 @ B4 )
     => ( ( ord_le1854472233513649201nnreal @ B4 @ C3 )
       => ( ( ( sigma_5465916536984168985nnreal @ A4 )
            = ( sigma_5465916536984168985nnreal @ C3 ) )
         => ( ( sigma_5465916536984168985nnreal @ B4 )
            = ( sigma_5465916536984168985nnreal @ A4 ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_921_sets__eq__iff__bounded,axiom,
    ! [A4: sigma_measure_o,B4: sigma_measure_o,C3: sigma_measure_o] :
      ( ( ord_le478349814012620405sure_o @ A4 @ B4 )
     => ( ( ord_le478349814012620405sure_o @ B4 @ C3 )
       => ( ( ( sigma_sets_o @ A4 )
            = ( sigma_sets_o @ C3 ) )
         => ( ( sigma_sets_o @ B4 )
            = ( sigma_sets_o @ A4 ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_922_sets__eq__iff__bounded,axiom,
    ! [A4: sigma_measure_nat,B4: sigma_measure_nat,C3: sigma_measure_nat] :
      ( ( ord_le2862109966718184649re_nat @ A4 @ B4 )
     => ( ( ord_le2862109966718184649re_nat @ B4 @ C3 )
       => ( ( ( sigma_sets_nat @ A4 )
            = ( sigma_sets_nat @ C3 ) )
         => ( ( sigma_sets_nat @ B4 )
            = ( sigma_sets_nat @ A4 ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_923_sigma__algebra_Osets__measure__of__eq,axiom,
    ! [Omega: set_real,M: set_set_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( sigma_1481383337440427903a_real @ Omega @ M )
     => ( ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega @ M @ Mu ) )
        = M ) ) ).

% sigma_algebra.sets_measure_of_eq
thf(fact_924_sigma__algebra_Osets__measure__of__eq,axiom,
    ! [Omega: set_Ex3793607809372303086nnreal,M: set_se4580700918925141924nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( sigma_2413694886200424843nnreal @ Omega @ M )
     => ( ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ M @ Mu ) )
        = M ) ) ).

% sigma_algebra.sets_measure_of_eq
thf(fact_925_sigma__algebra_Osets__measure__of__eq,axiom,
    ! [Omega: set_o,M: set_set_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( sigma_3687534776968752773ebra_o @ Omega @ M )
     => ( ( sigma_sets_o @ ( sigma_measure_of_o @ Omega @ M @ Mu ) )
        = M ) ) ).

% sigma_algebra.sets_measure_of_eq
thf(fact_926_sigma__algebra_Osets__measure__of__eq,axiom,
    ! [Omega: set_nat,M: set_set_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( sigma_8817008012692346403ra_nat @ Omega @ M )
     => ( ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega @ M @ Mu ) )
        = M ) ) ).

% sigma_algebra.sets_measure_of_eq
thf(fact_927_sigma__algebra_Ospace__measure__of__eq,axiom,
    ! [Omega: set_real,M: set_set_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( sigma_1481383337440427903a_real @ Omega @ M )
     => ( ( sigma_space_real @ ( sigma_2693083824694760531f_real @ Omega @ M @ Mu ) )
        = Omega ) ) ).

% sigma_algebra.space_measure_of_eq
thf(fact_928_sigma__algebra_Ospace__measure__of__eq,axiom,
    ! [Omega: set_Ex3793607809372303086nnreal,M: set_se4580700918925141924nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( sigma_2413694886200424843nnreal @ Omega @ M )
     => ( ( sigma_3147302497200244656nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ M @ Mu ) )
        = Omega ) ) ).

% sigma_algebra.space_measure_of_eq
thf(fact_929_sigma__algebra_Ospace__measure__of__eq,axiom,
    ! [Omega: set_o,M: set_set_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( sigma_3687534776968752773ebra_o @ Omega @ M )
     => ( ( sigma_space_o @ ( sigma_measure_of_o @ Omega @ M @ Mu ) )
        = Omega ) ) ).

% sigma_algebra.space_measure_of_eq
thf(fact_930_sigma__algebra_Ospace__measure__of__eq,axiom,
    ! [Omega: set_nat,M: set_set_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( sigma_8817008012692346403ra_nat @ Omega @ M )
     => ( ( sigma_space_nat @ ( sigma_measure_of_nat @ Omega @ M @ Mu ) )
        = Omega ) ) ).

% sigma_algebra.space_measure_of_eq
thf(fact_931_Diff__insert,axiom,
    ! [A4: set_real,A: real,B4: set_real] :
      ( ( minus_minus_set_real @ A4 @ ( insert_real @ A @ B4 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A4 @ B4 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_932_Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_933_Diff__insert,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( minus_104578273773384135nnreal @ A4 @ ( insert7407984058720857448nnreal @ A @ B4 ) )
      = ( minus_104578273773384135nnreal @ ( minus_104578273773384135nnreal @ A4 @ B4 ) @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) ) ).

% Diff_insert
thf(fact_934_Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ B4 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_935_insert__Diff,axiom,
    ! [A: real > a,A4: set_real_a] :
      ( ( member_real_a @ A @ A4 )
     => ( ( insert_real_a @ A @ ( minus_6532636778494125008real_a @ A4 @ ( insert_real_a @ A @ bot_bot_set_real_a ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_936_insert__Diff,axiom,
    ! [A: $o > real,A4: set_o_real] :
      ( ( member_o_real @ A @ A4 )
     => ( ( insert_o_real @ A @ ( minus_2870878895999678972o_real @ A4 @ ( insert_o_real @ A @ bot_bot_set_o_real ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_937_insert__Diff,axiom,
    ! [A: nat > real,A4: set_nat_real] :
      ( ( member_nat_real @ A @ A4 )
     => ( ( insert_nat_real @ A @ ( minus_3492551254948764970t_real @ A4 @ ( insert_nat_real @ A @ bot_bot_set_nat_real ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_938_insert__Diff,axiom,
    ! [A: c > b,A4: set_c_b] :
      ( ( member_c_b @ A @ A4 )
     => ( ( insert_c_b @ A @ ( minus_minus_set_c_b @ A4 @ ( insert_c_b @ A @ bot_bot_set_c_b ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_939_insert__Diff,axiom,
    ! [A: a > b,A4: set_a_b] :
      ( ( member_a_b @ A @ A4 )
     => ( ( insert_a_b @ A @ ( minus_minus_set_a_b @ A4 @ ( insert_a_b @ A @ bot_bot_set_a_b ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_940_insert__Diff,axiom,
    ! [A: real,A4: set_real] :
      ( ( member_real @ A @ A4 )
     => ( ( insert_real @ A @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_941_insert__Diff,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_942_insert__Diff,axiom,
    ! [A: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A @ A4 )
     => ( ( insert7407984058720857448nnreal @ A @ ( minus_104578273773384135nnreal @ A4 @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_943_insert__Diff,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_944_Diff__insert2,axiom,
    ! [A4: set_real,A: real,B4: set_real] :
      ( ( minus_minus_set_real @ A4 @ ( insert_real @ A @ B4 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_945_Diff__insert2,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_946_Diff__insert2,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,A: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( minus_104578273773384135nnreal @ A4 @ ( insert7407984058720857448nnreal @ A @ B4 ) )
      = ( minus_104578273773384135nnreal @ ( minus_104578273773384135nnreal @ A4 @ ( insert7407984058720857448nnreal @ A @ bot_bo4854962954004695426nnreal ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_947_Diff__insert2,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_948_Diff__insert__absorb,axiom,
    ! [X3: real > a,A4: set_real_a] :
      ( ~ ( member_real_a @ X3 @ A4 )
     => ( ( minus_6532636778494125008real_a @ ( insert_real_a @ X3 @ A4 ) @ ( insert_real_a @ X3 @ bot_bot_set_real_a ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_949_Diff__insert__absorb,axiom,
    ! [X3: $o > real,A4: set_o_real] :
      ( ~ ( member_o_real @ X3 @ A4 )
     => ( ( minus_2870878895999678972o_real @ ( insert_o_real @ X3 @ A4 ) @ ( insert_o_real @ X3 @ bot_bot_set_o_real ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_950_Diff__insert__absorb,axiom,
    ! [X3: nat > real,A4: set_nat_real] :
      ( ~ ( member_nat_real @ X3 @ A4 )
     => ( ( minus_3492551254948764970t_real @ ( insert_nat_real @ X3 @ A4 ) @ ( insert_nat_real @ X3 @ bot_bot_set_nat_real ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_951_Diff__insert__absorb,axiom,
    ! [X3: c > b,A4: set_c_b] :
      ( ~ ( member_c_b @ X3 @ A4 )
     => ( ( minus_minus_set_c_b @ ( insert_c_b @ X3 @ A4 ) @ ( insert_c_b @ X3 @ bot_bot_set_c_b ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_952_Diff__insert__absorb,axiom,
    ! [X3: a > b,A4: set_a_b] :
      ( ~ ( member_a_b @ X3 @ A4 )
     => ( ( minus_minus_set_a_b @ ( insert_a_b @ X3 @ A4 ) @ ( insert_a_b @ X3 @ bot_bot_set_a_b ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_953_Diff__insert__absorb,axiom,
    ! [X3: real,A4: set_real] :
      ( ~ ( member_real @ X3 @ A4 )
     => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A4 ) @ ( insert_real @ X3 @ bot_bot_set_real ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_954_Diff__insert__absorb,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ~ ( member_nat @ X3 @ A4 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A4 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_955_Diff__insert__absorb,axiom,
    ! [X3: extend8495563244428889912nnreal,A4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ X3 @ A4 )
     => ( ( minus_104578273773384135nnreal @ ( insert7407984058720857448nnreal @ X3 @ A4 ) @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_956_Diff__insert__absorb,axiom,
    ! [X3: $o,A4: set_o] :
      ( ~ ( member_o @ X3 @ A4 )
     => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ ( insert_o @ X3 @ bot_bot_set_o ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_957_subset__Diff__insert,axiom,
    ! [A4: set_real,B4: set_real,X3: real,C3: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( minus_minus_set_real @ B4 @ ( insert_real @ X3 @ C3 ) ) )
      = ( ( ord_less_eq_set_real @ A4 @ ( minus_minus_set_real @ B4 @ C3 ) )
        & ~ ( member_real @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_958_subset__Diff__insert,axiom,
    ! [A4: set_nat,B4: set_nat,X3: nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X3 @ C3 ) ) )
      = ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ C3 ) )
        & ~ ( member_nat @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_959_subset__Diff__insert,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal,C3: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A4 @ ( minus_104578273773384135nnreal @ B4 @ ( insert7407984058720857448nnreal @ X3 @ C3 ) ) )
      = ( ( ord_le6787938422905777998nnreal @ A4 @ ( minus_104578273773384135nnreal @ B4 @ C3 ) )
        & ~ ( member7908768830364227535nnreal @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_960_subset__Diff__insert,axiom,
    ! [A4: set_o,B4: set_o,X3: $o,C3: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ ( insert_o @ X3 @ C3 ) ) )
      = ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ C3 ) )
        & ~ ( member_o @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_961_subset__Diff__insert,axiom,
    ! [A4: set_real_a,B4: set_real_a,X3: real > a,C3: set_real_a] :
      ( ( ord_le5743406823621094409real_a @ A4 @ ( minus_6532636778494125008real_a @ B4 @ ( insert_real_a @ X3 @ C3 ) ) )
      = ( ( ord_le5743406823621094409real_a @ A4 @ ( minus_6532636778494125008real_a @ B4 @ C3 ) )
        & ~ ( member_real_a @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_962_subset__Diff__insert,axiom,
    ! [A4: set_o_real,B4: set_o_real,X3: $o > real,C3: set_o_real] :
      ( ( ord_le3251842697534426805o_real @ A4 @ ( minus_2870878895999678972o_real @ B4 @ ( insert_o_real @ X3 @ C3 ) ) )
      = ( ( ord_le3251842697534426805o_real @ A4 @ ( minus_2870878895999678972o_real @ B4 @ C3 ) )
        & ~ ( member_o_real @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_963_subset__Diff__insert,axiom,
    ! [A4: set_nat_real,B4: set_nat_real,X3: nat > real,C3: set_nat_real] :
      ( ( ord_le2908806416726583473t_real @ A4 @ ( minus_3492551254948764970t_real @ B4 @ ( insert_nat_real @ X3 @ C3 ) ) )
      = ( ( ord_le2908806416726583473t_real @ A4 @ ( minus_3492551254948764970t_real @ B4 @ C3 ) )
        & ~ ( member_nat_real @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_964_subset__Diff__insert,axiom,
    ! [A4: set_c_b,B4: set_c_b,X3: c > b,C3: set_c_b] :
      ( ( ord_less_eq_set_c_b @ A4 @ ( minus_minus_set_c_b @ B4 @ ( insert_c_b @ X3 @ C3 ) ) )
      = ( ( ord_less_eq_set_c_b @ A4 @ ( minus_minus_set_c_b @ B4 @ C3 ) )
        & ~ ( member_c_b @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_965_subset__Diff__insert,axiom,
    ! [A4: set_a_b,B4: set_a_b,X3: a > b,C3: set_a_b] :
      ( ( ord_less_eq_set_a_b @ A4 @ ( minus_minus_set_a_b @ B4 @ ( insert_a_b @ X3 @ C3 ) ) )
      = ( ( ord_less_eq_set_a_b @ A4 @ ( minus_minus_set_a_b @ B4 @ C3 ) )
        & ~ ( member_a_b @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_966_measure__of__subset,axiom,
    ! [M: set_set_real,Omega: set_real,M2: set_set_real,Mu: set_real > extend8495563244428889912nnreal,Mu2: set_real > extend8495563244428889912nnreal] :
      ( ( ord_le3558479182127378552t_real @ M @ ( pow_real @ Omega ) )
     => ( ( ord_le3558479182127378552t_real @ M2 @ M )
       => ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega @ M2 @ Mu ) ) @ ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega @ M @ Mu2 ) ) ) ) ) ).

% measure_of_subset
thf(fact_967_measure__of__subset,axiom,
    ! [M: set_se4580700918925141924nnreal,Omega: set_Ex3793607809372303086nnreal,M2: set_se4580700918925141924nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal,Mu2: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( ord_le3366939622266546180nnreal @ M @ ( pow_Ex5372160365422184283nnreal @ Omega ) )
     => ( ( ord_le3366939622266546180nnreal @ M2 @ M )
       => ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ M2 @ Mu ) ) @ ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega @ M @ Mu2 ) ) ) ) ) ).

% measure_of_subset
thf(fact_968_measure__of__subset,axiom,
    ! [M: set_set_o,Omega: set_o,M2: set_set_o,Mu: set_o > extend8495563244428889912nnreal,Mu2: set_o > extend8495563244428889912nnreal] :
      ( ( ord_le4374716579403074808_set_o @ M @ ( pow_o @ Omega ) )
     => ( ( ord_le4374716579403074808_set_o @ M2 @ M )
       => ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ ( sigma_measure_of_o @ Omega @ M2 @ Mu ) ) @ ( sigma_sets_o @ ( sigma_measure_of_o @ Omega @ M @ Mu2 ) ) ) ) ) ).

% measure_of_subset
thf(fact_969_measure__of__subset,axiom,
    ! [M: set_set_nat,Omega: set_nat,M2: set_set_nat,Mu: set_nat > extend8495563244428889912nnreal,Mu2: set_nat > extend8495563244428889912nnreal] :
      ( ( ord_le6893508408891458716et_nat @ M @ ( pow_nat @ Omega ) )
     => ( ( ord_le6893508408891458716et_nat @ M2 @ M )
       => ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega @ M2 @ Mu ) ) @ ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega @ M @ Mu2 ) ) ) ) ) ).

% measure_of_subset
thf(fact_970_subset__insert__iff,axiom,
    ! [A4: set_real_a,X3: real > a,B4: set_real_a] :
      ( ( ord_le5743406823621094409real_a @ A4 @ ( insert_real_a @ X3 @ B4 ) )
      = ( ( ( member_real_a @ X3 @ A4 )
         => ( ord_le5743406823621094409real_a @ ( minus_6532636778494125008real_a @ A4 @ ( insert_real_a @ X3 @ bot_bot_set_real_a ) ) @ B4 ) )
        & ( ~ ( member_real_a @ X3 @ A4 )
         => ( ord_le5743406823621094409real_a @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_971_subset__insert__iff,axiom,
    ! [A4: set_o_real,X3: $o > real,B4: set_o_real] :
      ( ( ord_le3251842697534426805o_real @ A4 @ ( insert_o_real @ X3 @ B4 ) )
      = ( ( ( member_o_real @ X3 @ A4 )
         => ( ord_le3251842697534426805o_real @ ( minus_2870878895999678972o_real @ A4 @ ( insert_o_real @ X3 @ bot_bot_set_o_real ) ) @ B4 ) )
        & ( ~ ( member_o_real @ X3 @ A4 )
         => ( ord_le3251842697534426805o_real @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_972_subset__insert__iff,axiom,
    ! [A4: set_nat_real,X3: nat > real,B4: set_nat_real] :
      ( ( ord_le2908806416726583473t_real @ A4 @ ( insert_nat_real @ X3 @ B4 ) )
      = ( ( ( member_nat_real @ X3 @ A4 )
         => ( ord_le2908806416726583473t_real @ ( minus_3492551254948764970t_real @ A4 @ ( insert_nat_real @ X3 @ bot_bot_set_nat_real ) ) @ B4 ) )
        & ( ~ ( member_nat_real @ X3 @ A4 )
         => ( ord_le2908806416726583473t_real @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_973_subset__insert__iff,axiom,
    ! [A4: set_c_b,X3: c > b,B4: set_c_b] :
      ( ( ord_less_eq_set_c_b @ A4 @ ( insert_c_b @ X3 @ B4 ) )
      = ( ( ( member_c_b @ X3 @ A4 )
         => ( ord_less_eq_set_c_b @ ( minus_minus_set_c_b @ A4 @ ( insert_c_b @ X3 @ bot_bot_set_c_b ) ) @ B4 ) )
        & ( ~ ( member_c_b @ X3 @ A4 )
         => ( ord_less_eq_set_c_b @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_974_subset__insert__iff,axiom,
    ! [A4: set_a_b,X3: a > b,B4: set_a_b] :
      ( ( ord_less_eq_set_a_b @ A4 @ ( insert_a_b @ X3 @ B4 ) )
      = ( ( ( member_a_b @ X3 @ A4 )
         => ( ord_less_eq_set_a_b @ ( minus_minus_set_a_b @ A4 @ ( insert_a_b @ X3 @ bot_bot_set_a_b ) ) @ B4 ) )
        & ( ~ ( member_a_b @ X3 @ A4 )
         => ( ord_less_eq_set_a_b @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_975_subset__insert__iff,axiom,
    ! [A4: set_real,X3: real,B4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( insert_real @ X3 @ B4 ) )
      = ( ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B4 ) )
        & ( ~ ( member_real @ X3 @ A4 )
         => ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_976_subset__insert__iff,axiom,
    ! [A4: set_nat,X3: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X3 @ B4 ) )
      = ( ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B4 ) )
        & ( ~ ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_977_subset__insert__iff,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A4 @ ( insert7407984058720857448nnreal @ X3 @ B4 ) )
      = ( ( ( member7908768830364227535nnreal @ X3 @ A4 )
         => ( ord_le6787938422905777998nnreal @ ( minus_104578273773384135nnreal @ A4 @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) @ B4 ) )
        & ( ~ ( member7908768830364227535nnreal @ X3 @ A4 )
         => ( ord_le6787938422905777998nnreal @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_978_subset__insert__iff,axiom,
    ! [A4: set_o,X3: $o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
      = ( ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B4 ) )
        & ( ~ ( member_o @ X3 @ A4 )
         => ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_979_Diff__single__insert,axiom,
    ! [A4: set_real,X3: real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B4 )
     => ( ord_less_eq_set_real @ A4 @ ( insert_real @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_980_Diff__single__insert,axiom,
    ! [A4: set_nat,X3: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B4 )
     => ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_981_Diff__single__insert,axiom,
    ! [A4: set_Ex3793607809372303086nnreal,X3: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ ( minus_104578273773384135nnreal @ A4 @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) @ B4 )
     => ( ord_le6787938422905777998nnreal @ A4 @ ( insert7407984058720857448nnreal @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_982_Diff__single__insert,axiom,
    ! [A4: set_o,X3: $o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B4 )
     => ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_983_le__measureD2,axiom,
    ! [A4: sigma_measure_real,B4: sigma_measure_real] :
      ( ( ord_le487379304121309861e_real @ A4 @ B4 )
     => ( ( ( sigma_space_real @ A4 )
          = ( sigma_space_real @ B4 ) )
       => ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A4 ) @ ( sigma_sets_real @ B4 ) ) ) ) ).

