TPTP Problem File: SLH0936^1.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/0001_QuasiBorel/prob_00113_003809__15128404_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1877 ( 531 unt; 597 typ;   0 def)
%            Number of atoms       : 3825 (1100 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 11420 ( 297   ~;  41   |; 430   &;9322   @)
%                                         (   0 <=>;1330  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   7 avg)
%            Number of types       :  120 ( 119 usr)
%            Number of type conns  : 2461 (2461   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  481 ( 478 usr;  77 con; 0-4 aty)
%            Number of variables   : 3790 ( 756   ^;2976   !;  58   ?;3790   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:08:21.534
%------------------------------------------------------------------------------
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    quasi_borel_b: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_Itf__b_J,type,
    sigma_measure_b: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
    set_set_b: $tType ).

thf(ty_n_t__Sigma____Algebra__Omeasure_I_Eo_J,type,
    sigma_measure_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_Mtf__b_J_J,type,
    set_b_b: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    set_set_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__b_M_Eo_J_J,type,
    set_b_o: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__b_J_J,type,
    set_o_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
    set_o_o: $tType ).

thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
    set_real: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_Itf__b_J,type,
    set_b: $tType ).

thf(ty_n_t__Set__Oset_I_Eo_J,type,
    set_o: $tType ).

thf(ty_n_t__Real__Oreal,type,
    real: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_tf__b,type,
    b: $tType ).

% Explicit typings (478)
thf(sy_c_Basic__BNFs_Osnds_001t__Set__Oset_Itf__b_J_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    basic_3503571567493806864real_b: produc4373655991844504306real_b > set_set_real_b ).

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_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    borel_7664774927250867717t_real: ( real > nat > real ) > sigma_measure_real > $o ).

thf(sy_c_Borel__Space_Ois__borel_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    borel_3689727235773987461al_nat: ( real > real > nat ) > 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_Otopological__space__class_Oborel_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    borel_5725509229735958141t_real: sigma_3396294578489551860t_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    borel_1750461538259077885al_nat: sigma_6586288717683155060al_nat ).

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__Nat__Onat,type,
    borel_8449730974584783410el_nat: sigma_measure_nat ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Real__Oreal_J_Mt__Nat__Onat_J,type,
    borel_8282316692848364138al_nat: sigma_5506949827314733513al_nat ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
    borel_5246341344715896774l_real: sigma_1811887234549217061l_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    borel_1467662793739727594t_real: sigma_3149601563892791881t_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    borel_2901874158916493343at_nat: sigma_5515648953823433982at_nat ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    borel_3052437322889528059t_real: sigma_5310753476256395226t_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    borel_7346254683739562566t_real: sigma_73088456041816485t_real ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    borel_8300761668267423611al_nat: sigma_8500747615449998426al_nat ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    borel_9155112475215227991l_real: sigma_2308072346491277622l_real ).

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_Eo,type,
    counta5976203206615340371able_o: set_o > $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_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_FuncSet_OPi_001_Eo_001_Eo,type,
    pi_o_o: set_o > ( $o > set_o ) > set_o_o ).

thf(sy_c_FuncSet_OPi_001_Eo_001t__Nat__Onat,type,
    pi_o_nat: set_o > ( $o > set_nat ) > set_o_nat ).

thf(sy_c_FuncSet_OPi_001_Eo_001t__Real__Oreal,type,
    pi_o_real: set_o > ( $o > set_real ) > set_o_real ).

thf(sy_c_FuncSet_OPi_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
    pi_o_set_nat: set_o > ( $o > set_set_nat ) > set_o_set_nat ).

thf(sy_c_FuncSet_OPi_001_Eo_001tf__b,type,
    pi_o_b: set_o > ( $o > set_b ) > set_o_b ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001_Eo,type,
    pi_nat_o: set_nat > ( nat > set_o ) > set_nat_o ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Nat__Onat,type,
    pi_nat_nat: set_nat > ( nat > set_nat ) > set_nat_nat ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Real__Oreal,type,
    pi_nat_real: set_nat > ( nat > set_real ) > set_nat_real ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    pi_nat_set_nat: set_nat > ( nat > set_set_nat ) > set_nat_set_nat ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__b,type,
    pi_nat_b: set_nat > ( nat > set_b ) > set_nat_b ).

thf(sy_c_FuncSet_OPi_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    pi_real_nat_real: set_real > ( real > set_nat_real ) > set_real_nat_real ).

thf(sy_c_FuncSet_OPi_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    pi_real_real_nat: set_real > ( real > set_real_nat ) > set_real_real_nat ).

thf(sy_c_FuncSet_OPi_001t__Real__Oreal_001_062_It__Real__Oreal_Mtf__b_J,type,
    pi_real_real_b: set_real > ( real > set_real_b ) > set_real_real_b ).

thf(sy_c_FuncSet_OPi_001t__Real__Oreal_001_Eo,type,
    pi_real_o: set_real > ( real > set_o ) > set_real_o ).

thf(sy_c_FuncSet_OPi_001t__Real__Oreal_001t__Nat__Onat,type,
    pi_real_nat: set_real > ( real > set_nat ) > set_real_nat ).

thf(sy_c_FuncSet_OPi_001t__Real__Oreal_001t__Real__Oreal,type,
    pi_real_real: set_real > ( real > set_real ) > set_real_real ).

thf(sy_c_FuncSet_OPi_001t__Real__Oreal_001t__Set__Oset_It__Nat__Onat_J,type,
    pi_real_set_nat: set_real > ( real > set_set_nat ) > set_real_set_nat ).

thf(sy_c_FuncSet_OPi_001t__Real__Oreal_001tf__b,type,
    pi_real_b: set_real > ( real > set_b ) > set_real_b ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    pi_set_nat_nat: set_set_nat > ( set_nat > set_nat ) > set_set_nat_nat ).

thf(sy_c_FuncSet_OPi_001t__Set__Oset_It__Nat__Onat_J_001tf__b,type,
    pi_set_nat_b: set_set_nat > ( set_nat > set_b ) > set_set_nat_b ).

thf(sy_c_FuncSet_OPi_001tf__b_001t__Nat__Onat,type,
    pi_b_nat: set_b > ( b > set_nat ) > set_b_nat ).

thf(sy_c_FuncSet_OPi_001tf__b_001t__Set__Oset_It__Nat__Onat_J,type,
    pi_b_set_nat: set_b > ( b > set_set_nat ) > set_b_set_nat ).

thf(sy_c_FuncSet_OPi_001tf__b_001tf__b,type,
    pi_b_b: set_b > ( b > set_b ) > set_b_b ).

thf(sy_c_HOL_OUniq_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    uniq_nat_real: ( ( nat > real ) > $o ) > $o ).

thf(sy_c_HOL_OUniq_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    uniq_real_nat_real: ( ( real > nat > real ) > $o ) > $o ).

thf(sy_c_HOL_OUniq_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    uniq_real_nat: ( ( real > nat ) > $o ) > $o ).

thf(sy_c_HOL_OUniq_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    uniq_real_set_nat: ( ( real > set_nat ) > $o ) > $o ).

thf(sy_c_HOL_OUniq_001_062_It__Real__Oreal_Mtf__b_J,type,
    uniq_real_b: ( ( real > b ) > $o ) > $o ).

thf(sy_c_HOL_OUniq_001_Eo,type,
    uniq_o: ( $o > $o ) > $o ).

thf(sy_c_HOL_OUniq_001t__Nat__Onat,type,
    uniq_nat: ( nat > $o ) > $o ).

thf(sy_c_HOL_OUniq_001t__Real__Oreal,type,
    uniq_real: ( real > $o ) > $o ).

thf(sy_c_HOL_OUniq_001t__Set__Oset_It__Nat__Onat_J,type,
    uniq_set_nat: ( set_nat > $o ) > $o ).

thf(sy_c_HOL_OUniq_001tf__b,type,
    uniq_b: ( b > $o ) > $o ).

thf(sy_c_If_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    if_nat_real: $o > ( nat > real ) > ( nat > real ) > nat > real ).

thf(sy_c_If_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    if_real_nat: $o > ( real > nat ) > ( real > nat ) > real > nat ).

thf(sy_c_If_001_062_It__Real__Oreal_Mtf__b_J,type,
    if_real_b: $o > ( real > b ) > ( real > b ) > real > b ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Real__Oreal,type,
    if_real: $o > real > real > real ).

thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
    if_set_nat: $o > set_nat > set_nat > set_nat ).

thf(sy_c_If_001tf__b,type,
    if_b: $o > b > b > b ).