% le_measureD2
thf(fact_984_le__measureD2,axiom,
    ! [A4: sigma_7234349610311085201nnreal,B4: sigma_7234349610311085201nnreal] :
      ( ( ord_le1854472233513649201nnreal @ A4 @ B4 )
     => ( ( ( sigma_3147302497200244656nnreal @ A4 )
          = ( sigma_3147302497200244656nnreal @ B4 ) )
       => ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ A4 ) @ ( sigma_5465916536984168985nnreal @ B4 ) ) ) ) ).

% le_measureD2
thf(fact_985_le__measureD2,axiom,
    ! [A4: sigma_measure_o,B4: sigma_measure_o] :
      ( ( ord_le478349814012620405sure_o @ A4 @ B4 )
     => ( ( ( sigma_space_o @ A4 )
          = ( sigma_space_o @ B4 ) )
       => ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ A4 ) @ ( sigma_sets_o @ B4 ) ) ) ) ).

% le_measureD2
thf(fact_986_le__measureD2,axiom,
    ! [A4: sigma_measure_nat,B4: sigma_measure_nat] :
      ( ( ord_le2862109966718184649re_nat @ A4 @ B4 )
     => ( ( ( sigma_space_nat @ A4 )
          = ( sigma_space_nat @ B4 ) )
       => ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ A4 ) @ ( sigma_sets_nat @ B4 ) ) ) ) ).

% le_measureD2
thf(fact_987_sigma__sets__singleton,axiom,
    ! [X2: set_real,S4: set_real] :
      ( ( ord_less_eq_set_real @ X2 @ S4 )
     => ( ( sigma_7195353284648819924s_real @ S4 @ ( insert_set_real @ X2 @ bot_bot_set_set_real ) )
        = ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ X2 @ ( insert_set_real @ ( minus_minus_set_real @ S4 @ X2 ) @ ( insert_set_real @ S4 @ bot_bot_set_set_real ) ) ) ) ) ) ).

% sigma_sets_singleton
thf(fact_988_sigma__sets__singleton,axiom,
    ! [X2: set_nat,S4: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ S4 )
     => ( ( sigma_sigma_sets_nat @ S4 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
        = ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X2 @ ( insert_set_nat @ ( minus_minus_set_nat @ S4 @ X2 ) @ ( insert_set_nat @ S4 @ bot_bot_set_set_nat ) ) ) ) ) ) ).

% sigma_sets_singleton
thf(fact_989_sigma__sets__singleton,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,S4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ X2 @ S4 )
     => ( ( sigma_7808855514367478112nnreal @ S4 @ ( insert1343806209672318238nnreal @ X2 @ bot_bo2988155216863113784nnreal ) )
        = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ X2 @ ( insert1343806209672318238nnreal @ ( minus_104578273773384135nnreal @ S4 @ X2 ) @ ( insert1343806209672318238nnreal @ S4 @ bot_bo2988155216863113784nnreal ) ) ) ) ) ) ).

% sigma_sets_singleton
thf(fact_990_sigma__sets__singleton,axiom,
    ! [X2: set_o,S4: set_o] :
      ( ( ord_less_eq_set_o @ X2 @ S4 )
     => ( ( sigma_sigma_sets_o @ S4 @ ( insert_set_o @ X2 @ bot_bot_set_set_o ) )
        = ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ X2 @ ( insert_set_o @ ( minus_minus_set_o @ S4 @ X2 ) @ ( insert_set_o @ S4 @ bot_bot_set_set_o ) ) ) ) ) ) ).

% sigma_sets_singleton
thf(fact_991_diff__shunt__var,axiom,
    ! [X3: set_real,Y2: set_real] :
      ( ( ( minus_minus_set_real @ X3 @ Y2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X3 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_992_diff__shunt__var,axiom,
    ! [X3: set_nat,Y2: set_nat] :
      ( ( ( minus_minus_set_nat @ X3 @ Y2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X3 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_993_diff__shunt__var,axiom,
    ! [X3: set_Ex3793607809372303086nnreal,Y2: set_Ex3793607809372303086nnreal] :
      ( ( ( minus_104578273773384135nnreal @ X3 @ Y2 )
        = bot_bo4854962954004695426nnreal )
      = ( ord_le6787938422905777998nnreal @ X3 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_994_diff__shunt__var,axiom,
    ! [X3: set_o,Y2: set_o] :
      ( ( ( minus_minus_set_o @ X3 @ Y2 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X3 @ Y2 ) ) ).

% diff_shunt_var
thf(fact_995_remove__def,axiom,
    ( remove_real
    = ( ^ [X5: real,A5: set_real] : ( minus_minus_set_real @ A5 @ ( insert_real @ X5 @ bot_bot_set_real ) ) ) ) ).

% remove_def
thf(fact_996_remove__def,axiom,
    ( remove_nat
    = ( ^ [X5: nat,A5: set_nat] : ( minus_minus_set_nat @ A5 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ).

% remove_def
thf(fact_997_remove__def,axiom,
    ( remove159653468147250451nnreal
    = ( ^ [X5: extend8495563244428889912nnreal,A5: set_Ex3793607809372303086nnreal] : ( minus_104578273773384135nnreal @ A5 @ ( insert7407984058720857448nnreal @ X5 @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% remove_def
thf(fact_998_remove__def,axiom,
    ( remove_o
    = ( ^ [X5: $o,A5: set_o] : ( minus_minus_set_o @ A5 @ ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ).

% remove_def
thf(fact_999_algebra__single__set,axiom,
    ! [X2: set_real,S4: set_real] :
      ( ( ord_less_eq_set_real @ X2 @ S4 )
     => ( sigma_algebra_real @ S4 @ ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ X2 @ ( insert_set_real @ ( minus_minus_set_real @ S4 @ X2 ) @ ( insert_set_real @ S4 @ bot_bot_set_set_real ) ) ) ) ) ) ).

% algebra_single_set
thf(fact_1000_algebra__single__set,axiom,
    ! [X2: set_nat,S4: set_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ S4 )
     => ( sigma_algebra_nat @ S4 @ ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X2 @ ( insert_set_nat @ ( minus_minus_set_nat @ S4 @ X2 ) @ ( insert_set_nat @ S4 @ bot_bot_set_set_nat ) ) ) ) ) ) ).

% algebra_single_set
thf(fact_1001_algebra__single__set,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,S4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ X2 @ S4 )
     => ( sigma_5981082695875523474nnreal @ S4 @ ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ X2 @ ( insert1343806209672318238nnreal @ ( minus_104578273773384135nnreal @ S4 @ X2 ) @ ( insert1343806209672318238nnreal @ S4 @ bot_bo2988155216863113784nnreal ) ) ) ) ) ) ).

% algebra_single_set
thf(fact_1002_algebra__single__set,axiom,
    ! [X2: set_o,S4: set_o] :
      ( ( ord_less_eq_set_o @ X2 @ S4 )
     => ( sigma_algebra_o @ S4 @ ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ X2 @ ( insert_set_o @ ( minus_minus_set_o @ S4 @ X2 ) @ ( insert_set_o @ S4 @ bot_bot_set_set_o ) ) ) ) ) ) ).

% algebra_single_set
thf(fact_1003_standard__borel__space__UNIV__axioms__def,axiom,
    ( standa602082540683807836nnreal
    = ( ^ [M4: sigma_7234349610311085201nnreal] :
          ( ( sigma_3147302497200244656nnreal @ M4 )
          = top_to7994903218803871134nnreal ) ) ) ).

% standard_borel_space_UNIV_axioms_def
thf(fact_1004_standard__borel__space__UNIV__axioms__def,axiom,
    ( standa4898135366436483316ms_nat
    = ( ^ [M4: sigma_measure_nat] :
          ( ( sigma_space_nat @ M4 )
          = top_top_set_nat ) ) ) ).

% standard_borel_space_UNIV_axioms_def
thf(fact_1005_standard__borel__space__UNIV__axioms__def,axiom,
    ( standa1498722272452280784s_real
    = ( ^ [M4: sigma_measure_real] :
          ( ( sigma_space_real @ M4 )
          = top_top_set_real ) ) ) ).

% standard_borel_space_UNIV_axioms_def
thf(fact_1006_standard__borel__space__UNIV__axioms__def,axiom,
    ( standa4575222554423029108ioms_o
    = ( ^ [M4: sigma_measure_o] :
          ( ( sigma_space_o @ M4 )
          = top_top_set_o ) ) ) ).

% standard_borel_space_UNIV_axioms_def
thf(fact_1007_standard__borel__space__UNIV__axioms_Ointro,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ M )
        = top_to7994903218803871134nnreal )
     => ( standa602082540683807836nnreal @ M ) ) ).

% standard_borel_space_UNIV_axioms.intro
thf(fact_1008_standard__borel__space__UNIV__axioms_Ointro,axiom,
    ! [M: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ M )
        = top_top_set_nat )
     => ( standa4898135366436483316ms_nat @ M ) ) ).

% standard_borel_space_UNIV_axioms.intro
thf(fact_1009_standard__borel__space__UNIV__axioms_Ointro,axiom,
    ! [M: sigma_measure_real] :
      ( ( ( sigma_space_real @ M )
        = top_top_set_real )
     => ( standa1498722272452280784s_real @ M ) ) ).

% standard_borel_space_UNIV_axioms.intro
thf(fact_1010_standard__borel__space__UNIV__axioms_Ointro,axiom,
    ! [M: sigma_measure_o] :
      ( ( ( sigma_space_o @ M )
        = top_top_set_o )
     => ( standa4575222554423029108ioms_o @ M ) ) ).

% standard_borel_space_UNIV_axioms.intro
thf(fact_1011_measure__eqI__countable_H,axiom,
    ! [M: sigma_measure_real_a,A4: set_real_a,N: sigma_measure_real_a] :
      ( ( ( sigma_space_real_a @ M )
        = A4 )
     => ( ( ( sigma_space_real_a @ N )
          = A4 )
       => ( ! [X: real > a] :
              ( ( member_real_a @ X @ A4 )
             => ( member_set_real_a @ ( insert_real_a @ X @ bot_bot_set_real_a ) @ ( sigma_sets_real_a @ M ) ) )
         => ( ! [X: real > a] :
                ( ( member_real_a @ X @ A4 )
               => ( member_set_real_a @ ( insert_real_a @ X @ bot_bot_set_real_a ) @ ( sigma_sets_real_a @ N ) ) )
           => ( ( counta6639396083684174020real_a @ A4 )
             => ( ! [A3: real > a] :
                    ( ( member_real_a @ A3 @ A4 )
                   => ( ( sigma_6502373073922819808real_a @ M @ ( insert_real_a @ A3 @ bot_bot_set_real_a ) )
                      = ( sigma_6502373073922819808real_a @ N @ ( insert_real_a @ A3 @ bot_bot_set_real_a ) ) ) )
               => ( M = N ) ) ) ) ) ) ) ).

% measure_eqI_countable'
thf(fact_1012_measure__eqI__countable_H,axiom,
    ! [M: sigma_measure_o_real,A4: set_o_real,N: sigma_measure_o_real] :
      ( ( ( sigma_space_o_real @ M )
        = A4 )
     => ( ( ( sigma_space_o_real @ N )
          = A4 )
       => ( ! [X: $o > real] :
              ( ( member_o_real @ X @ A4 )
             => ( member_set_o_real @ ( insert_o_real @ X @ bot_bot_set_o_real ) @ ( sigma_sets_o_real @ M ) ) )
         => ( ! [X: $o > real] :
                ( ( member_o_real @ X @ A4 )
               => ( member_set_o_real @ ( insert_o_real @ X @ bot_bot_set_o_real ) @ ( sigma_sets_o_real @ N ) ) )
           => ( ( counta8783200249485735024o_real @ A4 )
             => ( ! [A3: $o > real] :
                    ( ( member_o_real @ A3 @ A4 )
                   => ( ( sigma_4433523422001307788o_real @ M @ ( insert_o_real @ A3 @ bot_bot_set_o_real ) )
                      = ( sigma_4433523422001307788o_real @ N @ ( insert_o_real @ A3 @ bot_bot_set_o_real ) ) ) )
               => ( M = N ) ) ) ) ) ) ) ).

% measure_eqI_countable'
thf(fact_1013_measure__eqI__countable_H,axiom,
    ! [M: sigma_3396294578489551860t_real,A4: set_nat_real,N: sigma_3396294578489551860t_real] :
      ( ( ( sigma_space_nat_real @ M )
        = A4 )
     => ( ( ( sigma_space_nat_real @ N )
          = A4 )
       => ( ! [X: nat > real] :
              ( ( member_nat_real @ X @ A4 )
             => ( member_set_nat_real @ ( insert_nat_real @ X @ bot_bot_set_nat_real ) @ ( sigma_sets_nat_real @ M ) ) )
         => ( ! [X: nat > real] :
                ( ( member_nat_real @ X @ A4 )
               => ( member_set_nat_real @ ( insert_nat_real @ X @ bot_bot_set_nat_real ) @ ( sigma_sets_nat_real @ N ) ) )
           => ( ( counta2162411829015494944t_real @ A4 )
             => ( ! [A3: nat > real] :
                    ( ( member_nat_real @ A3 @ A4 )
                   => ( ( sigma_2433462726372594692t_real @ M @ ( insert_nat_real @ A3 @ bot_bot_set_nat_real ) )
                      = ( sigma_2433462726372594692t_real @ N @ ( insert_nat_real @ A3 @ bot_bot_set_nat_real ) ) ) )
               => ( M = N ) ) ) ) ) ) ) ).

% measure_eqI_countable'
thf(fact_1014_measure__eqI__countable_H,axiom,
    ! [M: sigma_measure_c_b,A4: set_c_b,N: sigma_measure_c_b] :
      ( ( ( sigma_space_c_b @ M )
        = A4 )
     => ( ( ( sigma_space_c_b @ N )
          = A4 )
       => ( ! [X: c > b] :
              ( ( member_c_b @ X @ A4 )
             => ( member_set_c_b @ ( insert_c_b @ X @ bot_bot_set_c_b ) @ ( sigma_sets_c_b @ M ) ) )
         => ( ! [X: c > b] :
                ( ( member_c_b @ X @ A4 )
               => ( member_set_c_b @ ( insert_c_b @ X @ bot_bot_set_c_b ) @ ( sigma_sets_c_b @ N ) ) )
           => ( ( counta2657777928882154345le_c_b @ A4 )
             => ( ! [A3: c > b] :
                    ( ( member_c_b @ A3 @ A4 )
                   => ( ( sigma_emeasure_c_b @ M @ ( insert_c_b @ A3 @ bot_bot_set_c_b ) )
                      = ( sigma_emeasure_c_b @ N @ ( insert_c_b @ A3 @ bot_bot_set_c_b ) ) ) )
               => ( M = N ) ) ) ) ) ) ) ).

% measure_eqI_countable'
thf(fact_1015_measure__eqI__countable_H,axiom,
    ! [M: sigma_measure_a_b,A4: set_a_b,N: sigma_measure_a_b] :
      ( ( ( sigma_space_a_b @ M )
        = A4 )
     => ( ( ( sigma_space_a_b @ N )
          = A4 )
       => ( ! [X: a > b] :
              ( ( member_a_b @ X @ A4 )
             => ( member_set_a_b @ ( insert_a_b @ X @ bot_bot_set_a_b ) @ ( sigma_sets_a_b @ M ) ) )
         => ( ! [X: a > b] :
                ( ( member_a_b @ X @ A4 )
               => ( member_set_a_b @ ( insert_a_b @ X @ bot_bot_set_a_b ) @ ( sigma_sets_a_b @ N ) ) )
           => ( ( counta8232689092827506411le_a_b @ A4 )
             => ( ! [A3: a > b] :
                    ( ( member_a_b @ A3 @ A4 )
                   => ( ( sigma_emeasure_a_b @ M @ ( insert_a_b @ A3 @ bot_bot_set_a_b ) )
                      = ( sigma_emeasure_a_b @ N @ ( insert_a_b @ A3 @ bot_bot_set_a_b ) ) ) )
               => ( M = N ) ) ) ) ) ) ) ).

% measure_eqI_countable'
thf(fact_1016_measure__eqI__countable_H,axiom,
    ! [M: sigma_measure_real,A4: set_real,N: sigma_measure_real] :
      ( ( ( sigma_space_real @ M )
        = A4 )
     => ( ( ( sigma_space_real @ N )
          = A4 )
       => ( ! [X: real] :
              ( ( member_real @ X @ A4 )
             => ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
         => ( ! [X: real] :
                ( ( member_real @ X @ A4 )
               => ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ N ) ) )
           => ( ( counta7319604579010473777e_real @ A4 )
             => ( ! [A3: real] :
                    ( ( member_real @ A3 @ A4 )
                   => ( ( sigma_emeasure_real @ M @ ( insert_real @ A3 @ bot_bot_set_real ) )
                      = ( sigma_emeasure_real @ N @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) )
               => ( M = N ) ) ) ) ) ) ) ).