thf(sy_c_Measure__Space_Oincreasing_001t__Nat__Onat_001t__Nat__Onat,type,
    measur1302623347068717141at_nat: set_set_nat > ( set_nat > nat ) > $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 ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_M_Eo_J,type,
    bot_bo839430502184047061real_o: ( real > nat > real ) > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Real__Oreal_Mt__Nat__Onat_J_M_Eo_J,type,
    bot_bot_real_nat_o: ( real > nat ) > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    bot_bo2654456731858362666_nat_o: ( real > set_nat ) > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Real__Oreal_Mtf__b_J_M_Eo_J,type,
    bot_bot_real_b_o: ( real > b ) > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
    bot_bot_o_o: $o > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
    bot_bot_nat_o: nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
    bot_bot_real_o: real > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    bot_bot_set_nat_o: set_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
    bot_bot_b_o: b > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J,type,
    bot_bo5661300308423610259_set_o: produc7369051934464679207_set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bo5064604495688752833et_nat: produc133642713317727789et_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
    bot_bo6057198287672936477t_real: produc808471633252061577t_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_I_Eo_J_J,type,
    bot_bo6859089648663734907_set_o: produc1928127866292709863_set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bo3047382831089536473et_nat: produc7819656566062154093et_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
    bot_bo4718971705043381045t_real: produc4765104147712873673t_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_I_Eo_J_J,type,
    bot_bo3099358262589906591_set_o: produc7074003645023807499_set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bo4343341718608865973et_nat: produc4389474161278358601et_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
    bot_bo987450378295171601t_real: produc2118043781569182117t_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_Itf__b_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    bot_bo932696493702709598real_b: produc4373655991844504306real_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
    bot_bot_set_o_o: set_o_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
    bot_bot_set_o_nat: set_o_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
    bot_bot_set_o_real: set_o_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    bot_bot_set_nat_o2: set_nat_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bot_set_nat_nat: set_nat_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    bot_bot_set_nat_real: set_nat_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    bot_bo6533810469807102640t_real: set_real_nat_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_M_Eo_J_J,type,
    bot_bot_set_real_o: set_real_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    bot_bot_set_real_nat: set_real_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    bot_bo6767488733719836353l_real: set_real_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    bot_bo6814059168456595739et_nat: set_real_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    bot_bot_set_real_b: set_real_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    bot_bo5566352981089410870real_b: set_set_real_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    bot_bot_set_set_o: set_set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
    bot_bot_set_set_real: set_set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
    bot_bot_set_b: set_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_I_Eo_J,type,
    bot_bo5758314138661044393sure_o: sigma_measure_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
    bot_bo6718502177978453909re_nat: sigma_measure_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    bot_bo5982154664989874033e_real: sigma_measure_real ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
    ord_le7676461544873280788real_o: ( ( nat > real ) > $o ) > ( ( nat > real ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_M_Eo_J,type,
    ord_le2056620204158255881real_o: ( ( real > nat > real ) > $o ) > ( ( real > nat > real ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J_M_Eo_J,type,
    ord_le5498898627171184265_nat_o: ( ( real > real > nat ) > $o ) > ( ( real > real > nat ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J_M_Eo_J,type,
    ord_le7427477666976093192al_b_o: ( ( real > real > b ) > $o ) > ( ( real > real > b ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Real__Oreal_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le5694569607330361492_nat_o: ( ( real > nat ) > $o ) > ( ( real > nat ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    ord_le5255281653000370782_nat_o: ( ( real > set_nat ) > $o ) > ( ( real > set_nat ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Real__Oreal_Mtf__b_J_M_Eo_J,type,
    ord_less_eq_real_b_o: ( ( real > b ) > $o ) > ( ( real > b ) > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__b_M_Eo_J,type,
    ord_less_eq_b_o: ( b > $o ) > ( b > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    ord_le8460144461188290721at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_Itf__b_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    ord_le209527602574031506real_b: produc4373655991844504306real_b > produc4373655991844504306real_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: real > real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    ord_le2908806416726583473t_real: set_nat_real > set_nat_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    ord_le6912594875210535036t_real: set_real_nat_real > set_real_nat_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    ord_le491656556176229628al_nat: set_real_real_nat > set_real_real_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    ord_le227027178093677845real_b: set_real_real_b > set_real_real_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    ord_le6098800555920186673al_nat: set_real_nat > set_real_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    ord_le7035988643939837671et_nat: set_real_set_nat > set_real_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    ord_le5814440863667440394real_b: set_real_b > set_real_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
    ord_less_eq_set_o: set_o > set_o > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    ord_le4374716579403074808_set_o: set_set_o > set_set_o > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
    ord_le3558479182127378552t_real: set_set_real > set_set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__b_J,type,
    ord_less_eq_set_b: set_b > set_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_I_Eo_J,type,
    ord_le478349814012620405sure_o: sigma_measure_o > sigma_measure_o > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
    ord_le2862109966718184649re_nat: sigma_measure_nat > sigma_measure_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    ord_le487379304121309861e_real: sigma_measure_real > sigma_measure_real > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_M_Eo_J,type,
    top_top_nat_real_o: ( nat > real ) > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_M_Eo_J,type,
    top_to1109506300641978553real_o: ( real > nat > real ) > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Real__Oreal_Mt__Nat__Onat_J_M_Eo_J,type,
    top_top_real_nat_o: ( real > nat ) > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    top_to150443550350409742_nat_o: ( real > set_nat ) > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Real__Oreal_Mtf__b_J_M_Eo_J,type,
    top_top_real_b_o: ( real > b ) > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
    top_top_o_o: $o > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
    top_top_nat_o: nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
    top_top_real_o: real > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    top_top_set_nat_o: set_nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__b_M_Eo_J,type,
    top_top_b_o: b > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_I_Eo_J_J,type,
    top_to8356300885640301175_set_o: produc7369051934464679207_set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    top_to2343176657905263837et_nat: produc133642713317727789et_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_I_Eo_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
    top_to8585549949225041977t_real: produc808471633252061577t_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_I_Eo_J_J,type,
    top_to4137661810880245911_set_o: produc1928127866292709863_set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    top_to7593079806418470589et_nat: produc7819656566062154093et_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
    top_to8609433042786791961t_real: produc4765104147712873673t_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_I_Eo_J_J,type,
    top_to5627709924142012091_set_o: produc7074003645023807499_set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    top_to8233803056352276889et_nat: produc4389474161278358601et_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_It__Real__Oreal_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
    top_to1822899090350206709t_real: produc2118043781569182117t_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Product____Type__Oprod_It__Set__Oset_Itf__b_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    top_to1114480880377561154real_b: produc4373655991844504306real_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
    top_top_set_o_o: set_o_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
    top_top_set_o_nat: set_o_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
    top_top_set_o_real: set_o_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
    top_top_set_nat_o2: set_nat_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_top_set_nat_nat: set_nat_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    top_top_set_nat_real: set_nat_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    top_to6902130281023745740t_real: set_real_nat_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_M_Eo_J_J,type,
    top_top_set_real_o: set_real_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    top_top_set_real_nat: set_real_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    top_to2071711978144146653l_real: set_real_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    top_to245300144855000375et_nat: set_real_set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    top_top_set_real_b: set_real_b ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

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_Itf__b_J,type,
    top_top_set_b: set_b ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Real__Oreal_J_001t__Nat__Onat,type,
    produc157057340832456324al_nat: ( nat > real ) > nat > produc6623316681580375884al_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Real__Oreal_J_001t__Real__Oreal,type,
    produc7188456909510567776l_real: ( nat > real ) > real > produc7980284238819019688l_real ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    produc2678254361277162628t_real: nat > ( nat > real ) > produc9032034819326515148t_real ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Real__Oreal,type,
    produc7837566107596912789t_real: nat > real > produc7716430852924023517t_real ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    produc8719833358983803744t_real: real > ( nat > real ) > produc856825540987909672t_real ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
    produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    produc9006861556841674502t_real: set_nat_real > set_real_nat_real > produc5618539574968880590t_real ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    produc4583020555951415302al_nat: set_real_nat > set_real_real_nat > produc400598280281323214al_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    produc3140906811317515298real_b: set_real_b > set_real_real_b > produc5156796054771612850real_b ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    produc5838405689764958487_set_o: set_o > set_o > produc7369051934464679207_set_o ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc2212626075605038759et_nat: set_o > set_nat > produc133642713317727789et_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Real__Oreal_J,type,
    produc2850126452351882627t_real: set_o > set_real > produc808471633252061577t_real ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    produc3843710909396041968al_nat: set_nat > set_real_nat > produc3490415624965685560al_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
    produc2750217728217304729_set_o: set_nat > set_o > produc1928127866292709863_set_o ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Real__Oreal_J,type,
    produc3494877982625406337t_real: set_nat > set_real > produc4765104147712873673t_real ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_I_Eo_J,type,
    produc9171573333813886013_set_o: set_real > set_o > produc7074003645023807499_set_o ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4432056366243864833et_nat: set_real > set_nat > produc4389474161278358601et_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
    produc6377492392003646173t_real: set_real > set_real > produc2118043781569182117t_real ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc3412633259576239964et_nat: set_set_nat > set_real_set_nat > produc5636995695864575140et_nat ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_Itf__b_J_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    produc5260422477894090978real_b: set_b > set_real_b > produc4373655991844504306real_b ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J_001_Eo,type,
    produc6450457281045158333real_o: ( set_nat_real > set_real_nat_real > $o ) > produc5618539574968880590t_real > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J_J_001_Eo,type,
    produc5549966297984299197_nat_o: ( set_real_nat > set_real_real_nat > $o ) > produc400598280281323214al_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J_J_001_Eo,type,
    produc7687788952222807969al_b_o: ( set_real_b > set_real_real_b > $o ) > produc5156796054771612850real_b > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J_001_Eo,type,
    produc5377950512376928595_nat_o: ( set_nat > set_real_nat > $o ) > produc3490415624965685560al_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J_001_Eo,type,
    produc1937539024057733095_nat_o: ( set_set_nat > set_real_set_nat > $o ) > produc5636995695864575140et_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_Itf__b_J_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_001_Eo,type,
    produc9150283905253018337al_b_o: ( set_b > set_real_b > $o ) > produc4373655991844504306real_b > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_Itf__b_J_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    produc8482724652568146995real_b: ( set_b > set_real_b > set_real_b ) > produc4373655991844504306real_b > set_real_b ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    produc7833056394006324t_real: produc5618539574968880590t_real > set_real_nat_real ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    produc4807364092358522932al_nat: produc400598280281323214al_nat > set_real_real_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    produc5800820901218438480real_b: produc5156796054771612850real_b > set_real_real_b ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    produc8862089267992772894al_nat: produc3490415624965685560al_nat > set_real_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    produc1329678470257883018et_nat: produc5636995695864575140et_nat > set_real_set_nat ).