% measure_eqI_countable'
thf(fact_1017_measure__eqI__countable_H,axiom,
    ! [M: sigma_measure_nat,A4: set_nat,N: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ M )
        = A4 )
     => ( ( ( sigma_space_nat @ N )
          = A4 )
       => ( ! [X: nat] :
              ( ( member_nat @ X @ A4 )
             => ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) ) )
         => ( ! [X: nat] :
                ( ( member_nat @ X @ A4 )
               => ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ N ) ) )
           => ( ( counta1168086296615599829le_nat @ A4 )
             => ( ! [A3: nat] :
                    ( ( member_nat @ A3 @ A4 )
                   => ( ( sigma_emeasure_nat @ M @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
                      = ( sigma_emeasure_nat @ N @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) )
               => ( M = N ) ) ) ) ) ) ) ).

% measure_eqI_countable'
thf(fact_1018_measure__eqI__countable_H,axiom,
    ! [M: sigma_7234349610311085201nnreal,A4: set_Ex3793607809372303086nnreal,N: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ M )
        = A4 )
     => ( ( ( sigma_3147302497200244656nnreal @ N )
          = A4 )
       => ( ! [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ A4 )
             => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
         => ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ A4 )
               => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ N ) ) )
           => ( ( counta8439243037236335165nnreal @ A4 )
             => ( ! [A3: extend8495563244428889912nnreal] :
                    ( ( member7908768830364227535nnreal @ A3 @ A4 )
                   => ( ( sigma_6589832970846575905nnreal @ M @ ( insert7407984058720857448nnreal @ A3 @ bot_bo4854962954004695426nnreal ) )
                      = ( sigma_6589832970846575905nnreal @ N @ ( insert7407984058720857448nnreal @ A3 @ bot_bo4854962954004695426nnreal ) ) ) )
               => ( M = N ) ) ) ) ) ) ) ).

% measure_eqI_countable'
thf(fact_1019_measure__eqI__countable_H,axiom,
    ! [M: sigma_measure_o,A4: set_o,N: sigma_measure_o] :
      ( ( ( sigma_space_o @ M )
        = A4 )
     => ( ( ( sigma_space_o @ N )
          = A4 )
       => ( ! [X: $o] :
              ( ( member_o @ X @ A4 )
             => ( member_set_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) ) )
         => ( ! [X: $o] :
                ( ( member_o @ X @ A4 )
               => ( member_set_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( sigma_sets_o @ N ) ) )
           => ( ( counta5976203206615340371able_o @ A4 )
             => ( ! [A3: $o] :
                    ( ( member_o @ A3 @ A4 )
                   => ( ( sigma_emeasure_o @ M @ ( insert_o @ A3 @ bot_bot_set_o ) )
                      = ( sigma_emeasure_o @ N @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) )
               => ( M = N ) ) ) ) ) ) ) ).

% measure_eqI_countable'
thf(fact_1020_measurable__discrete__difference,axiom,
    ! [F: c > b,M: sigma_measure_c,N: sigma_measure_b,X2: set_c,G: c > b] :
      ( ( member_c_b @ F @ ( sigma_measurable_c_b @ M @ N ) )
     => ( ( counta4098120917673242427able_c @ X2 )
       => ( ! [X: c] :
              ( ( member_c @ X @ X2 )
             => ( member_set_c @ ( insert_c @ X @ bot_bot_set_c ) @ ( sigma_sets_c @ M ) ) )
         => ( ! [X: c] :
                ( ( member_c @ X @ X2 )
               => ( member_b @ ( G @ X ) @ ( sigma_space_b @ N ) ) )
           => ( ! [X: c] :
                  ( ( member_c @ X @ ( sigma_space_c @ M ) )
                 => ( ~ ( member_c @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member_c_b @ G @ ( sigma_measurable_c_b @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1021_measurable__discrete__difference,axiom,
    ! [F: a > b,M: sigma_measure_a,N: sigma_measure_b,X2: set_a,G: a > b] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ N ) )
     => ( ( counta4098120917673242425able_a @ X2 )
       => ( ! [X: a] :
              ( ( member_a @ X @ X2 )
             => ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( sigma_sets_a @ M ) ) )
         => ( ! [X: a] :
                ( ( member_a @ X @ X2 )
               => ( member_b @ ( G @ X ) @ ( sigma_space_b @ N ) ) )
           => ( ! [X: a] :
                  ( ( member_a @ X @ ( sigma_space_a @ M ) )
                 => ( ~ ( member_a @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member_a_b @ G @ ( sigma_measurable_a_b @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1022_measurable__discrete__difference,axiom,
    ! [F: real > a,M: sigma_measure_real,N: sigma_measure_a,X2: set_real,G: real > a] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
     => ( ( counta7319604579010473777e_real @ X2 )
       => ( ! [X: real] :
              ( ( member_real @ X @ X2 )
             => ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
         => ( ! [X: real] :
                ( ( member_real @ X @ X2 )
               => ( member_a @ ( G @ X ) @ ( sigma_space_a @ N ) ) )
           => ( ! [X: real] :
                  ( ( member_real @ X @ ( sigma_space_real @ M ) )
                 => ( ~ ( member_real @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1023_measurable__discrete__difference,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,N: sigma_7234349610311085201nnreal,X2: set_real,G: real > extend8495563244428889912nnreal] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
     => ( ( counta7319604579010473777e_real @ X2 )
       => ( ! [X: real] :
              ( ( member_real @ X @ X2 )
             => ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
         => ( ! [X: real] :
                ( ( member_real @ X @ X2 )
               => ( member7908768830364227535nnreal @ ( G @ X ) @ ( sigma_3147302497200244656nnreal @ N ) ) )
           => ( ! [X: real] :
                  ( ( member_real @ X @ ( sigma_space_real @ M ) )
                 => ( ~ ( member_real @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member2919562650594848410nnreal @ G @ ( sigma_9017504469962657078nnreal @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1024_measurable__discrete__difference,axiom,
    ! [F: nat > extend8495563244428889912nnreal,M: sigma_measure_nat,N: sigma_7234349610311085201nnreal,X2: set_nat,G: nat > extend8495563244428889912nnreal] :
      ( ( member8283130129095025342nnreal @ F @ ( sigma_6306161311797543642nnreal @ M @ N ) )
     => ( ( counta1168086296615599829le_nat @ X2 )
       => ( ! [X: nat] :
              ( ( member_nat @ X @ X2 )
             => ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) ) )
         => ( ! [X: nat] :
                ( ( member_nat @ X @ X2 )
               => ( member7908768830364227535nnreal @ ( G @ X ) @ ( sigma_3147302497200244656nnreal @ N ) ) )
           => ( ! [X: nat] :
                  ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
                 => ( ~ ( member_nat @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member8283130129095025342nnreal @ G @ ( sigma_6306161311797543642nnreal @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1025_measurable__discrete__difference,axiom,
    ! [F: nat > $o,M: sigma_measure_nat,N: sigma_measure_o,X2: set_nat,G: nat > $o] :
      ( ( member_nat_o @ F @ ( sigma_5101835498682829686_nat_o @ M @ N ) )
     => ( ( counta1168086296615599829le_nat @ X2 )
       => ( ! [X: nat] :
              ( ( member_nat @ X @ X2 )
             => ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) ) )
         => ( ! [X: nat] :
                ( ( member_nat @ X @ X2 )
               => ( member_o @ ( G @ X ) @ ( sigma_space_o @ N ) ) )
           => ( ! [X: nat] :
                  ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
                 => ( ~ ( member_nat @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member_nat_o @ G @ ( sigma_5101835498682829686_nat_o @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1026_measurable__discrete__difference,axiom,
    ! [F: nat > nat,M: sigma_measure_nat,N: sigma_measure_nat,X2: set_nat,G: nat > nat] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ N ) )
     => ( ( counta1168086296615599829le_nat @ X2 )
       => ( ! [X: nat] :
              ( ( member_nat @ X @ X2 )
             => ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) ) )
         => ( ! [X: nat] :
                ( ( member_nat @ X @ X2 )
               => ( member_nat @ ( G @ X ) @ ( sigma_space_nat @ N ) ) )
           => ( ! [X: nat] :
                  ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
                 => ( ~ ( member_nat @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member_nat_nat @ G @ ( sigma_4350458207664084850at_nat @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1027_measurable__discrete__difference,axiom,
    ! [F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,N: sigma_measure_real,X2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > real] :
      ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ N ) )
     => ( ( counta8439243037236335165nnreal @ X2 )
       => ( ! [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ X2 )
             => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
         => ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ X2 )
               => ( member_real @ ( G @ X ) @ ( sigma_space_real @ N ) ) )
           => ( ! [X: extend8495563244428889912nnreal] :
                  ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
                 => ( ~ ( member7908768830364227535nnreal @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member2874014351250825754l_real @ G @ ( sigma_7049758200512112822l_real @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1028_measurable__discrete__difference,axiom,
    ! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,X2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
      ( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ N ) )
     => ( ( counta8439243037236335165nnreal @ X2 )
       => ( ! [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ X2 )
             => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
         => ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ X2 )
               => ( member7908768830364227535nnreal @ ( G @ X ) @ ( sigma_3147302497200244656nnreal @ N ) ) )
           => ( ! [X: extend8495563244428889912nnreal] :
                  ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
                 => ( ~ ( member7908768830364227535nnreal @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member8329810500450651686nnreal @ G @ ( sigma_7926153774531450434nnreal @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1029_measurable__discrete__difference,axiom,
    ! [F: extend8495563244428889912nnreal > $o,M: sigma_7234349610311085201nnreal,N: sigma_measure_o,X2: set_Ex3793607809372303086nnreal,G: extend8495563244428889912nnreal > $o] :
      ( ( member8095236870201115968real_o @ F @ ( sigma_6279906219187228174real_o @ M @ N ) )
     => ( ( counta8439243037236335165nnreal @ X2 )
       => ( ! [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ X2 )
             => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
         => ( ! [X: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ X @ X2 )
               => ( member_o @ ( G @ X ) @ ( sigma_space_o @ N ) ) )
           => ( ! [X: extend8495563244428889912nnreal] :
                  ( ( member7908768830364227535nnreal @ X @ ( sigma_3147302497200244656nnreal @ M ) )
                 => ( ~ ( member7908768830364227535nnreal @ X @ X2 )
                   => ( ( F @ X )
                      = ( G @ X ) ) ) )
             => ( member8095236870201115968real_o @ G @ ( sigma_6279906219187228174real_o @ M @ N ) ) ) ) ) ) ) ).

% measurable_discrete_difference
thf(fact_1030_member__remove,axiom,
    ! [X3: real > a,Y2: real > a,A4: set_real_a] :
      ( ( member_real_a @ X3 @ ( remove_real_a @ Y2 @ A4 ) )
      = ( ( member_real_a @ X3 @ A4 )
        & ( X3 != Y2 ) ) ) ).

% member_remove
thf(fact_1031_member__remove,axiom,
    ! [X3: $o > real,Y2: $o > real,A4: set_o_real] :
      ( ( member_o_real @ X3 @ ( remove_o_real @ Y2 @ A4 ) )
      = ( ( member_o_real @ X3 @ A4 )
        & ( X3 != Y2 ) ) ) ).

% member_remove
thf(fact_1032_member__remove,axiom,
    ! [X3: nat > real,Y2: nat > real,A4: set_nat_real] :
      ( ( member_nat_real @ X3 @ ( remove_nat_real @ Y2 @ A4 ) )
      = ( ( member_nat_real @ X3 @ A4 )
        & ( X3 != Y2 ) ) ) ).

% member_remove
thf(fact_1033_member__remove,axiom,
    ! [X3: c > b,Y2: c > b,A4: set_c_b] :
      ( ( member_c_b @ X3 @ ( remove_c_b @ Y2 @ A4 ) )
      = ( ( member_c_b @ X3 @ A4 )
        & ( X3 != Y2 ) ) ) ).

% member_remove
thf(fact_1034_member__remove,axiom,
    ! [X3: a > b,Y2: a > b,A4: set_a_b] :
      ( ( member_a_b @ X3 @ ( remove_a_b @ Y2 @ A4 ) )
      = ( ( member_a_b @ X3 @ A4 )
        & ( X3 != Y2 ) ) ) ).

% member_remove
thf(fact_1035_measurable__cong__sets,axiom,
    ! [M: sigma_measure_real,M2: sigma_measure_real,N: sigma_7234349610311085201nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M2 ) )
     => ( ( ( sigma_5465916536984168985nnreal @ N )
          = ( sigma_5465916536984168985nnreal @ N2 ) )
       => ( ( sigma_9017504469962657078nnreal @ M @ N )
          = ( sigma_9017504469962657078nnreal @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1036_measurable__cong__sets,axiom,
    ! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal,N: sigma_measure_real,N2: sigma_measure_real] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( sigma_5465916536984168985nnreal @ M2 ) )
     => ( ( ( sigma_sets_real @ N )
          = ( sigma_sets_real @ N2 ) )
       => ( ( sigma_7049758200512112822l_real @ M @ N )
          = ( sigma_7049758200512112822l_real @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1037_measurable__cong__sets,axiom,
    ! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( sigma_5465916536984168985nnreal @ M2 ) )
     => ( ( ( sigma_5465916536984168985nnreal @ N )
          = ( sigma_5465916536984168985nnreal @ N2 ) )
       => ( ( sigma_7926153774531450434nnreal @ M @ N )
          = ( sigma_7926153774531450434nnreal @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1038_measurable__cong__sets,axiom,
    ! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal,N: sigma_measure_o,N2: sigma_measure_o] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( sigma_5465916536984168985nnreal @ M2 ) )
     => ( ( ( sigma_sets_o @ N )
          = ( sigma_sets_o @ N2 ) )
       => ( ( sigma_6279906219187228174real_o @ M @ N )
          = ( sigma_6279906219187228174real_o @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1039_measurable__cong__sets,axiom,
    ! [M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal,N: sigma_measure_nat,N2: sigma_measure_nat] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( sigma_5465916536984168985nnreal @ M2 ) )
     => ( ( ( sigma_sets_nat @ N )
          = ( sigma_sets_nat @ N2 ) )
       => ( ( sigma_1856489715609627354al_nat @ M @ N )
          = ( sigma_1856489715609627354al_nat @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1040_measurable__cong__sets,axiom,
    ! [M: sigma_measure_o,M2: sigma_measure_o,N: sigma_7234349610311085201nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ M2 ) )
     => ( ( ( sigma_5465916536984168985nnreal @ N )
          = ( sigma_5465916536984168985nnreal @ N2 ) )
       => ( ( sigma_6459699357617223168nnreal @ M @ N )
          = ( sigma_6459699357617223168nnreal @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1041_measurable__cong__sets,axiom,
    ! [M: sigma_measure_o,M2: sigma_measure_o,N: sigma_measure_o,N2: sigma_measure_o] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ M2 ) )
     => ( ( ( sigma_sets_o @ N )
          = ( sigma_sets_o @ N2 ) )
       => ( ( sigma_measurable_o_o @ M @ N )
          = ( sigma_measurable_o_o @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1042_measurable__cong__sets,axiom,
    ! [M: sigma_measure_o,M2: sigma_measure_o,N: sigma_measure_nat,N2: sigma_measure_nat] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ M2 ) )
     => ( ( ( sigma_sets_nat @ N )
          = ( sigma_sets_nat @ N2 ) )
       => ( ( sigma_1999164137574644376_o_nat @ M @ N )
          = ( sigma_1999164137574644376_o_nat @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1043_measurable__cong__sets,axiom,
    ! [M: sigma_measure_nat,M2: sigma_measure_nat,N: sigma_7234349610311085201nnreal,N2: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ M2 ) )
     => ( ( ( sigma_5465916536984168985nnreal @ N )
          = ( sigma_5465916536984168985nnreal @ N2 ) )
       => ( ( sigma_6306161311797543642nnreal @ M @ N )
          = ( sigma_6306161311797543642nnreal @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1044_measurable__cong__sets,axiom,
    ! [M: sigma_measure_nat,M2: sigma_measure_nat,N: sigma_measure_o,N2: sigma_measure_o] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ M2 ) )
     => ( ( ( sigma_sets_o @ N )
          = ( sigma_sets_o @ N2 ) )
       => ( ( sigma_5101835498682829686_nat_o @ M @ N )
          = ( sigma_5101835498682829686_nat_o @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_1045_measure__eqI,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ N ) )
     => ( ! [A8: set_real] :
            ( ( member_set_real @ A8 @ ( sigma_sets_real @ M ) )
           => ( ( sigma_emeasure_real @ M @ A8 )
              = ( sigma_emeasure_real @ N @ A8 ) ) )
       => ( M = N ) ) ) ).

% measure_eqI
thf(fact_1046_measure__eqI,axiom,
    ! [M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( sigma_5465916536984168985nnreal @ N ) )
     => ( ! [A8: set_Ex3793607809372303086nnreal] :
            ( ( member603777416030116741nnreal @ A8 @ ( sigma_5465916536984168985nnreal @ M ) )
           => ( ( sigma_6589832970846575905nnreal @ M @ A8 )
              = ( sigma_6589832970846575905nnreal @ N @ A8 ) ) )
       => ( M = N ) ) ) ).

% measure_eqI
thf(fact_1047_measure__eqI,axiom,
    ! [M: sigma_measure_o,N: sigma_measure_o] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ N ) )
     => ( ! [A8: set_o] :
            ( ( member_set_o @ A8 @ ( sigma_sets_o @ M ) )
           => ( ( sigma_emeasure_o @ M @ A8 )
              = ( sigma_emeasure_o @ N @ A8 ) ) )
       => ( M = N ) ) ) ).

% measure_eqI
thf(fact_1048_measure__eqI,axiom,
    ! [M: sigma_measure_nat,N: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ N ) )
     => ( ! [A8: set_nat] :
            ( ( member_set_nat @ A8 @ ( sigma_sets_nat @ M ) )
           => ( ( sigma_emeasure_nat @ M @ A8 )
              = ( sigma_emeasure_nat @ N @ A8 ) ) )
       => ( M = N ) ) ) ).

% measure_eqI
thf(fact_1049_le__measureD3,axiom,
    ! [A4: sigma_measure_real,B4: sigma_measure_real,X2: set_real] :
      ( ( ord_le487379304121309861e_real @ A4 @ B4 )
     => ( ( ( sigma_sets_real @ A4 )
          = ( sigma_sets_real @ B4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ A4 @ X2 ) @ ( sigma_emeasure_real @ B4 @ X2 ) ) ) ) ).

% le_measureD3
thf(fact_1050_le__measureD3,axiom,
    ! [A4: sigma_7234349610311085201nnreal,B4: sigma_7234349610311085201nnreal,X2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le1854472233513649201nnreal @ A4 @ B4 )
     => ( ( ( sigma_5465916536984168985nnreal @ A4 )
          = ( sigma_5465916536984168985nnreal @ B4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ A4 @ X2 ) @ ( sigma_6589832970846575905nnreal @ B4 @ X2 ) ) ) ) ).