thf(sy_c_Product__Type_Oprod_Osnd_001t__Set__Oset_Itf__b_J_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    produc1187263614985732624real_b: produc4373655991844504306real_b > set_real_b ).

thf(sy_c_QuasiBorel_Ois__quasi__borel_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    is_qua7292199676159343652t_real: set_nat_real > set_real_nat_real > $o ).

thf(sy_c_QuasiBorel_Ois__quasi__borel_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    is_qua3317151984682463396al_nat: set_real_nat > set_real_real_nat > $o ).

thf(sy_c_QuasiBorel_Ois__quasi__borel_001_062_It__Real__Oreal_Mtf__b_J,type,
    is_qua1212226743203608129real_b: set_real_b > set_real_real_b > $o ).

thf(sy_c_QuasiBorel_Ois__quasi__borel_001t__Nat__Onat,type,
    is_quasi_borel_nat: set_nat > set_real_nat > $o ).

thf(sy_c_QuasiBorel_Ois__quasi__borel_001t__Set__Oset_It__Nat__Onat_J,type,
    is_qua1626198732325083535et_nat: set_set_nat > set_real_set_nat > $o ).

thf(sy_c_QuasiBorel_Ois__quasi__borel_001tf__b,type,
    is_quasi_borel_b: set_b > set_real_b > $o ).

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_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    qbs_Mx_real_nat_real: quasi_4498916059141695318t_real > set_re1338850055584481895t_real ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    qbs_Mx_real_real_nat: quasi_7301349776962165718al_nat > set_re3347183479607654119al_nat ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J,type,
    qbs_Mx_real_real_b: quasi_4416045583689693371real_b > set_real_real_real_b ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    qbs_Mx_real_nat: quasi_borel_real_nat > set_real_real_nat ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    qbs_Mx_real_set_nat: quasi_1649577361549849025et_nat > set_re5613859201272075090et_nat ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001_062_It__Real__Oreal_Mtf__b_J,type,
    qbs_Mx_real_b: quasi_borel_real_b > set_real_real_b ).

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__Set__Oset_It__Nat__Onat_J,type,
    qbs_Mx_set_nat: quasi_borel_set_nat > set_real_set_nat ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001tf__b,type,
    qbs_Mx_b: quasi_borel_b > set_real_b ).

thf(sy_c_QuasiBorel_Oqbs__closed1_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_closed1_nat_real: set_real_nat_real > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed1_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    qbs_closed1_real_nat: set_real_real_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed1_001_062_It__Real__Oreal_Mtf__b_J,type,
    qbs_closed1_real_b: set_real_real_b > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed1_001t__Nat__Onat,type,
    qbs_closed1_nat: set_real_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed1_001tf__b,type,
    qbs_closed1_b: set_real_b > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_closed2_nat_real: set_nat_real > set_real_nat_real > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    qbs_cl6586241553476272112t_real: set_real_nat_real > set_re1338850055584481895t_real > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    qbs_cl552863655815099504al_nat: set_real_real_nat > set_re3347183479607654119al_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J,type,
    qbs_cl646577709987169675real_b: set_real_real_b > set_real_real_real_b > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    qbs_closed2_real_nat: set_real_nat > set_real_real_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    qbs_cl1079945222394894811et_nat: set_real_set_nat > set_re5613859201272075090et_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001_062_It__Real__Oreal_Mtf__b_J,type,
    qbs_closed2_real_b: set_real_b > set_real_real_b > $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__Set__Oset_It__Nat__Onat_J,type,
    qbs_closed2_set_nat: set_set_nat > set_real_set_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed2_001tf__b,type,
    qbs_closed2_b: set_b > set_real_b > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed3_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    qbs_closed3_nat_real: set_real_nat_real > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed3_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    qbs_closed3_real_nat: set_real_real_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed3_001_062_It__Real__Oreal_Mtf__b_J,type,
    qbs_closed3_real_b: set_real_real_b > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed3_001t__Nat__Onat,type,
    qbs_closed3_nat: set_real_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed3_001t__Set__Oset_It__Nat__Onat_J,type,
    qbs_closed3_set_nat: set_real_set_nat > $o ).

thf(sy_c_QuasiBorel_Oqbs__closed3_001tf__b,type,
    qbs_closed3_b: set_real_b > $o ).

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_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    qbs_sp1807256676205042352t_real: quasi_4498916059141695318t_real > set_real_nat_real ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    qbs_sp4997250815398645552al_nat: quasi_7301349776962165718al_nat > set_real_real_nat ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J,type,
    qbs_sp4769540098391857867real_b: quasi_4416045583689693371real_b > set_real_real_b ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    qbs_space_real_nat: quasi_borel_real_nat > set_real_nat ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    qbs_sp2552676844468483227et_nat: quasi_1649577361549849025et_nat > set_real_set_nat ).

thf(sy_c_QuasiBorel_Oqbs__space_001_062_It__Real__Oreal_Mtf__b_J,type,
    qbs_space_real_b: quasi_borel_real_b > set_real_b ).

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__Set__Oset_It__Nat__Onat_J,type,
    qbs_space_set_nat: quasi_borel_set_nat > set_set_nat ).

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

thf(sy_c_QuasiBorel_Oquasi__borel_OAbs__quasi__borel_001tf__b,type,
    quasi_2002468295286565185orel_b: produc4373655991844504306real_b > quasi_borel_b ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    quasi_8146262523719627044t_real: quasi_borel_nat_real > produc5618539574968880590t_real ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    quasi_4171214832242746788al_nat: quasi_borel_real_nat > produc400598280281323214al_nat ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001_062_It__Real__Oreal_Mtf__b_J,type,
    quasi_347859643032404289real_b: quasi_borel_real_b > produc5156796054771612850real_b ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Nat__Onat,type,
    quasi_2638801612224710361el_nat: quasi_borel_nat > produc3490415624965685560al_nat ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001t__Set__Oset_It__Nat__Onat_J,type,
    quasi_853997576918650511et_nat: quasi_borel_set_nat > produc5636995695864575140et_nat ).

thf(sy_c_QuasiBorel_Oquasi__borel_ORep__quasi__borel_001tf__b,type,
    quasi_4958298574314430518orel_b: quasi_borel_b > produc4373655991844504306real_b ).

thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    collect_nat_real: ( ( nat > real ) > $o ) > set_nat_real ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    collec336724336074912571t_real: ( ( real > nat > real ) > $o ) > set_real_nat_real ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    collec3526718475268515771al_nat: ( ( real > real > nat ) > $o ) > set_real_real_nat ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J,type,
    collect_real_real_b: ( ( real > real > b ) > $o ) > set_real_real_b ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    collect_real_nat: ( ( real > nat ) > $o ) > set_real_nat ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    collect_real_set_nat: ( ( real > set_nat ) > $o ) > set_real_set_nat ).

thf(sy_c_Set_OCollect_001_062_It__Real__Oreal_Mtf__b_J,type,
    collect_real_b: ( ( real > b ) > $o ) > set_real_b ).

thf(sy_c_Set_OCollect_001_Eo,type,
    collect_o: ( $o > $o ) > set_o ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J_Mt__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J_J,type,
    collec3396274693757851065t_real: ( produc5618539574968880590t_real > $o ) > set_Pr1996641493173216942t_real ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J_Mt__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J_J_J,type,
    collec7401705435925069497al_nat: ( produc400598280281323214al_nat > $o ) > set_Pr8267393451869278126al_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_Mt__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J_J_J,type,
    collec1032871213651684125real_b: ( produc5156796054771612850real_b > $o ) > set_Pr4282867346099416338real_b ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    collec8700111531230907683al_nat: ( produc3490415624965685560al_nat > $o ) > set_Pr445334604831195416al_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    collec6231571016085039247et_nat: ( produc5636995695864575140et_nat > $o ) > set_Pr4307940823654323844et_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Set__Oset_Itf__b_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    collec5860786303674561885real_b: ( produc4373655991844504306real_b > $o ) > set_Pr100679515247169362real_b ).

thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
    collect_real: ( real > $o ) > set_real ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(sy_c_Set_OCollect_001tf__b,type,
    collect_b: ( b > $o ) > set_b ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
    image_nat_real: ( nat > real ) > set_nat > set_real ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Nat__Onat,type,
    image_real_nat: ( real > nat ) > set_real > set_nat ).

thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    insert_nat_real: ( nat > real ) > set_nat_real > set_nat_real ).

thf(sy_c_Set_Oinsert_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    insert_real_nat_real: ( real > nat > real ) > set_real_nat_real > set_real_nat_real ).

thf(sy_c_Set_Oinsert_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    insert_real_nat: ( real > nat ) > set_real_nat > set_real_nat ).

thf(sy_c_Set_Oinsert_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    insert_real_set_nat: ( real > set_nat ) > set_real_set_nat > set_real_set_nat ).

thf(sy_c_Set_Oinsert_001_062_It__Real__Oreal_Mtf__b_J,type,
    insert_real_b: ( real > b ) > set_real_b > set_real_b ).

thf(sy_c_Set_Oinsert_001_Eo,type,
    insert_o: $o > set_o > set_o ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: real > set_real > set_real ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    insert_set_real_b: set_real_b > set_set_real_b > set_set_real_b ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
    insert_set_o: set_o > set_set_o > set_set_o ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Real__Oreal_J,type,
    insert_set_real: set_real > set_set_real > set_set_real ).

thf(sy_c_Set_Oinsert_001tf__b,type,
    insert_b: b > set_b > set_b ).

thf(sy_c_Set_Ois__empty_001_Eo,type,
    is_empty_o: set_o > $o ).

thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
    is_empty_nat: set_nat > $o ).

thf(sy_c_Set_Ois__empty_001t__Real__Oreal,type,
    is_empty_real: set_real > $o ).

thf(sy_c_Set_Ois__singleton_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    is_sin3816863272566379559t_real: set_nat_real > $o ).

thf(sy_c_Set_Ois__singleton_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    is_sin4536856481422437234t_real: set_real_nat_real > $o ).

thf(sy_c_Set_Ois__singleton_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    is_sin9065187617944275111al_nat: set_real_nat > $o ).

thf(sy_c_Set_Ois__singleton_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    is_sin5719733656810162525et_nat: set_real_set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001_062_It__Real__Oreal_Mtf__b_J,type,
    is_singleton_real_b: set_real_b > $o ).

thf(sy_c_Set_Ois__singleton_001_Eo,type,
    is_singleton_o: set_o > $o ).

thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
    is_singleton_nat: set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
    is_singleton_real: set_real > $o ).

thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
    is_singleton_set_nat: set_set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001tf__b,type,
    is_singleton_b: set_b > $o ).

thf(sy_c_Set_Othe__elem_001_Eo,type,
    the_elem_o: set_o > $o ).

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

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

thf(sy_c_Set_Ovimage_001_Eo_001_Eo,type,
    vimage_o_o: ( $o > $o ) > set_o > set_o ).

thf(sy_c_Set_Ovimage_001_Eo_001t__Nat__Onat,type,
    vimage_o_nat: ( $o > nat ) > set_nat > set_o ).

thf(sy_c_Set_Ovimage_001_Eo_001t__Real__Oreal,type,
    vimage_o_real: ( $o > real ) > set_real > set_o ).

thf(sy_c_Set_Ovimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
    vimage_o_set_nat: ( $o > set_nat ) > set_set_nat > set_o ).

thf(sy_c_Set_Ovimage_001_Eo_001tf__b,type,
    vimage_o_b: ( $o > b ) > set_b > set_o ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    vimage_nat_nat_real: ( nat > nat > real ) > set_nat_real > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    vimage264186217122334195t_real: ( nat > real > nat > real ) > set_real_nat_real > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J,type,
    vimage3471692816685529672real_b: ( nat > real > real > b ) > set_real_real_b > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    vimage_nat_real_nat: ( nat > real > nat ) > set_real_nat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    vimage965819199455225310et_nat: ( nat > real > set_nat ) > set_real_set_nat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_062_It__Real__Oreal_Mtf__b_J,type,
    vimage_nat_real_b: ( nat > real > b ) > set_real_b > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_Eo,type,
    vimage_nat_o: ( nat > $o ) > set_o > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Real__Oreal,type,
    vimage_nat_real: ( nat > real ) > set_real > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    vimage_nat_set_nat: ( nat > set_nat ) > set_set_nat > set_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001tf__b,type,
    vimage_nat_b: ( nat > b ) > set_b > set_nat ).