% le_measureD3
thf(fact_1051_le__measureD3,axiom,
    ! [A4: sigma_measure_o,B4: sigma_measure_o,X2: set_o] :
      ( ( ord_le478349814012620405sure_o @ A4 @ B4 )
     => ( ( ( sigma_sets_o @ A4 )
          = ( sigma_sets_o @ B4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ A4 @ X2 ) @ ( sigma_emeasure_o @ B4 @ X2 ) ) ) ) ).

% le_measureD3
thf(fact_1052_le__measureD3,axiom,
    ! [A4: sigma_measure_nat,B4: sigma_measure_nat,X2: set_nat] :
      ( ( ord_le2862109966718184649re_nat @ A4 @ B4 )
     => ( ( ( sigma_sets_nat @ A4 )
          = ( sigma_sets_nat @ B4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ A4 @ X2 ) @ ( sigma_emeasure_nat @ B4 @ X2 ) ) ) ) ).

% le_measureD3
thf(fact_1053_le__measure,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ N ) )
     => ( ( ord_le487379304121309861e_real @ M @ N )
        = ( ! [X5: set_real] :
              ( ( member_set_real @ X5 @ ( sigma_sets_real @ M ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ M @ X5 ) @ ( sigma_emeasure_real @ N @ X5 ) ) ) ) ) ) ).

% le_measure
thf(fact_1054_le__measure,axiom,
    ! [M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( sigma_5465916536984168985nnreal @ N ) )
     => ( ( ord_le1854472233513649201nnreal @ M @ N )
        = ( ! [X5: set_Ex3793607809372303086nnreal] :
              ( ( member603777416030116741nnreal @ X5 @ ( sigma_5465916536984168985nnreal @ M ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ M @ X5 ) @ ( sigma_6589832970846575905nnreal @ N @ X5 ) ) ) ) ) ) ).

% le_measure
thf(fact_1055_le__measure,axiom,
    ! [M: sigma_measure_o,N: sigma_measure_o] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ N ) )
     => ( ( ord_le478349814012620405sure_o @ M @ N )
        = ( ! [X5: set_o] :
              ( ( member_set_o @ X5 @ ( sigma_sets_o @ M ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ M @ X5 ) @ ( sigma_emeasure_o @ N @ X5 ) ) ) ) ) ) ).

% le_measure
thf(fact_1056_le__measure,axiom,
    ! [M: sigma_measure_nat,N: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ N ) )
     => ( ( ord_le2862109966718184649re_nat @ M @ N )
        = ( ! [X5: set_nat] :
              ( ( member_set_nat @ X5 @ ( sigma_sets_nat @ M ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ M @ X5 ) @ ( sigma_emeasure_nat @ N @ X5 ) ) ) ) ) ) ).

% le_measure
thf(fact_1057_measurable__cong,axiom,
    ! [M: sigma_measure_c,F: c > b,G: c > b,M2: sigma_measure_b] :
      ( ! [W: c] :
          ( ( member_c @ W @ ( sigma_space_c @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_c_b @ F @ ( sigma_measurable_c_b @ M @ M2 ) )
        = ( member_c_b @ G @ ( sigma_measurable_c_b @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1058_measurable__cong,axiom,
    ! [M: sigma_measure_a,F: a > b,G: a > b,M2: sigma_measure_b] :
      ( ! [W: a] :
          ( ( member_a @ W @ ( sigma_space_a @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ M2 ) )
        = ( member_a_b @ G @ ( sigma_measurable_a_b @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1059_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > a,G: real > a,M2: sigma_measure_a] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M2 ) )
        = ( member_real_a @ G @ ( sigma_523072396149930112real_a @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1060_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > real,G: real > real,M2: sigma_measure_real] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
        = ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1061_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > $o,G: real > $o,M2: sigma_measure_o] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ M2 ) )
        = ( member_real_o @ G @ ( sigma_3939073009482781210real_o @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1062_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > nat,G: real > nat,M2: sigma_measure_nat] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ M2 ) )
        = ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1063_measurable__cong,axiom,
    ! [M: sigma_measure_o,F: $o > real,G: $o > real,M2: sigma_measure_real] :
      ( ! [W: $o] :
          ( ( member_o @ W @ ( sigma_space_o @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ M2 ) )
        = ( member_o_real @ G @ ( sigma_2430008634441611636o_real @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1064_measurable__cong,axiom,
    ! [M: sigma_measure_nat,F: nat > real,G: nat > real,M2: sigma_measure_real] :
      ( ! [W: nat] :
          ( ( member_nat @ W @ ( sigma_space_nat @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ M2 ) )
        = ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1065_measurable__space,axiom,
    ! [F: c > b,M: sigma_measure_c,A4: sigma_measure_b,X3: c] :
      ( ( member_c_b @ F @ ( sigma_measurable_c_b @ M @ A4 ) )
     => ( ( member_c @ X3 @ ( sigma_space_c @ M ) )
       => ( member_b @ ( F @ X3 ) @ ( sigma_space_b @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1066_measurable__space,axiom,
    ! [F: a > b,M: sigma_measure_a,A4: sigma_measure_b,X3: a] :
      ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ A4 ) )
     => ( ( member_a @ X3 @ ( sigma_space_a @ M ) )
       => ( member_b @ ( F @ X3 ) @ ( sigma_space_b @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1067_measurable__space,axiom,
    ! [F: real > a,M: sigma_measure_real,A4: sigma_measure_a,X3: real] :
      ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ A4 ) )
     => ( ( member_real @ X3 @ ( sigma_space_real @ M ) )
       => ( member_a @ ( F @ X3 ) @ ( sigma_space_a @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1068_measurable__space,axiom,
    ! [F: real > extend8495563244428889912nnreal,M: sigma_measure_real,A4: sigma_7234349610311085201nnreal,X3: real] :
      ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ A4 ) )
     => ( ( member_real @ X3 @ ( sigma_space_real @ M ) )
       => ( member7908768830364227535nnreal @ ( F @ X3 ) @ ( sigma_3147302497200244656nnreal @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1069_measurable__space,axiom,
    ! [F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal,A4: sigma_measure_real,X3: extend8495563244428889912nnreal] :
      ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ A4 ) )
     => ( ( member7908768830364227535nnreal @ X3 @ ( sigma_3147302497200244656nnreal @ M ) )
       => ( member_real @ ( F @ X3 ) @ ( sigma_space_real @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1070_measurable__space,axiom,
    ! [F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A4: sigma_7234349610311085201nnreal,X3: extend8495563244428889912nnreal] :
      ( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ A4 ) )
     => ( ( member7908768830364227535nnreal @ X3 @ ( sigma_3147302497200244656nnreal @ M ) )
       => ( member7908768830364227535nnreal @ ( F @ X3 ) @ ( sigma_3147302497200244656nnreal @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1071_measurable__space,axiom,
    ! [F: extend8495563244428889912nnreal > $o,M: sigma_7234349610311085201nnreal,A4: sigma_measure_o,X3: extend8495563244428889912nnreal] :
      ( ( member8095236870201115968real_o @ F @ ( sigma_6279906219187228174real_o @ M @ A4 ) )
     => ( ( member7908768830364227535nnreal @ X3 @ ( sigma_3147302497200244656nnreal @ M ) )
       => ( member_o @ ( F @ X3 ) @ ( sigma_space_o @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1072_measurable__space,axiom,
    ! [F: extend8495563244428889912nnreal > nat,M: sigma_7234349610311085201nnreal,A4: sigma_measure_nat,X3: extend8495563244428889912nnreal] :
      ( ( member6436672275262627518al_nat @ F @ ( sigma_1856489715609627354al_nat @ M @ A4 ) )
     => ( ( member7908768830364227535nnreal @ X3 @ ( sigma_3147302497200244656nnreal @ M ) )
       => ( member_nat @ ( F @ X3 ) @ ( sigma_space_nat @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1073_measurable__space,axiom,
    ! [F: $o > extend8495563244428889912nnreal,M: sigma_measure_o,A4: sigma_7234349610311085201nnreal,X3: $o] :
      ( ( member5265953103328284778nnreal @ F @ ( sigma_6459699357617223168nnreal @ M @ A4 ) )
     => ( ( member_o @ X3 @ ( sigma_space_o @ M ) )
       => ( member7908768830364227535nnreal @ ( F @ X3 ) @ ( sigma_3147302497200244656nnreal @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1074_measurable__space,axiom,
    ! [F: $o > $o,M: sigma_measure_o,A4: sigma_measure_o,X3: $o] :
      ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ A4 ) )
     => ( ( member_o @ X3 @ ( sigma_space_o @ M ) )
       => ( member_o @ ( F @ X3 ) @ ( sigma_space_o @ A4 ) ) ) ) ).

% measurable_space
thf(fact_1075_measurable__cong__simp,axiom,
    ! [M: sigma_measure_c,N: sigma_measure_c,M2: sigma_measure_b,N2: sigma_measure_b,F: c > b,G: c > b] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: c] :
              ( ( member_c @ W @ ( sigma_space_c @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_c_b @ F @ ( sigma_measurable_c_b @ M @ M2 ) )
            = ( member_c_b @ G @ ( sigma_measurable_c_b @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1076_measurable__cong__simp,axiom,
    ! [M: sigma_measure_a,N: sigma_measure_a,M2: sigma_measure_b,N2: sigma_measure_b,F: a > b,G: a > b] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: a] :
              ( ( member_a @ W @ ( sigma_space_a @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ M2 ) )
            = ( member_a_b @ G @ ( sigma_measurable_a_b @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1077_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_measure_a,N2: sigma_measure_a,F: real > a,G: real > a] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ M2 ) )
            = ( member_real_a @ G @ ( sigma_523072396149930112real_a @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1078_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_measure_real,N2: sigma_measure_real,F: real > real,G: real > real] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ M2 ) )
            = ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1079_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_measure_o,N2: sigma_measure_o,F: real > $o,G: real > $o] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ M2 ) )
            = ( member_real_o @ G @ ( sigma_3939073009482781210real_o @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1080_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_measure_nat,N2: sigma_measure_nat,F: real > nat,G: real > nat] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ M2 ) )
            = ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1081_measurable__cong__simp,axiom,
    ! [M: sigma_measure_o,N: sigma_measure_o,M2: sigma_measure_real,N2: sigma_measure_real,F: $o > real,G: $o > real] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: $o] :
              ( ( member_o @ W @ ( sigma_space_o @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ M2 ) )
            = ( member_o_real @ G @ ( sigma_2430008634441611636o_real @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1082_measurable__cong__simp,axiom,
    ! [M: sigma_measure_nat,N: sigma_measure_nat,M2: sigma_measure_real,N2: sigma_measure_real,F: nat > real,G: nat > real] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: nat] :
              ( ( member_nat @ W @ ( sigma_space_nat @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ M2 ) )
            = ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1083_emeasure__mono,axiom,
    ! [A: set_real,B: set_real,M: sigma_measure_real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ( member_set_real @ B @ ( sigma_sets_real @ M ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ M @ A ) @ ( sigma_emeasure_real @ M @ B ) ) ) ) ).

% emeasure_mono
thf(fact_1084_emeasure__mono,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B )
     => ( ( member603777416030116741nnreal @ B @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ M @ A ) @ ( sigma_6589832970846575905nnreal @ M @ B ) ) ) ) ).

% emeasure_mono
thf(fact_1085_emeasure__mono,axiom,
    ! [A: set_o,B: set_o,M: sigma_measure_o] :
      ( ( ord_less_eq_set_o @ A @ B )
     => ( ( member_set_o @ B @ ( sigma_sets_o @ M ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ M @ A ) @ ( sigma_emeasure_o @ M @ B ) ) ) ) ).

% emeasure_mono
thf(fact_1086_emeasure__mono,axiom,
    ! [A: set_nat,B: set_nat,M: sigma_measure_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_set_nat @ B @ ( sigma_sets_nat @ M ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ M @ A ) @ ( sigma_emeasure_nat @ M @ B ) ) ) ) ).

% emeasure_mono
thf(fact_1087_measurable__empty__iff,axiom,
    ! [N: sigma_measure_b,F: c > b,M: sigma_measure_c] :
      ( ( ( sigma_space_b @ N )
        = bot_bot_set_b )
     => ( ( member_c_b @ F @ ( sigma_measurable_c_b @ M @ N ) )
        = ( ( sigma_space_c @ M )
          = bot_bot_set_c ) ) ) ).

% measurable_empty_iff
thf(fact_1088_measurable__empty__iff,axiom,
    ! [N: sigma_measure_b,F: a > b,M: sigma_measure_a] :
      ( ( ( sigma_space_b @ N )
        = bot_bot_set_b )
     => ( ( member_a_b @ F @ ( sigma_measurable_a_b @ M @ N ) )
        = ( ( sigma_space_a @ M )
          = bot_bot_set_a ) ) ) ).

% measurable_empty_iff
thf(fact_1089_measurable__empty__iff,axiom,
    ! [N: sigma_measure_a,F: real > a,M: sigma_measure_real] :
      ( ( ( sigma_space_a @ N )
        = bot_bot_set_a )
     => ( ( member_real_a @ F @ ( sigma_523072396149930112real_a @ M @ N ) )
        = ( ( sigma_space_real @ M )
          = bot_bot_set_real ) ) ) ).

% measurable_empty_iff
thf(fact_1090_measurable__empty__iff,axiom,
    ! [N: sigma_measure_real,F: extend8495563244428889912nnreal > real,M: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_space_real @ N )
        = bot_bot_set_real )
     => ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ N ) )
        = ( ( sigma_3147302497200244656nnreal @ M )
          = bot_bo4854962954004695426nnreal ) ) ) ).

% measurable_empty_iff
thf(fact_1091_measurable__empty__iff,axiom,
    ! [N: sigma_measure_nat,F: nat > nat,M: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ N )
        = bot_bot_set_nat )
     => ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ N ) )
        = ( ( sigma_space_nat @ M )
          = bot_bot_set_nat ) ) ) ).

% measurable_empty_iff
thf(fact_1092_measurable__empty__iff,axiom,
    ! [N: sigma_measure_nat,F: extend8495563244428889912nnreal > nat,M: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_space_nat @ N )
        = bot_bot_set_nat )
     => ( ( member6436672275262627518al_nat @ F @ ( sigma_1856489715609627354al_nat @ M @ N ) )
        = ( ( sigma_3147302497200244656nnreal @ M )
          = bot_bo4854962954004695426nnreal ) ) ) ).

% measurable_empty_iff
thf(fact_1093_measurable__empty__iff,axiom,
    ! [N: sigma_measure_nat,F: $o > nat,M: sigma_measure_o] :
      ( ( ( sigma_space_nat @ N )
        = bot_bot_set_nat )
     => ( ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M @ N ) )
        = ( ( sigma_space_o @ M )
          = bot_bot_set_o ) ) ) ).

% measurable_empty_iff
thf(fact_1094_measurable__empty__iff,axiom,
    ! [N: sigma_7234349610311085201nnreal,F: real > extend8495563244428889912nnreal,M: sigma_measure_real] :
      ( ( ( sigma_3147302497200244656nnreal @ N )
        = bot_bo4854962954004695426nnreal )
     => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ N ) )
        = ( ( sigma_space_real @ M )
          = bot_bot_set_real ) ) ) ).

% measurable_empty_iff
thf(fact_1095_measurable__empty__iff,axiom,
    ! [N: sigma_7234349610311085201nnreal,F: nat > extend8495563244428889912nnreal,M: sigma_measure_nat] :
      ( ( ( sigma_3147302497200244656nnreal @ N )
        = bot_bo4854962954004695426nnreal )
     => ( ( member8283130129095025342nnreal @ F @ ( sigma_6306161311797543642nnreal @ M @ N ) )
        = ( ( sigma_space_nat @ M )
          = bot_bot_set_nat ) ) ) ).

% measurable_empty_iff
thf(fact_1096_measurable__empty__iff,axiom,
    ! [N: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ N )
        = bot_bo4854962954004695426nnreal )
     => ( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ N ) )
        = ( ( sigma_3147302497200244656nnreal @ M )
          = bot_bo4854962954004695426nnreal ) ) ) ).

% measurable_empty_iff
thf(fact_1097_less__eq__measure_Ointros_I3_J,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real] :
      ( ( ( sigma_space_real @ M )
        = ( sigma_space_real @ N ) )
     => ( ( ( sigma_sets_real @ M )
          = ( sigma_sets_real @ N ) )
       => ( ( ord_le637582726751450265nnreal @ ( sigma_emeasure_real @ M ) @ ( sigma_emeasure_real @ N ) )
         => ( ord_le487379304121309861e_real @ M @ N ) ) ) ) ).

% less_eq_measure.intros(3)
thf(fact_1098_less__eq__measure_Ointros_I3_J,axiom,
    ! [M: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ M )
        = ( sigma_3147302497200244656nnreal @ N ) )
     => ( ( ( sigma_5465916536984168985nnreal @ M )
          = ( sigma_5465916536984168985nnreal @ N ) )
       => ( ( ord_le8913848522597308453nnreal @ ( sigma_6589832970846575905nnreal @ M ) @ ( sigma_6589832970846575905nnreal @ N ) )
         => ( ord_le1854472233513649201nnreal @ M @ N ) ) ) ) ).

% less_eq_measure.intros(3)
thf(fact_1099_less__eq__measure_Ointros_I3_J,axiom,
    ! [M: sigma_measure_o,N: sigma_measure_o] :
      ( ( ( sigma_space_o @ M )
        = ( sigma_space_o @ N ) )
     => ( ( ( sigma_sets_o @ M )
          = ( sigma_sets_o @ N ) )
       => ( ( ord_le4997716739388385473nnreal @ ( sigma_emeasure_o @ M ) @ ( sigma_emeasure_o @ N ) )
         => ( ord_le478349814012620405sure_o @ M @ N ) ) ) ) ).

% less_eq_measure.intros(3)
thf(fact_1100_less__eq__measure_Ointros_I3_J,axiom,
    ! [M: sigma_measure_nat,N: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ M )
        = ( sigma_space_nat @ N ) )
     => ( ( ( sigma_sets_nat @ M )
          = ( sigma_sets_nat @ N ) )
       => ( ( ord_le9125165017735938237nnreal @ ( sigma_emeasure_nat @ M ) @ ( sigma_emeasure_nat @ N ) )
         => ( ord_le2862109966718184649re_nat @ M @ N ) ) ) ) ).

% less_eq_measure.intros(3)
thf(fact_1101_sets_Oalgebra__axioms,axiom,
    ! [M: sigma_measure_real] : ( sigma_algebra_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).

% sets.algebra_axioms
thf(fact_1102_sets_Oalgebra__axioms,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( sigma_5981082695875523474nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.algebra_axioms
thf(fact_1103_sets_Oalgebra__axioms,axiom,
    ! [M: sigma_measure_o] : ( sigma_algebra_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) ).

% sets.algebra_axioms
thf(fact_1104_sets_Oalgebra__axioms,axiom,
    ! [M: sigma_measure_nat] : ( sigma_algebra_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) ).

% sets.algebra_axioms
thf(fact_1105_le__emeasure__sup__measure_H1,axiom,
    ! [B4: sigma_measure_real,A4: sigma_measure_real,X2: set_real] :
      ( ( ( sigma_sets_real @ B4 )
        = ( sigma_sets_real @ A4 ) )
     => ( ( member_set_real @ X2 @ ( sigma_sets_real @ A4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ A4 @ X2 ) @ ( sigma_emeasure_real @ ( measur2147279183506585690e_real @ A4 @ B4 ) @ X2 ) ) ) ) ).

% le_emeasure_sup_measure'1
thf(fact_1106_le__emeasure__sup__measure_H1,axiom,
    ! [B4: sigma_7234349610311085201nnreal,A4: sigma_7234349610311085201nnreal,X2: set_Ex3793607809372303086nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ B4 )
        = ( sigma_5465916536984168985nnreal @ A4 ) )
     => ( ( member603777416030116741nnreal @ X2 @ ( sigma_5465916536984168985nnreal @ A4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ A4 @ X2 ) @ ( sigma_6589832970846575905nnreal @ ( measur4473656680840910822nnreal @ A4 @ B4 ) @ X2 ) ) ) ) ).

% le_emeasure_sup_measure'1
thf(fact_1107_le__emeasure__sup__measure_H1,axiom,
    ! [B4: sigma_measure_o,A4: sigma_measure_o,X2: set_o] :
      ( ( ( sigma_sets_o @ B4 )
        = ( sigma_sets_o @ A4 ) )
     => ( ( member_set_o @ X2 @ ( sigma_sets_o @ A4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ A4 @ X2 ) @ ( sigma_emeasure_o @ ( measur4529518739368704874sure_o @ A4 @ B4 ) @ X2 ) ) ) ) ).

% le_emeasure_sup_measure'1
thf(fact_1108_le__emeasure__sup__measure_H1,axiom,
    ! [B4: sigma_measure_nat,A4: sigma_measure_nat,X2: set_nat] :
      ( ( ( sigma_sets_nat @ B4 )
        = ( sigma_sets_nat @ A4 ) )
     => ( ( member_set_nat @ X2 @ ( sigma_sets_nat @ A4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ A4 @ X2 ) @ ( sigma_emeasure_nat @ ( measur876423496291765374re_nat @ A4 @ B4 ) @ X2 ) ) ) ) ).

% le_emeasure_sup_measure'1
thf(fact_1109_le__emeasure__sup__measure_H2,axiom,
    ! [B4: sigma_measure_real,A4: sigma_measure_real,X2: set_real] :
      ( ( ( sigma_sets_real @ B4 )
        = ( sigma_sets_real @ A4 ) )
     => ( ( member_set_real @ X2 @ ( sigma_sets_real @ A4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ B4 @ X2 ) @ ( sigma_emeasure_real @ ( measur2147279183506585690e_real @ A4 @ B4 ) @ X2 ) ) ) ) ).

% le_emeasure_sup_measure'2
thf(fact_1110_le__emeasure__sup__measure_H2,axiom,
    ! [B4: sigma_7234349610311085201nnreal,A4: sigma_7234349610311085201nnreal,X2: set_Ex3793607809372303086nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ B4 )
        = ( sigma_5465916536984168985nnreal @ A4 ) )
     => ( ( member603777416030116741nnreal @ X2 @ ( sigma_5465916536984168985nnreal @ A4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ B4 @ X2 ) @ ( sigma_6589832970846575905nnreal @ ( measur4473656680840910822nnreal @ A4 @ B4 ) @ X2 ) ) ) ) ).

% le_emeasure_sup_measure'2
thf(fact_1111_le__emeasure__sup__measure_H2,axiom,
    ! [B4: sigma_measure_o,A4: sigma_measure_o,X2: set_o] :
      ( ( ( sigma_sets_o @ B4 )
        = ( sigma_sets_o @ A4 ) )
     => ( ( member_set_o @ X2 @ ( sigma_sets_o @ A4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ B4 @ X2 ) @ ( sigma_emeasure_o @ ( measur4529518739368704874sure_o @ A4 @ B4 ) @ X2 ) ) ) ) ).

% le_emeasure_sup_measure'2
thf(fact_1112_le__emeasure__sup__measure_H2,axiom,
    ! [B4: sigma_measure_nat,A4: sigma_measure_nat,X2: set_nat] :
      ( ( ( sigma_sets_nat @ B4 )
        = ( sigma_sets_nat @ A4 ) )
     => ( ( member_set_nat @ X2 @ ( sigma_sets_nat @ A4 ) )
       => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ B4 @ X2 ) @ ( sigma_emeasure_nat @ ( measur876423496291765374re_nat @ A4 @ B4 ) @ X2 ) ) ) ) ).

% le_emeasure_sup_measure'2
thf(fact_1113_measurable__mono,axiom,
    ! [N2: sigma_measure_real,N: sigma_measure_real,M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N2 ) @ ( sigma_sets_real @ N ) )
     => ( ( ( sigma_space_real @ N )
          = ( sigma_space_real @ N2 ) )
       => ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M2 ) )
         => ( ( ( sigma_3147302497200244656nnreal @ M )
              = ( sigma_3147302497200244656nnreal @ M2 ) )
           => ( ord_le2792513217584188441l_real @ ( sigma_7049758200512112822l_real @ M @ N ) @ ( sigma_7049758200512112822l_real @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1114_measurable__mono,axiom,
    ! [N2: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,M: sigma_measure_real,M2: sigma_measure_real] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N2 ) @ ( sigma_5465916536984168985nnreal @ N ) )
     => ( ( ( sigma_3147302497200244656nnreal @ N )
          = ( sigma_3147302497200244656nnreal @ N2 ) )
       => ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ M2 ) )
         => ( ( ( sigma_space_real @ M )
              = ( sigma_space_real @ M2 ) )
           => ( ord_le2462468573666744473nnreal @ ( sigma_9017504469962657078nnreal @ M @ N ) @ ( sigma_9017504469962657078nnreal @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1115_measurable__mono,axiom,
    ! [N2: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N2 ) @ ( sigma_5465916536984168985nnreal @ N ) )
     => ( ( ( sigma_3147302497200244656nnreal @ N )
          = ( sigma_3147302497200244656nnreal @ N2 ) )
       => ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M2 ) )
         => ( ( ( sigma_3147302497200244656nnreal @ M )
              = ( sigma_3147302497200244656nnreal @ M2 ) )
           => ( ord_le2847260637007690789nnreal @ ( sigma_7926153774531450434nnreal @ M @ N ) @ ( sigma_7926153774531450434nnreal @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1116_measurable__mono,axiom,
    ! [N2: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,M: sigma_measure_o,M2: sigma_measure_o] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N2 ) @ ( sigma_5465916536984168985nnreal @ N ) )
     => ( ( ( sigma_3147302497200244656nnreal @ N )
          = ( sigma_3147302497200244656nnreal @ N2 ) )
       => ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ M ) @ ( sigma_sets_o @ M2 ) )
         => ( ( ( sigma_space_o @ M )
              = ( sigma_space_o @ M2 ) )
           => ( ord_le83118845223365825nnreal @ ( sigma_6459699357617223168nnreal @ M @ N ) @ ( sigma_6459699357617223168nnreal @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1117_measurable__mono,axiom,
    ! [N2: sigma_7234349610311085201nnreal,N: sigma_7234349610311085201nnreal,M: sigma_measure_nat,M2: sigma_measure_nat] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N2 ) @ ( sigma_5465916536984168985nnreal @ N ) )
     => ( ( ( sigma_3147302497200244656nnreal @ N )
          = ( sigma_3147302497200244656nnreal @ N2 ) )
       => ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ M ) @ ( sigma_sets_nat @ M2 ) )
         => ( ( ( sigma_space_nat @ M )
              = ( sigma_space_nat @ M2 ) )
           => ( ord_le7339261118519895229nnreal @ ( sigma_6306161311797543642nnreal @ M @ N ) @ ( sigma_6306161311797543642nnreal @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1118_measurable__mono,axiom,
    ! [N2: sigma_measure_o,N: sigma_measure_o,M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ N2 ) @ ( sigma_sets_o @ N ) )
     => ( ( ( sigma_space_o @ N )
          = ( sigma_space_o @ N2 ) )
       => ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M2 ) )
         => ( ( ( sigma_3147302497200244656nnreal @ M )
              = ( sigma_3147302497200244656nnreal @ M2 ) )
           => ( ord_le5553135326598321815real_o @ ( sigma_6279906219187228174real_o @ M @ N ) @ ( sigma_6279906219187228174real_o @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1119_measurable__mono,axiom,
    ! [N2: sigma_measure_o,N: sigma_measure_o,M: sigma_measure_o,M2: sigma_measure_o] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ N2 ) @ ( sigma_sets_o @ N ) )
     => ( ( ( sigma_space_o @ N )
          = ( sigma_space_o @ N2 ) )
       => ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ M ) @ ( sigma_sets_o @ M2 ) )
         => ( ( ( sigma_space_o @ M )
              = ( sigma_space_o @ M2 ) )
           => ( ord_less_eq_set_o_o @ ( sigma_measurable_o_o @ M @ N ) @ ( sigma_measurable_o_o @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1120_measurable__mono,axiom,
    ! [N2: sigma_measure_o,N: sigma_measure_o,M: sigma_measure_nat,M2: sigma_measure_nat] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ N2 ) @ ( sigma_sets_o @ N ) )
     => ( ( ( sigma_space_o @ N )
          = ( sigma_space_o @ N2 ) )
       => ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ M ) @ ( sigma_sets_nat @ M2 ) )
         => ( ( ( sigma_space_nat @ M )
              = ( sigma_space_nat @ M2 ) )
           => ( ord_le6029213668185085951_nat_o @ ( sigma_5101835498682829686_nat_o @ M @ N ) @ ( sigma_5101835498682829686_nat_o @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1121_measurable__mono,axiom,
    ! [N2: sigma_measure_nat,N: sigma_measure_nat,M: sigma_7234349610311085201nnreal,M2: sigma_7234349610311085201nnreal] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ N2 ) @ ( sigma_sets_nat @ N ) )
     => ( ( ( sigma_space_nat @ N )
          = ( sigma_space_nat @ N2 ) )
       => ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M2 ) )
         => ( ( ( sigma_3147302497200244656nnreal @ M )
              = ( sigma_3147302497200244656nnreal @ M2 ) )
           => ( ord_le1497719195298685117al_nat @ ( sigma_1856489715609627354al_nat @ M @ N ) @ ( sigma_1856489715609627354al_nat @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1122_measurable__mono,axiom,
    ! [N2: sigma_measure_nat,N: sigma_measure_nat,M: sigma_measure_o,M2: sigma_measure_o] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ N2 ) @ ( sigma_sets_nat @ N ) )
     => ( ( ( sigma_space_nat @ N )
          = ( sigma_space_nat @ N2 ) )
       => ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ M ) @ ( sigma_sets_o @ M2 ) )
         => ( ( ( sigma_space_o @ M )
              = ( sigma_space_o @ M2 ) )
           => ( ord_le4981610546006782297_o_nat @ ( sigma_1999164137574644376_o_nat @ M @ N ) @ ( sigma_1999164137574644376_o_nat @ M2 @ N2 ) ) ) ) ) ) ).

% measurable_mono
thf(fact_1123_measure__of__of__measure,axiom,
    ! [M: sigma_measure_real] :
      ( ( sigma_2693083824694760531f_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) @ ( sigma_emeasure_real @ M ) )
      = M ) ).

% measure_of_of_measure
thf(fact_1124_measure__of__of__measure,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( sigma_8167827323036178527nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_6589832970846575905nnreal @ M ) )
      = M ) ).

% measure_of_of_measure
thf(fact_1125_measure__of__of__measure,axiom,
    ! [M: sigma_measure_o] :
      ( ( sigma_measure_of_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) @ ( sigma_emeasure_o @ M ) )
      = M ) ).

% measure_of_of_measure
thf(fact_1126_measure__of__of__measure,axiom,
    ! [M: sigma_measure_nat] :
      ( ( sigma_measure_of_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) @ ( sigma_emeasure_nat @ M ) )
      = M ) ).

% measure_of_of_measure
thf(fact_1127_measurable__mono1,axiom,
    ! [M2: set_set_real,Omega: set_real,M: set_set_real,Mu: set_real > extend8495563244428889912nnreal,N: sigma_measure_real,Mu2: set_real > extend8495563244428889912nnreal] :
      ( ( ord_le3558479182127378552t_real @ M2 @ ( pow_real @ Omega ) )
     => ( ( ord_le3558479182127378552t_real @ M @ M2 )
       => ( ord_le4198349162570665613l_real @ ( sigma_5267869275261027754l_real @ ( sigma_2693083824694760531f_real @ Omega @ M @ Mu ) @ N ) @ ( sigma_5267869275261027754l_real @ ( sigma_2693083824694760531f_real @ Omega @ M2 @ Mu2 ) @ N ) ) ) ) ).

% measurable_mono1
thf(fact_1128_measurable__mono1,axiom,
    ! [M2: set_set_real,Omega: set_real,M: set_set_real,Mu: set_real > extend8495563244428889912nnreal,N: sigma_measure_o,Mu2: set_real > extend8495563244428889912nnreal] :
      ( ( ord_le3558479182127378552t_real @ M2 @ ( pow_real @ Omega ) )
     => ( ( ord_le3558479182127378552t_real @ M @ M2 )
       => ( ord_le1615110227528160547real_o @ ( sigma_3939073009482781210real_o @ ( sigma_2693083824694760531f_real @ Omega @ M @ Mu ) @ N ) @ ( sigma_3939073009482781210real_o @ ( sigma_2693083824694760531f_real @ Omega @ M2 @ Mu2 ) @ N ) ) ) ) ).

% measurable_mono1
thf(fact_1129_measurable__mono1,axiom,
    ! [M2: set_set_real,Omega: set_real,M: set_set_real,Mu: set_real > extend8495563244428889912nnreal,N: sigma_measure_nat,Mu2: set_real > extend8495563244428889912nnreal] :
      ( ( ord_le3558479182127378552t_real @ M2 @ ( pow_real @ Omega ) )
     => ( ( ord_le3558479182127378552t_real @ M @ M2 )
       => ( ord_le6098800555920186673al_nat @ ( sigma_6315060578831106510al_nat @ ( sigma_2693083824694760531f_real @ Omega @ M @ Mu ) @ N ) @ ( sigma_6315060578831106510al_nat @ ( sigma_2693083824694760531f_real @ Omega @ M2 @ Mu2 ) @ N ) ) ) ) ).

% measurable_mono1
thf(fact_1130_measurable__mono1,axiom,
    ! [M2: set_set_o,Omega: set_o,M: set_set_o,Mu: set_o > extend8495563244428889912nnreal,N: sigma_measure_real,Mu2: set_o > extend8495563244428889912nnreal] :
      ( ( ord_le4374716579403074808_set_o @ M2 @ ( pow_o @ Omega ) )
     => ( ( ord_le4374716579403074808_set_o @ M @ M2 )
       => ( ord_le3251842697534426805o_real @ ( sigma_2430008634441611636o_real @ ( sigma_measure_of_o @ Omega @ M @ Mu ) @ N ) @ ( sigma_2430008634441611636o_real @ ( sigma_measure_of_o @ Omega @ M2 @ Mu2 ) @ N ) ) ) ) ).

% measurable_mono1
thf(fact_1131_measurable__mono1,axiom,
    ! [M2: set_set_nat,Omega: set_nat,M: set_set_nat,Mu: set_nat > extend8495563244428889912nnreal,N: sigma_measure_real,Mu2: set_nat > extend8495563244428889912nnreal] :
      ( ( ord_le6893508408891458716et_nat @ M2 @ ( pow_nat @ Omega ) )
     => ( ( ord_le6893508408891458716et_nat @ M @ M2 )
       => ( ord_le2908806416726583473t_real @ ( sigma_1747752005702207822t_real @ ( sigma_measure_of_nat @ Omega @ M @ Mu ) @ N ) @ ( sigma_1747752005702207822t_real @ ( sigma_measure_of_nat @ Omega @ M2 @ Mu2 ) @ N ) ) ) ) ).

% measurable_mono1
thf(fact_1132_emeasure__sup__measure_H__le2,axiom,
    ! [B4: sigma_measure_real,C3: sigma_measure_real,A4: sigma_measure_real,X2: set_real] :
      ( ( ( sigma_sets_real @ B4 )
        = ( sigma_sets_real @ C3 ) )
     => ( ( ( sigma_sets_real @ A4 )
          = ( sigma_sets_real @ C3 ) )
       => ( ( member_set_real @ X2 @ ( sigma_sets_real @ C3 ) )
         => ( ! [Y6: set_real] :
                ( ( ord_less_eq_set_real @ Y6 @ X2 )
               => ( ( member_set_real @ Y6 @ ( sigma_sets_real @ A4 ) )
                 => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ A4 @ Y6 ) @ ( sigma_emeasure_real @ C3 @ Y6 ) ) ) )
           => ( ! [Y6: set_real] :
                  ( ( ord_less_eq_set_real @ Y6 @ X2 )
                 => ( ( member_set_real @ Y6 @ ( sigma_sets_real @ A4 ) )
                   => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ B4 @ Y6 ) @ ( sigma_emeasure_real @ C3 @ Y6 ) ) ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_real @ ( measur2147279183506585690e_real @ A4 @ B4 ) @ X2 ) @ ( sigma_emeasure_real @ C3 @ X2 ) ) ) ) ) ) ) ).

% emeasure_sup_measure'_le2
thf(fact_1133_emeasure__sup__measure_H__le2,axiom,
    ! [B4: sigma_7234349610311085201nnreal,C3: sigma_7234349610311085201nnreal,A4: sigma_7234349610311085201nnreal,X2: set_Ex3793607809372303086nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ B4 )
        = ( sigma_5465916536984168985nnreal @ C3 ) )
     => ( ( ( sigma_5465916536984168985nnreal @ A4 )
          = ( sigma_5465916536984168985nnreal @ C3 ) )
       => ( ( member603777416030116741nnreal @ X2 @ ( sigma_5465916536984168985nnreal @ C3 ) )
         => ( ! [Y6: set_Ex3793607809372303086nnreal] :
                ( ( ord_le6787938422905777998nnreal @ Y6 @ X2 )
               => ( ( member603777416030116741nnreal @ Y6 @ ( sigma_5465916536984168985nnreal @ A4 ) )
                 => ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ A4 @ Y6 ) @ ( sigma_6589832970846575905nnreal @ C3 @ Y6 ) ) ) )
           => ( ! [Y6: set_Ex3793607809372303086nnreal] :
                  ( ( ord_le6787938422905777998nnreal @ Y6 @ X2 )
                 => ( ( member603777416030116741nnreal @ Y6 @ ( sigma_5465916536984168985nnreal @ A4 ) )
                   => ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ B4 @ Y6 ) @ ( sigma_6589832970846575905nnreal @ C3 @ Y6 ) ) ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_6589832970846575905nnreal @ ( measur4473656680840910822nnreal @ A4 @ B4 ) @ X2 ) @ ( sigma_6589832970846575905nnreal @ C3 @ X2 ) ) ) ) ) ) ) ).

% emeasure_sup_measure'_le2
thf(fact_1134_emeasure__sup__measure_H__le2,axiom,
    ! [B4: sigma_measure_o,C3: sigma_measure_o,A4: sigma_measure_o,X2: set_o] :
      ( ( ( sigma_sets_o @ B4 )
        = ( sigma_sets_o @ C3 ) )
     => ( ( ( sigma_sets_o @ A4 )
          = ( sigma_sets_o @ C3 ) )
       => ( ( member_set_o @ X2 @ ( sigma_sets_o @ C3 ) )
         => ( ! [Y6: set_o] :
                ( ( ord_less_eq_set_o @ Y6 @ X2 )
               => ( ( member_set_o @ Y6 @ ( sigma_sets_o @ A4 ) )
                 => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ A4 @ Y6 ) @ ( sigma_emeasure_o @ C3 @ Y6 ) ) ) )
           => ( ! [Y6: set_o] :
                  ( ( ord_less_eq_set_o @ Y6 @ X2 )
                 => ( ( member_set_o @ Y6 @ ( sigma_sets_o @ A4 ) )
                   => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ B4 @ Y6 ) @ ( sigma_emeasure_o @ C3 @ Y6 ) ) ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_o @ ( measur4529518739368704874sure_o @ A4 @ B4 ) @ X2 ) @ ( sigma_emeasure_o @ C3 @ X2 ) ) ) ) ) ) ) ).

% emeasure_sup_measure'_le2
thf(fact_1135_emeasure__sup__measure_H__le2,axiom,
    ! [B4: sigma_measure_nat,C3: sigma_measure_nat,A4: sigma_measure_nat,X2: set_nat] :
      ( ( ( sigma_sets_nat @ B4 )
        = ( sigma_sets_nat @ C3 ) )
     => ( ( ( sigma_sets_nat @ A4 )
          = ( sigma_sets_nat @ C3 ) )
       => ( ( member_set_nat @ X2 @ ( sigma_sets_nat @ C3 ) )
         => ( ! [Y6: set_nat] :
                ( ( ord_less_eq_set_nat @ Y6 @ X2 )
               => ( ( member_set_nat @ Y6 @ ( sigma_sets_nat @ A4 ) )
                 => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ A4 @ Y6 ) @ ( sigma_emeasure_nat @ C3 @ Y6 ) ) ) )
           => ( ! [Y6: set_nat] :
                  ( ( ord_less_eq_set_nat @ Y6 @ X2 )
                 => ( ( member_set_nat @ Y6 @ ( sigma_sets_nat @ A4 ) )
                   => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ B4 @ Y6 ) @ ( sigma_emeasure_nat @ C3 @ Y6 ) ) ) )
             => ( ord_le3935885782089961368nnreal @ ( sigma_emeasure_nat @ ( measur876423496291765374re_nat @ A4 @ B4 ) @ X2 ) @ ( sigma_emeasure_nat @ C3 @ X2 ) ) ) ) ) ) ) ).

% emeasure_sup_measure'_le2
thf(fact_1136_le__measure__iff,axiom,
    ( ord_le487379304121309861e_real
    = ( ^ [M4: sigma_measure_real,N3: sigma_measure_real] :
          ( ( ( ( sigma_space_real @ M4 )
              = ( sigma_space_real @ N3 ) )
           => ( ( ( ( sigma_sets_real @ M4 )
                  = ( sigma_sets_real @ N3 ) )
               => ( ord_le637582726751450265nnreal @ ( sigma_emeasure_real @ M4 ) @ ( sigma_emeasure_real @ N3 ) ) )
              & ( ( ( sigma_sets_real @ M4 )
                 != ( sigma_sets_real @ N3 ) )
               => ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M4 ) @ ( sigma_sets_real @ N3 ) ) ) ) )
          & ( ( ( sigma_space_real @ M4 )
             != ( sigma_space_real @ N3 ) )
           => ( ord_less_eq_set_real @ ( sigma_space_real @ M4 ) @ ( sigma_space_real @ N3 ) ) ) ) ) ) ).

% le_measure_iff
thf(fact_1137_le__measure__iff,axiom,
    ( ord_le1854472233513649201nnreal
    = ( ^ [M4: sigma_7234349610311085201nnreal,N3: sigma_7234349610311085201nnreal] :
          ( ( ( ( sigma_3147302497200244656nnreal @ M4 )
              = ( sigma_3147302497200244656nnreal @ N3 ) )
           => ( ( ( ( sigma_5465916536984168985nnreal @ M4 )
                  = ( sigma_5465916536984168985nnreal @ N3 ) )
               => ( ord_le8913848522597308453nnreal @ ( sigma_6589832970846575905nnreal @ M4 ) @ ( sigma_6589832970846575905nnreal @ N3 ) ) )
              & ( ( ( sigma_5465916536984168985nnreal @ M4 )
                 != ( sigma_5465916536984168985nnreal @ N3 ) )
               => ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M4 ) @ ( sigma_5465916536984168985nnreal @ N3 ) ) ) ) )
          & ( ( ( sigma_3147302497200244656nnreal @ M4 )
             != ( sigma_3147302497200244656nnreal @ N3 ) )
           => ( ord_le6787938422905777998nnreal @ ( sigma_3147302497200244656nnreal @ M4 ) @ ( sigma_3147302497200244656nnreal @ N3 ) ) ) ) ) ) ).

% le_measure_iff
thf(fact_1138_le__measure__iff,axiom,
    ( ord_le478349814012620405sure_o
    = ( ^ [M4: sigma_measure_o,N3: sigma_measure_o] :
          ( ( ( ( sigma_space_o @ M4 )
              = ( sigma_space_o @ N3 ) )
           => ( ( ( ( sigma_sets_o @ M4 )
                  = ( sigma_sets_o @ N3 ) )
               => ( ord_le4997716739388385473nnreal @ ( sigma_emeasure_o @ M4 ) @ ( sigma_emeasure_o @ N3 ) ) )
              & ( ( ( sigma_sets_o @ M4 )
                 != ( sigma_sets_o @ N3 ) )
               => ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ M4 ) @ ( sigma_sets_o @ N3 ) ) ) ) )
          & ( ( ( sigma_space_o @ M4 )
             != ( sigma_space_o @ N3 ) )
           => ( ord_less_eq_set_o @ ( sigma_space_o @ M4 ) @ ( sigma_space_o @ N3 ) ) ) ) ) ) ).

% le_measure_iff
thf(fact_1139_le__measure__iff,axiom,
    ( ord_le2862109966718184649re_nat
    = ( ^ [M4: sigma_measure_nat,N3: sigma_measure_nat] :
          ( ( ( ( sigma_space_nat @ M4 )
              = ( sigma_space_nat @ N3 ) )
           => ( ( ( ( sigma_sets_nat @ M4 )
                  = ( sigma_sets_nat @ N3 ) )
               => ( ord_le9125165017735938237nnreal @ ( sigma_emeasure_nat @ M4 ) @ ( sigma_emeasure_nat @ N3 ) ) )
              & ( ( ( sigma_sets_nat @ M4 )
                 != ( sigma_sets_nat @ N3 ) )
               => ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ M4 ) @ ( sigma_sets_nat @ N3 ) ) ) ) )
          & ( ( ( sigma_space_nat @ M4 )
             != ( sigma_space_nat @ N3 ) )
           => ( ord_less_eq_set_nat @ ( sigma_space_nat @ M4 ) @ ( sigma_space_nat @ N3 ) ) ) ) ) ) ).

% le_measure_iff
thf(fact_1140_borel__measurable__subalgebra,axiom,
    ! [N: sigma_7234349610311085201nnreal,M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > real] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N ) @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( ( sigma_3147302497200244656nnreal @ N )
          = ( sigma_3147302497200244656nnreal @ M ) )
       => ( ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ N @ borel_5078946678739801102l_real ) )
         => ( member2874014351250825754l_real @ F @ ( sigma_7049758200512112822l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1141_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_real,M: sigma_measure_real,F: real > real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
     => ( ( ( sigma_space_real @ N )
          = ( sigma_space_real @ M ) )
       => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ N @ borel_5078946678739801102l_real ) )
         => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1142_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_o,M: sigma_measure_o,F: $o > real] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ N ) @ ( sigma_sets_o @ M ) )
     => ( ( ( sigma_space_o @ N )
          = ( sigma_space_o @ M ) )
       => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ N @ borel_5078946678739801102l_real ) )
         => ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1143_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_nat,M: sigma_measure_nat,F: nat > real] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ N ) @ ( sigma_sets_nat @ M ) )
     => ( ( ( sigma_space_nat @ N )
          = ( sigma_space_nat @ M ) )
       => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ N @ borel_5078946678739801102l_real ) )
         => ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1144_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_real,M: sigma_measure_real,F: real > extend8495563244428889912nnreal] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ N ) @ ( sigma_sets_real @ M ) )
     => ( ( ( sigma_space_real @ N )
          = ( sigma_space_real @ M ) )
       => ( ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ N @ borel_6524799422816628122nnreal ) )
         => ( member2919562650594848410nnreal @ F @ ( sigma_9017504469962657078nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1145_borel__measurable__subalgebra,axiom,
    ! [N: sigma_7234349610311085201nnreal,M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N ) @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( ( sigma_3147302497200244656nnreal @ N )
          = ( sigma_3147302497200244656nnreal @ M ) )
       => ( ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ N @ borel_6524799422816628122nnreal ) )
         => ( member8329810500450651686nnreal @ F @ ( sigma_7926153774531450434nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1146_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_o,M: sigma_measure_o,F: $o > extend8495563244428889912nnreal] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ N ) @ ( sigma_sets_o @ M ) )
     => ( ( ( sigma_space_o @ N )
          = ( sigma_space_o @ M ) )
       => ( ( member5265953103328284778nnreal @ F @ ( sigma_6459699357617223168nnreal @ N @ borel_6524799422816628122nnreal ) )
         => ( member5265953103328284778nnreal @ F @ ( sigma_6459699357617223168nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1147_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_nat,M: sigma_measure_nat,F: nat > extend8495563244428889912nnreal] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ N ) @ ( sigma_sets_nat @ M ) )
     => ( ( ( sigma_space_nat @ N )
          = ( sigma_space_nat @ M ) )
       => ( ( member8283130129095025342nnreal @ F @ ( sigma_6306161311797543642nnreal @ N @ borel_6524799422816628122nnreal ) )
         => ( member8283130129095025342nnreal @ F @ ( sigma_6306161311797543642nnreal @ M @ borel_6524799422816628122nnreal ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1148_borel__measurable__subalgebra,axiom,
    ! [N: sigma_7234349610311085201nnreal,M: sigma_7234349610311085201nnreal,F: extend8495563244428889912nnreal > $o] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ N ) @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( ( sigma_3147302497200244656nnreal @ N )
          = ( sigma_3147302497200244656nnreal @ M ) )
       => ( ( member8095236870201115968real_o @ F @ ( sigma_6279906219187228174real_o @ N @ borel_5500255247093592246orel_o ) )
         => ( member8095236870201115968real_o @ F @ ( sigma_6279906219187228174real_o @ M @ borel_5500255247093592246orel_o ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1149_borel__measurable__subalgebra,axiom,
    ! [N: sigma_measure_o,M: sigma_measure_o,F: $o > $o] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ N ) @ ( sigma_sets_o @ M ) )
     => ( ( ( sigma_space_o @ N )
          = ( sigma_space_o @ M ) )
       => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ N @ borel_5500255247093592246orel_o ) )
         => ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ borel_5500255247093592246orel_o ) ) ) ) ) ).

% borel_measurable_subalgebra
thf(fact_1150_measure__eqI__countable,axiom,
    ! [M: sigma_measure_real_a,A4: set_real_a,N: sigma_measure_real_a] :
      ( ( ( sigma_sets_real_a @ M )
        = ( pow_real_a @ A4 ) )
     => ( ( ( sigma_sets_real_a @ N )
          = ( pow_real_a @ A4 ) )
       => ( ( counta6639396083684174020real_a @ A4 )
         => ( ! [A3: real > a] :
                ( ( member_real_a @ A3 @ A4 )
               => ( ( sigma_6502373073922819808real_a @ M @ ( insert_real_a @ A3 @ bot_bot_set_real_a ) )
                  = ( sigma_6502373073922819808real_a @ N @ ( insert_real_a @ A3 @ bot_bot_set_real_a ) ) ) )
           => ( M = N ) ) ) ) ) ).

% measure_eqI_countable
thf(fact_1151_measure__eqI__countable,axiom,
    ! [M: sigma_measure_o_real,A4: set_o_real,N: sigma_measure_o_real] :
      ( ( ( sigma_sets_o_real @ M )
        = ( pow_o_real @ A4 ) )
     => ( ( ( sigma_sets_o_real @ N )
          = ( pow_o_real @ A4 ) )
       => ( ( counta8783200249485735024o_real @ A4 )
         => ( ! [A3: $o > real] :
                ( ( member_o_real @ A3 @ A4 )
               => ( ( sigma_4433523422001307788o_real @ M @ ( insert_o_real @ A3 @ bot_bot_set_o_real ) )
                  = ( sigma_4433523422001307788o_real @ N @ ( insert_o_real @ A3 @ bot_bot_set_o_real ) ) ) )
           => ( M = N ) ) ) ) ) ).

% measure_eqI_countable
thf(fact_1152_measure__eqI__countable,axiom,
    ! [M: sigma_3396294578489551860t_real,A4: set_nat_real,N: sigma_3396294578489551860t_real] :
      ( ( ( sigma_sets_nat_real @ M )
        = ( pow_nat_real @ A4 ) )
     => ( ( ( sigma_sets_nat_real @ N )
          = ( pow_nat_real @ A4 ) )
       => ( ( counta2162411829015494944t_real @ A4 )
         => ( ! [A3: nat > real] :
                ( ( member_nat_real @ A3 @ A4 )
               => ( ( sigma_2433462726372594692t_real @ M @ ( insert_nat_real @ A3 @ bot_bot_set_nat_real ) )
                  = ( sigma_2433462726372594692t_real @ N @ ( insert_nat_real @ A3 @ bot_bot_set_nat_real ) ) ) )
           => ( M = N ) ) ) ) ) ).

% measure_eqI_countable
thf(fact_1153_measure__eqI__countable,axiom,
    ! [M: sigma_measure_c_b,A4: set_c_b,N: sigma_measure_c_b] :
      ( ( ( sigma_sets_c_b @ M )
        = ( pow_c_b @ A4 ) )
     => ( ( ( sigma_sets_c_b @ N )
          = ( pow_c_b @ A4 ) )
       => ( ( counta2657777928882154345le_c_b @ A4 )
         => ( ! [A3: c > b] :
                ( ( member_c_b @ A3 @ A4 )
               => ( ( sigma_emeasure_c_b @ M @ ( insert_c_b @ A3 @ bot_bot_set_c_b ) )
                  = ( sigma_emeasure_c_b @ N @ ( insert_c_b @ A3 @ bot_bot_set_c_b ) ) ) )
           => ( M = N ) ) ) ) ) ).

% measure_eqI_countable
thf(fact_1154_measure__eqI__countable,axiom,
    ! [M: sigma_measure_a_b,A4: set_a_b,N: sigma_measure_a_b] :
      ( ( ( sigma_sets_a_b @ M )
        = ( pow_a_b @ A4 ) )
     => ( ( ( sigma_sets_a_b @ N )
          = ( pow_a_b @ A4 ) )
       => ( ( counta8232689092827506411le_a_b @ A4 )
         => ( ! [A3: a > b] :
                ( ( member_a_b @ A3 @ A4 )
               => ( ( sigma_emeasure_a_b @ M @ ( insert_a_b @ A3 @ bot_bot_set_a_b ) )
                  = ( sigma_emeasure_a_b @ N @ ( insert_a_b @ A3 @ bot_bot_set_a_b ) ) ) )
           => ( M = N ) ) ) ) ) ).

% measure_eqI_countable
thf(fact_1155_measure__eqI__countable,axiom,
    ! [M: sigma_measure_real,A4: set_real,N: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( pow_real @ A4 ) )
     => ( ( ( sigma_sets_real @ N )
          = ( pow_real @ A4 ) )
       => ( ( counta7319604579010473777e_real @ A4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ( sigma_emeasure_real @ M @ ( insert_real @ A3 @ bot_bot_set_real ) )
                  = ( sigma_emeasure_real @ N @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) )
           => ( M = N ) ) ) ) ) ).

% measure_eqI_countable
thf(fact_1156_measure__eqI__countable,axiom,
    ! [M: sigma_measure_nat,A4: set_nat,N: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ M )
        = ( pow_nat @ A4 ) )
     => ( ( ( sigma_sets_nat @ N )
          = ( pow_nat @ A4 ) )
       => ( ( counta1168086296615599829le_nat @ A4 )
         => ( ! [A3: nat] :
                ( ( member_nat @ A3 @ A4 )
               => ( ( sigma_emeasure_nat @ M @ ( insert_nat @ A3 @ bot_bot_set_nat ) )
                  = ( sigma_emeasure_nat @ N @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) )
           => ( M = N ) ) ) ) ) ).

% measure_eqI_countable
thf(fact_1157_measure__eqI__countable,axiom,
    ! [M: sigma_7234349610311085201nnreal,A4: set_Ex3793607809372303086nnreal,N: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( pow_Ex5372160365422184283nnreal @ A4 ) )
     => ( ( ( sigma_5465916536984168985nnreal @ N )
          = ( pow_Ex5372160365422184283nnreal @ A4 ) )
       => ( ( counta8439243037236335165nnreal @ A4 )
         => ( ! [A3: extend8495563244428889912nnreal] :
                ( ( member7908768830364227535nnreal @ A3 @ A4 )
               => ( ( sigma_6589832970846575905nnreal @ M @ ( insert7407984058720857448nnreal @ A3 @ bot_bo4854962954004695426nnreal ) )
                  = ( sigma_6589832970846575905nnreal @ N @ ( insert7407984058720857448nnreal @ A3 @ bot_bo4854962954004695426nnreal ) ) ) )
           => ( M = N ) ) ) ) ) ).

% measure_eqI_countable
thf(fact_1158_measure__eqI__countable,axiom,
    ! [M: sigma_measure_o,A4: set_o,N: sigma_measure_o] :
      ( ( ( sigma_sets_o @ M )
        = ( pow_o @ A4 ) )
     => ( ( ( sigma_sets_o @ N )
          = ( pow_o @ A4 ) )
       => ( ( counta5976203206615340371able_o @ A4 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A4 )
               => ( ( sigma_emeasure_o @ M @ ( insert_o @ A3 @ bot_bot_set_o ) )
                  = ( sigma_emeasure_o @ N @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) )
           => ( M = N ) ) ) ) ) ).

% measure_eqI_countable
thf(fact_1159_is__borel__def,axiom,
    ( borel_236569967776329622l_real
    = ( ^ [F2: real > real,M4: sigma_measure_real] : ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ M4 @ borel_5078946678739801102l_real ) ) ) ) ).

% is_borel_def
thf(fact_1160_is__borel__def,axiom,
    ( borel_2269360593130276488o_real
    = ( ^ [F2: $o > real,M4: sigma_measure_o] : ( member_o_real @ F2 @ ( sigma_2430008634441611636o_real @ M4 @ borel_5078946678739801102l_real ) ) ) ) ).

% is_borel_def
thf(fact_1161_is__borel__def,axiom,
    ( borel_9213571707143006522t_real
    = ( ^ [F2: nat > real,M4: sigma_measure_nat] : ( member_nat_real @ F2 @ ( sigma_1747752005702207822t_real @ M4 @ borel_5078946678739801102l_real ) ) ) ) ).

% is_borel_def
thf(fact_1162_is__borel__def,axiom,
    ( borel_3778424968171446062real_o
    = ( ^ [F2: real > $o,M4: sigma_measure_real] : ( member_real_o @ F2 @ ( sigma_3939073009482781210real_o @ M4 @ borel_5500255247093592246orel_o ) ) ) ) ).

% is_borel_def
thf(fact_1163_is__borel__def,axiom,
    ( borel_4557508243417129402al_nat
    = ( ^ [F2: real > nat,M4: sigma_measure_real] : ( member_real_nat @ F2 @ ( sigma_6315060578831106510al_nat @ M4 @ borel_8449730974584783410el_nat ) ) ) ) ).

% is_borel_def
thf(fact_1164_R__qbs__closed2,axiom,
    ! [X2: sigma_7234349610311085201nnreal] : ( qbs_cl7346018279885218754nnreal @ ( sigma_3147302497200244656nnreal @ X2 ) @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ X2 ) ) ).

% R_qbs_closed2
thf(fact_1165_R__qbs__closed2,axiom,
    ! [X2: sigma_measure_real] : ( qbs_closed2_real @ ( sigma_space_real @ X2 ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ X2 ) ) ).

% R_qbs_closed2
thf(fact_1166_R__qbs__closed2,axiom,
    ! [X2: sigma_measure_o] : ( qbs_closed2_o @ ( sigma_space_o @ X2 ) @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ X2 ) ) ).

% R_qbs_closed2
thf(fact_1167_R__qbs__closed2,axiom,
    ! [X2: sigma_measure_nat] : ( qbs_closed2_nat @ ( sigma_space_nat @ X2 ) @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ X2 ) ) ).

% R_qbs_closed2
thf(fact_1168_L__max__of__measurables,axiom,
    ! [M: sigma_measure_a,X2: quasi_borel_a] :
      ( ( ( sigma_space_a @ M )
        = ( qbs_space_a @ X2 ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ M ) )
       => ( ord_le3724670747650509150_set_a @ ( sigma_sets_a @ M ) @ ( sigma_sets_a @ ( measur7857763439677503898sure_a @ X2 ) ) ) ) ) ).

% L_max_of_measurables
thf(fact_1169_L__max__of__measurables,axiom,
    ! [M: sigma_measure_c,X2: quasi_borel_c] :
      ( ( ( sigma_space_c @ M )
        = ( qbs_space_c @ X2 ) )
     => ( ( ord_le5885474903713786379real_c @ ( qbs_Mx_c @ X2 ) @ ( sigma_523072396149930114real_c @ borel_5078946678739801102l_real @ M ) )
       => ( ord_le3866738827743201120_set_c @ ( sigma_sets_c @ M ) @ ( sigma_sets_c @ ( measur7857763439677503900sure_c @ X2 ) ) ) ) ) ).

% L_max_of_measurables
thf(fact_1170_L__max__of__measurables,axiom,
    ! [M: sigma_7234349610311085201nnreal,X2: quasi_9015997321629101608nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ M )
        = ( qbs_sp175953267596557954nnreal @ X2 ) )
     => ( ( ord_le2462468573666744473nnreal @ ( qbs_Mx6523938229262583809nnreal @ X2 ) @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ M ) )
       => ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ ( measur7384687747506661788nnreal @ X2 ) ) ) ) ) ).

% L_max_of_measurables
thf(fact_1171_L__max__of__measurables,axiom,
    ! [M: sigma_measure_real,X2: quasi_borel_real] :
      ( ( ( sigma_space_real @ M )
        = ( qbs_space_real @ X2 ) )
     => ( ( ord_le4198349162570665613l_real @ ( qbs_Mx_real @ X2 ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ M ) )
       => ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( sigma_sets_real @ ( measur1733462625046462224e_real @ X2 ) ) ) ) ) ).

% L_max_of_measurables
thf(fact_1172_L__max__of__measurables,axiom,
    ! [M: sigma_measure_o,X2: quasi_borel_o] :
      ( ( ( sigma_space_o @ M )
        = ( qbs_space_o @ X2 ) )
     => ( ( ord_le1615110227528160547real_o @ ( qbs_Mx_o @ X2 ) @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ M ) )
       => ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ M ) @ ( sigma_sets_o @ ( measur2926627334652526644sure_o @ X2 ) ) ) ) ) ).

% L_max_of_measurables
thf(fact_1173_L__max__of__measurables,axiom,
    ! [M: sigma_measure_nat,X2: quasi_borel_nat] :
      ( ( ( sigma_space_nat @ M )
        = ( qbs_space_nat @ X2 ) )
     => ( ( ord_le6098800555920186673al_nat @ ( qbs_Mx_nat @ X2 ) @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ M ) )
       => ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ M ) @ ( sigma_sets_nat @ ( measur7418878410283781684re_nat @ X2 ) ) ) ) ) ).

% L_max_of_measurables
thf(fact_1174_space__L,axiom,
    ! [X2: quasi_borel_real] :
      ( ( sigma_space_real @ ( measur1733462625046462224e_real @ X2 ) )
      = ( qbs_space_real @ X2 ) ) ).

% space_L
thf(fact_1175_space__L,axiom,
    ! [X2: quasi_9015997321629101608nnreal] :
      ( ( sigma_3147302497200244656nnreal @ ( measur7384687747506661788nnreal @ X2 ) )
      = ( qbs_sp175953267596557954nnreal @ X2 ) ) ).

% space_L
thf(fact_1176_space__L,axiom,
    ! [X2: quasi_borel_o] :
      ( ( sigma_space_o @ ( measur2926627334652526644sure_o @ X2 ) )
      = ( qbs_space_o @ X2 ) ) ).

% space_L
thf(fact_1177_space__L,axiom,
    ! [X2: quasi_borel_nat] :
      ( ( sigma_space_nat @ ( measur7418878410283781684re_nat @ X2 ) )
      = ( qbs_space_nat @ X2 ) ) ).

% space_L
thf(fact_1178_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > a,X2: quasi_borel_a] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X2 ) )
     => ( member_real_a @ Alpha @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( measur7857763439677503898sure_a @ X2 ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1179_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > c,X2: quasi_borel_c] :
      ( ( member_real_c @ Alpha @ ( qbs_Mx_c @ X2 ) )
     => ( member_real_c @ Alpha @ ( sigma_523072396149930114real_c @ borel_5078946678739801102l_real @ ( measur7857763439677503900sure_c @ X2 ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1180_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > real,X2: quasi_borel_real] :
      ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X2 ) )
     => ( member_real_real @ Alpha @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ ( measur1733462625046462224e_real @ X2 ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1181_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > $o,X2: quasi_borel_o] :
      ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X2 ) )
     => ( member_real_o @ Alpha @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ ( measur2926627334652526644sure_o @ X2 ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1182_qbs__Mx__are__measurable,axiom,
    ! [Alpha: real > nat,X2: quasi_borel_nat] :
      ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X2 ) )
     => ( member_real_nat @ Alpha @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ ( measur7418878410283781684re_nat @ X2 ) ) ) ) ).

% qbs_Mx_are_measurable
thf(fact_1183_l__preserves__morphisms,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_b] : ( ord_less_eq_set_a_b @ ( qbs_morphism_a_b @ X2 @ Y ) @ ( sigma_measurable_a_b @ ( measur7857763439677503898sure_a @ X2 ) @ ( measur7857763439677503899sure_b @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1184_l__preserves__morphisms,axiom,
    ! [X2: quasi_borel_c,Y: quasi_borel_b] : ( ord_less_eq_set_c_b @ ( qbs_morphism_c_b @ X2 @ Y ) @ ( sigma_measurable_c_b @ ( measur7857763439677503900sure_c @ X2 ) @ ( measur7857763439677503899sure_b @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1185_l__preserves__morphisms,axiom,
    ! [X2: quasi_borel_real,Y: quasi_borel_real] : ( ord_le4198349162570665613l_real @ ( qbs_mo5229770564518008146l_real @ X2 @ Y ) @ ( sigma_5267869275261027754l_real @ ( measur1733462625046462224e_real @ X2 ) @ ( measur1733462625046462224e_real @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1186_l__preserves__morphisms,axiom,
    ! [X2: quasi_borel_real,Y: quasi_borel_o] : ( ord_le1615110227528160547real_o @ ( qbs_morphism_real_o @ X2 @ Y ) @ ( sigma_3939073009482781210real_o @ ( measur1733462625046462224e_real @ X2 ) @ ( measur2926627334652526644sure_o @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1187_l__preserves__morphisms,axiom,
    ! [X2: quasi_borel_real,Y: quasi_borel_nat] : ( ord_le6098800555920186673al_nat @ ( qbs_mo6567951568834356598al_nat @ X2 @ Y ) @ ( sigma_6315060578831106510al_nat @ ( measur1733462625046462224e_real @ X2 ) @ ( measur7418878410283781684re_nat @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1188_l__preserves__morphisms,axiom,
    ! [X2: quasi_borel_o,Y: quasi_borel_real] : ( ord_le3251842697534426805o_real @ ( qbs_morphism_o_real @ X2 @ Y ) @ ( sigma_2430008634441611636o_real @ ( measur2926627334652526644sure_o @ X2 ) @ ( measur1733462625046462224e_real @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1189_l__preserves__morphisms,axiom,
    ! [X2: quasi_borel_nat,Y: quasi_borel_real] : ( ord_le2908806416726583473t_real @ ( qbs_mo2000642995705457910t_real @ X2 @ Y ) @ ( sigma_1747752005702207822t_real @ ( measur7418878410283781684re_nat @ X2 ) @ ( measur1733462625046462224e_real @ Y ) ) ) ).

% l_preserves_morphisms
thf(fact_1190_qbs__Mx__subset__of__measurable,axiom,
    ! [X2: quasi_borel_a] : ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( measur7857763439677503898sure_a @ X2 ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1191_qbs__Mx__subset__of__measurable,axiom,
    ! [X2: quasi_borel_c] : ( ord_le5885474903713786379real_c @ ( qbs_Mx_c @ X2 ) @ ( sigma_523072396149930114real_c @ borel_5078946678739801102l_real @ ( measur7857763439677503900sure_c @ X2 ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1192_qbs__Mx__subset__of__measurable,axiom,
    ! [X2: quasi_borel_real] : ( ord_le4198349162570665613l_real @ ( qbs_Mx_real @ X2 ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ ( measur1733462625046462224e_real @ X2 ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1193_qbs__Mx__subset__of__measurable,axiom,
    ! [X2: quasi_borel_o] : ( ord_le1615110227528160547real_o @ ( qbs_Mx_o @ X2 ) @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ ( measur2926627334652526644sure_o @ X2 ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1194_qbs__Mx__subset__of__measurable,axiom,
    ! [X2: quasi_borel_nat] : ( ord_le6098800555920186673al_nat @ ( qbs_Mx_nat @ X2 ) @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ ( measur7418878410283781684re_nat @ X2 ) ) ) ).

% qbs_Mx_subset_of_measurable
thf(fact_1195_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
    ( top_to3356475028079756884nnreal
    = ( sigma_5465916536984168985nnreal @ ( nonneg1394255657502361022nnreal @ top_to7994903218803871134nnreal ) ) ) ).

% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_1196_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_1197_sets__uniform__count__measure__eq__UNIV_I2_J,axiom,
    ( top_top_set_set_real
    = ( sigma_sets_real @ ( nonneg387815094551837234e_real @ top_top_set_real ) ) ) ).

% sets_uniform_count_measure_eq_UNIV(2)
thf(fact_1198_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_1199_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
    ( ( sigma_5465916536984168985nnreal @ ( nonneg1394255657502361022nnreal @ top_to7994903218803871134nnreal ) )
    = top_to3356475028079756884nnreal ) ).

% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_1200_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_1201_sets__uniform__count__measure__eq__UNIV_I1_J,axiom,
    ( ( sigma_sets_real @ ( nonneg387815094551837234e_real @ top_top_set_real ) )
    = top_top_set_set_real ) ).

% sets_uniform_count_measure_eq_UNIV(1)
thf(fact_1202_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_1203_qbs__Mx__sigma__Mx__contra,axiom,
    ! [X2: quasi_borel_a,Y: quasi_borel_a] :
      ( ( ( qbs_space_a @ X2 )
        = ( qbs_space_a @ Y ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( qbs_Mx_a @ Y ) )
       => ( ord_le3724670747650509150_set_a @ ( measur1355555235234291375a_Mx_a @ Y ) @ ( measur1355555235234291375a_Mx_a @ X2 ) ) ) ) ).

% qbs_Mx_sigma_Mx_contra
thf(fact_1204_qbs__Mx__sigma__Mx__contra,axiom,
    ! [X2: quasi_borel_c,Y: quasi_borel_c] :
      ( ( ( qbs_space_c @ X2 )
        = ( qbs_space_c @ Y ) )
     => ( ( ord_le5885474903713786379real_c @ ( qbs_Mx_c @ X2 ) @ ( qbs_Mx_c @ Y ) )
       => ( ord_le3866738827743201120_set_c @ ( measur1355555235234291377a_Mx_c @ Y ) @ ( measur1355555235234291377a_Mx_c @ X2 ) ) ) ) ).

% qbs_Mx_sigma_Mx_contra
thf(fact_1205_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_a,F: a > b,Y: quasi_borel_b] :
      ( ( counta4098120917673242425able_a @ ( qbs_space_a @ X2 ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( sigma_count_space_a @ ( qbs_space_a @ X2 ) ) ) )
       => ( ! [I3: a] :
              ( ( member_a @ I3 @ ( qbs_space_a @ X2 ) )
             => ( member_b @ ( F @ I3 ) @ ( qbs_space_b @ Y ) ) )
         => ( member_a_b @ F @ ( qbs_morphism_a_b @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1206_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_c,F: c > b,Y: quasi_borel_b] :
      ( ( counta4098120917673242427able_c @ ( qbs_space_c @ X2 ) )
     => ( ( ord_le5885474903713786379real_c @ ( qbs_Mx_c @ X2 ) @ ( sigma_523072396149930114real_c @ borel_5078946678739801102l_real @ ( sigma_count_space_c @ ( qbs_space_c @ X2 ) ) ) )
       => ( ! [I3: c] :
              ( ( member_c @ I3 @ ( qbs_space_c @ X2 ) )
             => ( member_b @ ( F @ I3 ) @ ( qbs_space_b @ Y ) ) )
         => ( member_c_b @ F @ ( qbs_morphism_c_b @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1207_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_o,F: $o > real,Y: quasi_borel_real] :
      ( ( counta5976203206615340371able_o @ ( qbs_space_o @ X2 ) )
     => ( ( ord_le1615110227528160547real_o @ ( qbs_Mx_o @ X2 ) @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ ( sigma_count_space_o @ ( qbs_space_o @ X2 ) ) ) )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ ( qbs_space_o @ X2 ) )
             => ( member_real @ ( F @ I3 ) @ ( qbs_space_real @ Y ) ) )
         => ( member_o_real @ F @ ( qbs_morphism_o_real @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1208_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_nat,F: nat > real,Y: quasi_borel_real] :
      ( ( counta1168086296615599829le_nat @ ( qbs_space_nat @ X2 ) )
     => ( ( ord_le6098800555920186673al_nat @ ( qbs_Mx_nat @ X2 ) @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ ( sigma_7685844798829912695ce_nat @ ( qbs_space_nat @ X2 ) ) ) )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ ( qbs_space_nat @ X2 ) )
             => ( member_real @ ( F @ I3 ) @ ( qbs_space_real @ Y ) ) )
         => ( member_nat_real @ F @ ( qbs_mo2000642995705457910t_real @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1209_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_real,F: real > a,Y: quasi_borel_a] :
      ( ( counta7319604579010473777e_real @ ( qbs_space_real @ X2 ) )
     => ( ( ord_le4198349162570665613l_real @ ( qbs_Mx_real @ X2 ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ ( sigma_8508918144308765139e_real @ ( qbs_space_real @ X2 ) ) ) )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ ( qbs_space_real @ X2 ) )
             => ( member_a @ ( F @ I3 ) @ ( qbs_space_a @ Y ) ) )
         => ( member_real_a @ F @ ( qbs_morphism_real_a @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1210_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_a,F: a > real > a,Y: quasi_borel_real_a] :
      ( ( counta4098120917673242425able_a @ ( qbs_space_a @ X2 ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( sigma_count_space_a @ ( qbs_space_a @ X2 ) ) ) )
       => ( ! [I3: a] :
              ( ( member_a @ I3 @ ( qbs_space_a @ X2 ) )
             => ( member_real_a @ ( F @ I3 ) @ ( qbs_space_real_a @ Y ) ) )
         => ( member_a_real_a @ F @ ( qbs_mo2545572719379674883real_a @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1211_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_a,F: a > $o > real,Y: quasi_borel_o_real] :
      ( ( counta4098120917673242425able_a @ ( qbs_space_a @ X2 ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( sigma_count_space_a @ ( qbs_space_a @ X2 ) ) ) )
       => ( ! [I3: a] :
              ( ( member_a @ I3 @ ( qbs_space_a @ X2 ) )
             => ( member_o_real @ ( F @ I3 ) @ ( qbs_space_o_real @ Y ) ) )
         => ( member_a_o_real @ F @ ( qbs_mo7370372776400040495o_real @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1212_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_a,F: a > nat > real,Y: quasi_borel_nat_real] :
      ( ( counta4098120917673242425able_a @ ( qbs_space_a @ X2 ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( sigma_count_space_a @ ( qbs_space_a @ X2 ) ) ) )
       => ( ! [I3: a] :
              ( ( member_a @ I3 @ ( qbs_space_a @ X2 ) )
             => ( member_nat_real @ ( F @ I3 ) @ ( qbs_space_nat_real @ Y ) ) )
         => ( member_a_nat_real @ F @ ( qbs_mo5829272867083514145t_real @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1213_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_a,F: a > c > b,Y: quasi_borel_c_b] :
      ( ( counta4098120917673242425able_a @ ( qbs_space_a @ X2 ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( sigma_count_space_a @ ( qbs_space_a @ X2 ) ) ) )
       => ( ! [I3: a] :
              ( ( member_a @ I3 @ ( qbs_space_a @ X2 ) )
             => ( member_c_b @ ( F @ I3 ) @ ( qbs_space_c_b @ Y ) ) )
         => ( member_a_c_b @ F @ ( qbs_morphism_a_c_b @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1214_qbs__morphisn__from__countable,axiom,
    ! [X2: quasi_borel_a,F: a > a > b,Y: quasi_borel_a_b] :
      ( ( counta4098120917673242425able_a @ ( qbs_space_a @ X2 ) )
     => ( ( ord_le5743406823621094409real_a @ ( qbs_Mx_a @ X2 ) @ ( sigma_523072396149930112real_a @ borel_5078946678739801102l_real @ ( sigma_count_space_a @ ( qbs_space_a @ X2 ) ) ) )
       => ( ! [I3: a] :
              ( ( member_a @ I3 @ ( qbs_space_a @ X2 ) )
             => ( member_a_b @ ( F @ I3 ) @ ( qbs_space_a_b @ Y ) ) )
         => ( member_a_a_b @ F @ ( qbs_morphism_a_a_b @ X2 @ Y ) ) ) ) ) ).

% qbs_morphisn_from_countable
thf(fact_1215_sets__L,axiom,
    ! [X2: quasi_borel_real] :
      ( ( sigma_sets_real @ ( measur1733462625046462224e_real @ X2 ) )
      = ( measur7113710793995618619x_real @ X2 ) ) ).

% sets_L
thf(fact_1216_sets__L,axiom,
    ! [X2: quasi_9015997321629101608nnreal] :
      ( ( sigma_5465916536984168985nnreal @ ( measur7384687747506661788nnreal @ X2 ) )
      = ( measur4088046290863407431nnreal @ X2 ) ) ).

% sets_L
thf(fact_1217_sets__L,axiom,
    ! [X2: quasi_borel_o] :
      ( ( sigma_sets_o @ ( measur2926627334652526644sure_o @ X2 ) )
      = ( measur560158960551862217a_Mx_o @ X2 ) ) ).

% sets_L
thf(fact_1218_sets__L,axiom,
    ! [X2: quasi_borel_nat] :
      ( ( sigma_sets_nat @ ( measur7418878410283781684re_nat @ X2 ) )
      = ( measur4633199607246208479Mx_nat @ X2 ) ) ).

% sets_L
thf(fact_1219_space__count__space,axiom,
    ! [Omega: set_Ex3793607809372303086nnreal] :
      ( ( sigma_3147302497200244656nnreal @ ( sigma_7204664791115113951nnreal @ Omega ) )
      = Omega ) ).

% space_count_space
thf(fact_1220_space__count__space,axiom,
    ! [Omega: set_o] :
      ( ( sigma_space_o @ ( sigma_count_space_o @ Omega ) )
      = Omega ) ).

% space_count_space
thf(fact_1221_space__count__space,axiom,
    ! [Omega: set_nat] :
      ( ( sigma_space_nat @ ( sigma_7685844798829912695ce_nat @ Omega ) )
      = Omega ) ).

% space_count_space
thf(fact_1222_space__count__space,axiom,
    ! [Omega: set_real] :
      ( ( sigma_space_real @ ( sigma_8508918144308765139e_real @ Omega ) )
      = Omega ) ).

% space_count_space
thf(fact_1223_sets__count__space,axiom,
    ! [Omega: set_Ex3793607809372303086nnreal] :
      ( ( sigma_5465916536984168985nnreal @ ( sigma_7204664791115113951nnreal @ Omega ) )
      = ( pow_Ex5372160365422184283nnreal @ Omega ) ) ).

% sets_count_space
thf(fact_1224_sets__count__space,axiom,
    ! [Omega: set_o] :
      ( ( sigma_sets_o @ ( sigma_count_space_o @ Omega ) )
      = ( pow_o @ Omega ) ) ).

% sets_count_space
thf(fact_1225_sets__count__space,axiom,
    ! [Omega: set_nat] :
      ( ( sigma_sets_nat @ ( sigma_7685844798829912695ce_nat @ Omega ) )
      = ( pow_nat @ Omega ) ) ).

% sets_count_space
thf(fact_1226_sets__count__space,axiom,
    ! [Omega: set_real] :
      ( ( sigma_sets_real @ ( sigma_8508918144308765139e_real @ Omega ) )
      = ( pow_real @ Omega ) ) ).

% sets_count_space
thf(fact_1227_space__uniform__count__measure__empty__iff,axiom,
    ! [X2: set_real] :
      ( ( ( sigma_space_real @ ( nonneg387815094551837234e_real @ X2 ) )
        = bot_bot_set_real )
      = ( X2 = bot_bot_set_real ) ) ).

% space_uniform_count_measure_empty_iff
thf(fact_1228_space__uniform__count__measure__empty__iff,axiom,
    ! [X2: set_nat] :
      ( ( ( sigma_space_nat @ ( nonneg7031465154080143958re_nat @ X2 ) )
        = bot_bot_set_nat )
      = ( X2 = bot_bot_set_nat ) ) ).

% space_uniform_count_measure_empty_iff
thf(fact_1229_space__uniform__count__measure__empty__iff,axiom,
    ! [X2: set_Ex3793607809372303086nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ ( nonneg1394255657502361022nnreal @ X2 ) )
        = bot_bo4854962954004695426nnreal )
      = ( X2 = bot_bo4854962954004695426nnreal ) ) ).

% space_uniform_count_measure_empty_iff
thf(fact_1230_space__uniform__count__measure__empty__iff,axiom,
    ! [X2: set_o] :
      ( ( ( sigma_space_o @ ( nonneg5198678888045619090sure_o @ X2 ) )
        = bot_bot_set_o )
      = ( X2 = bot_bot_set_o ) ) ).

% space_uniform_count_measure_empty_iff
thf(fact_1231_sets__uniform__count__measure__count__space,axiom,
    ! [A4: set_Ex3793607809372303086nnreal] :
      ( ( sigma_5465916536984168985nnreal @ ( nonneg1394255657502361022nnreal @ A4 ) )
      = ( sigma_5465916536984168985nnreal @ ( sigma_7204664791115113951nnreal @ A4 ) ) ) ).

% sets_uniform_count_measure_count_space
thf(fact_1232_sets__uniform__count__measure__count__space,axiom,
    ! [A4: set_o] :
      ( ( sigma_sets_o @ ( nonneg5198678888045619090sure_o @ A4 ) )
      = ( sigma_sets_o @ ( sigma_count_space_o @ A4 ) ) ) ).

% sets_uniform_count_measure_count_space
thf(fact_1233_sets__uniform__count__measure__count__space,axiom,
    ! [A4: set_nat] :
      ( ( sigma_sets_nat @ ( nonneg7031465154080143958re_nat @ A4 ) )
      = ( sigma_sets_nat @ ( sigma_7685844798829912695ce_nat @ A4 ) ) ) ).

% sets_uniform_count_measure_count_space
thf(fact_1234_sets__uniform__count__measure__count__space,axiom,
    ! [A4: set_real] :
      ( ( sigma_sets_real @ ( nonneg387815094551837234e_real @ A4 ) )
      = ( sigma_sets_real @ ( sigma_8508918144308765139e_real @ A4 ) ) ) ).

% sets_uniform_count_measure_count_space
thf(fact_1235_real_Ocountable__space__discrete,axiom,
    ( ( counta7319604579010473777e_real @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
   => ( ( sigma_sets_real @ borel_5078946678739801102l_real )
      = ( sigma_sets_real @ ( sigma_8508918144308765139e_real @ ( sigma_space_real @ borel_5078946678739801102l_real ) ) ) ) ) ).

% real.countable_space_discrete
thf(fact_1236_real_Ostandard__borel__axioms,axiom,
    standard_borel_real @ borel_5078946678739801102l_real ).

% real.standard_borel_axioms
thf(fact_1237_real_Ostandard__borel__sets,axiom,
    ! [Y: sigma_measure_real] :
      ( ( ( sigma_sets_real @ borel_5078946678739801102l_real )
        = ( sigma_sets_real @ Y ) )
     => ( standard_borel_real @ Y ) ) ).

% real.standard_borel_sets
thf(fact_1238_uncountable__UNIV__real,axiom,
    ~ ( counta7319604579010473777e_real @ top_top_set_real ) ).

% uncountable_UNIV_real
thf(fact_1239_ennreal_Og__meas,axiom,
    member2919562650594848410nnreal @ ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal ) @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).

% ennreal.g_meas
thf(fact_1240_real_Og__meas,axiom,
    member_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% real.g_meas
thf(fact_1241_bool_Og__meas,axiom,
    member_real_o @ ( standard_g_o @ borel_5500255247093592246orel_o ) @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ borel_5500255247093592246orel_o ) ).

% bool.g_meas
thf(fact_1242_nat_Og__meas,axiom,
    member_real_nat @ ( standard_g_nat @ borel_8449730974584783410el_nat ) @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) ).

% nat.g_meas
thf(fact_1243_separate__measurable,axiom,
    ! [P3: real > nat] :
      ( ! [I3: nat] : ( member_set_real @ ( vimage_real_nat @ P3 @ ( insert_nat @ I3 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( member_real_nat @ P3 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) ) ) ).

% separate_measurable
thf(fact_1244_measurable__separate,axiom,
    ! [P3: real > nat,I2: nat] :
      ( ( member_real_nat @ P3 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( member_set_real @ ( vimage_real_nat @ P3 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% measurable_separate
thf(fact_1245_ennreal_Osingleton__sets,axiom,
    ! [X3: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ X3 @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
     => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).

% ennreal.singleton_sets
thf(fact_1246_bool_Osingleton__sets,axiom,
    ! [X3: $o] :
      ( ( member_o @ X3 @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
     => ( member_set_o @ ( insert_o @ X3 @ bot_bot_set_o ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% bool.singleton_sets
thf(fact_1247_nat_Osingleton__sets,axiom,
    ! [X3: nat] :
      ( ( member_nat @ X3 @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
     => ( member_set_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% nat.singleton_sets
thf(fact_1248_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_1249_open__minus__countable,axiom,
    ! [A4: set_real,S4: set_real] :
      ( ( counta7319604579010473777e_real @ A4 )
     => ( ( S4 != bot_bot_set_real )
       => ( ( topolo4860482606490270245n_real @ S4 )
         => ? [X: real] :
              ( ( member_real @ X @ S4 )
              & ~ ( member_real @ X @ A4 ) ) ) ) ) ).

% open_minus_countable
thf(fact_1250_real_Ogf__comp__id_I2_J,axiom,
    ! [X3: real] :
      ( ( member_real @ X3 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( ( standard_g_real @ borel_5078946678739801102l_real @ ( standard_f_real @ borel_5078946678739801102l_real @ X3 ) )
        = X3 ) ) ).

% real.gf_comp_id(2)
thf(fact_1251_real_Ogf__comp__id_H_I2_J,axiom,
    ! [X3: real] :
      ( ( standard_g_real @ borel_5078946678739801102l_real @ ( standard_f_real @ borel_5078946678739801102l_real @ X3 ) )
      = X3 ) ).

% real.gf_comp_id'(2)
thf(fact_1252_real_Of__meas,axiom,
    member_real_real @ ( standard_f_real @ borel_5078946678739801102l_real ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% real.f_meas
thf(fact_1253_r01__to__r01__r01__snd__measurable,axiom,
    member_real_real @ r01_to_r01_r01_snd @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% r01_to_r01_r01_snd_measurable
thf(fact_1254_r01__to__r01__r01__fst__measurable,axiom,
    member_real_real @ r01_to_r01_r01_fst @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% r01_to_r01_r01_fst_measurable
thf(fact_1255_ennreal_Of__meas,axiom,
    member2874014351250825754l_real @ ( standa4501783974915749827nnreal @ borel_6524799422816628122nnreal ) @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ borel_5078946678739801102l_real ) ).

% ennreal.f_meas
thf(fact_1256_nat_Of__meas,axiom,
    member_nat_real @ ( standard_f_nat @ borel_8449730974584783410el_nat ) @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) ).

% nat.f_meas
thf(fact_1257_bool_Of__meas,axiom,
    member_o_real @ ( standard_f_o @ borel_5500255247093592246orel_o ) @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ borel_5078946678739801102l_real ) ).

% bool.f_meas
thf(fact_1258_real_Ogf__comp__id_I1_J,axiom,
    ! [X3: real] :
      ( ( member_real @ X3 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( ( comp_real_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( standard_f_real @ borel_5078946678739801102l_real ) @ X3 )
        = X3 ) ) ).

% real.gf_comp_id(1)
thf(fact_1259_real_Oexist__fg,axiom,
    ? [X: real > real] :
      ( ( member_real_real @ X @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > real] :
          ( ( member_real_real @ Xa @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
          & ! [Xb: real] :
              ( ( member_real @ Xb @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
             => ( ( comp_real_real_real @ Xa @ X @ Xb )
                = Xb ) ) ) ) ).

% real.exist_fg
thf(fact_1260_ennreal_Oexist__fg,axiom,
    ? [X: extend8495563244428889912nnreal > real] :
      ( ( member2874014351250825754l_real @ X @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ Xa @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) )
          & ! [Xb: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ Xb @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
             => ( ( comp_r6281409797179841921nnreal @ Xa @ X @ Xb )
                = Xb ) ) ) ) ).

% ennreal.exist_fg
thf(fact_1261_nat_Oexist__fg,axiom,
    ? [X: nat > real] :
      ( ( member_nat_real @ X @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > nat] :
          ( ( member_real_nat @ Xa @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
          & ! [Xb: nat] :
              ( ( member_nat @ Xb @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
             => ( ( comp_real_nat_nat @ Xa @ X @ Xb )
                = Xb ) ) ) ) ).

% nat.exist_fg
thf(fact_1262_bool_Oexist__fg,axiom,
    ? [X: $o > real] :
      ( ( member_o_real @ X @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > $o] :
          ( ( member_real_o @ Xa @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ borel_5500255247093592246orel_o ) )
          & ! [Xb: $o] :
              ( ( member_o @ Xb @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
             => ( ( comp_real_o_o @ Xa @ X @ Xb )
                = Xb ) ) ) ) ).

% bool.exist_fg
thf(fact_1263_real_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_real_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( standard_f_real @ borel_5078946678739801102l_real ) )
    = id_real ) ).

% real.gf_comp_id'(1)
thf(fact_1264_nat__real_Opair__standard__borel__axioms,axiom,
    pair_s8264832550775477520t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ).

% nat_real.pair_standard_borel_axioms
thf(fact_1265_real_Ostandard__borel__space__UNIV__axioms,axiom,
    standa1306199911732814765V_real @ borel_5078946678739801102l_real ).

% real.standard_borel_space_UNIV_axioms
thf(fact_1266_nat__real_Opair__standard__borel__space__UNIV__axioms,axiom,
    pair_s5107880421860391064t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ).

% nat_real.pair_standard_borel_space_UNIV_axioms
thf(fact_1267_f01__borel__measurable,axiom,
    ! [F: real > real] :
      ( ( member_set_real @ ( vimage_real_real @ F @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( ( member_set_real @ ( vimage_real_real @ F @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
       => ( ! [R3: real] : ( member_real @ ( F @ R3 ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
         => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) ) ) ).

% f01_borel_measurable
thf(fact_1268_r01__to__r01__r01__snd_Hin01,axiom,
    ! [R2: real,N4: nat] : ( member_nat @ ( r01_to_r01_r01_snd2 @ R2 @ N4 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% r01_to_r01_r01_snd'in01
thf(fact_1269_r01__to__r01__r01__fst_Hin01,axiom,
    ! [R2: real,N4: nat] : ( member_nat @ ( r01_to_r01_r01_fst2 @ R2 @ N4 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% r01_to_r01_r01_fst'in01
thf(fact_1270_r01__r01__to__r01_Hin01,axiom,
    ! [Rs: produc2422161461964618553l_real,N4: nat] : ( member_nat @ ( r01_r01_to_r01 @ Rs @ N4 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% r01_r01_to_r01'in01
thf(fact_1271_biexp01__well__formedE,axiom,
    ! [A: nat > nat] :
      ( ( biexp01_well_formed @ A )
     => ( ! [N5: nat] : ( member_nat @ ( A @ N5 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
        & ! [N5: nat] :
          ? [M3: nat] :
            ( ( ord_less_eq_nat @ N5 @ M3 )
            & ( ( A @ M3 )
              = zero_zero_nat ) ) ) ) ).

% biexp01_well_formedE
thf(fact_1272_biexp01__well__formed__def,axiom,
    ( biexp01_well_formed
    = ( ^ [A6: nat > nat] :
          ( ! [N6: nat] : ( member_nat @ ( A6 @ N6 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
          & ! [N6: nat] :
            ? [M5: nat] :
              ( ( ord_less_eq_nat @ N6 @ M5 )
              & ( ( A6 @ M5 )
                = zero_zero_nat ) ) ) ) ) ).

% biexp01_well_formed_def
thf(fact_1273_biexp01__well__formedI,axiom,
    ! [A: nat > nat] :
      ( ! [N7: nat] : ( member_nat @ ( A @ N7 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
     => ( ! [N7: nat] :
          ? [M6: nat] :
            ( ( ord_less_eq_nat @ N7 @ M6 )
            & ( ( A @ M6 )
              = zero_zero_nat ) )
       => ( biexp01_well_formed @ A ) ) ) ).

% biexp01_well_formedI
thf(fact_1274_real01__binary__expansion_H__0or1,axiom,
    ! [R2: real,N4: nat] : ( member_nat @ ( r01_binary_expansion @ R2 @ N4 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% real01_binary_expansion'_0or1
thf(fact_1275_measurable__ennreal,axiom,
    member2919562650594848410nnreal @ extend7643940197134561352nnreal @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).

% measurable_ennreal
thf(fact_1276_set__borel__integrable__singleton,axiom,
    ! [X3: real,F: real > real] : ( set_se5970144800844511125l_real @ lebesgue_lborel_real @ ( insert_real @ X3 @ bot_bot_set_real ) @ F ) ).

% set_borel_integrable_singleton

% Helper facts (3)
thf(help_If_3_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_T,axiom,
    ! [P3: $o] :
      ( ( P3 = $true )
      | ( P3 = $false ) ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_T,axiom,
    ! [X3: sum_sum_a_c,Y2: sum_sum_a_c] :
      ( ( if_Sum_sum_a_c @ $false @ X3 @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_T,axiom,
    ! [X3: sum_sum_a_c,Y2: sum_sum_a_c] :
      ( ( if_Sum_sum_a_c @ $true @ X3 @ Y2 )
      = X3 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [Alpha_1: real > a] :
      ( ( ( member_real_a @ Alpha_1 @ ( qbs_Mx_a @ x ) )
        & ( alpha
          = ( ^ [R: real] : ( sum_Inl_a_c @ ( Alpha_1 @ R ) ) ) ) )
     => thesis ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------