thf(sy_c_Set_Ovimage_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    vimage_real_real_nat: ( real > real > nat ) > set_real_nat > set_real ).

thf(sy_c_Set_Ovimage_001t__Real__Oreal_001_Eo,type,
    vimage_real_o: ( real > $o ) > set_o > set_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_Ovimage_001t__Real__Oreal_001t__Set__Oset_It__Nat__Onat_J,type,
    vimage_real_set_nat: ( real > set_nat ) > set_set_nat > set_real ).

thf(sy_c_Set_Ovimage_001t__Real__Oreal_001tf__b,type,
    vimage_real_b: ( real > b ) > set_b > set_real ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Nat__Onat_J_001_Eo,type,
    vimage_set_nat_o: ( set_nat > $o ) > set_o > set_set_nat ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    vimage_set_nat_nat: ( set_nat > nat ) > set_nat > set_set_nat ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
    vimage_set_nat_real: ( set_nat > real ) > set_real > set_set_nat ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    vimage4765135879290611145et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).

thf(sy_c_Set_Ovimage_001t__Set__Oset_It__Nat__Onat_J_001tf__b,type,
    vimage_set_nat_b: ( set_nat > b ) > set_b > set_set_nat ).

thf(sy_c_Set_Ovimage_001tf__b_001_Eo,type,
    vimage_b_o: ( b > $o ) > set_o > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001t__Nat__Onat,type,
    vimage_b_nat: ( b > nat ) > set_nat > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001t__Real__Oreal,type,
    vimage_b_real: ( b > real ) > set_real > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001t__Set__Oset_It__Nat__Onat_J,type,
    vimage_b_set_nat: ( b > set_nat ) > set_set_nat > set_b ).

thf(sy_c_Set_Ovimage_001tf__b_001tf__b,type,
    vimage_b_b: ( b > b ) > set_b > set_b ).

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__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_Omeasurable_001_062_It__Nat__Onat_Mt__Real__Oreal_J_001t__Real__Oreal,type,
    sigma_8475865534613037593l_real: sigma_3396294578489551860t_real > sigma_measure_real > set_nat_real_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_001t__Real__Oreal,type,
    sigma_7540513786767735268l_real: sigma_4069289485848494527t_real > sigma_measure_real > set_re7175981164672885223l_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Real__Oreal_Mt__Nat__Onat_J_001t__Real__Oreal,type,
    sigma_2883472102289741465t_real: sigma_6586288717683155060al_nat > sigma_measure_real > set_real_nat_real2 ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_001t__Real__Oreal,type,
    sigma_2288790926702686287t_real: sigma_7120471488171859498et_nat > sigma_measure_real > set_re2661618181930661074t_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001_062_It__Real__Oreal_Mtf__b_J_001t__Real__Oreal,type,
    sigma_4595654474610825574b_real: sigma_measure_real_b > sigma_measure_real > set_real_b_real ).

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__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_001_Eo_001tf__b,type,
    sigma_measurable_o_b: sigma_measure_o > sigma_measure_b > set_o_b ).

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__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__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    sigma_8750250338757262779t_real: sigma_measure_nat > sigma_5310753476256395226t_real > set_na4391433010951109924t_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    sigma_4775202647280382523al_nat: sigma_measure_nat > sigma_8500747615449998426al_nat > set_na7193866728771580324al_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Nat__Onat_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    sigma_5893925349391517463l_real: sigma_measure_nat > sigma_2308072346491277622l_real > set_na5185791421324906752l_real ).

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__Nat__Onat_001tf__b,type,
    sigma_4105081583803843549_nat_b: sigma_measure_nat > sigma_measure_b > set_nat_b ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    sigma_783869947231497753t_real: sigma_measure_real > sigma_3396294578489551860t_real > set_real_nat_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    sigma_6032194292609393305al_nat: sigma_measure_real > sigma_6586288717683155060al_nat > set_real_real_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001_062_It__Real__Oreal_Mtf__b_J,type,
    sigma_5735160446100821900real_b: sigma_measure_real > sigma_measure_real_b > set_real_real_b ).

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__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__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Real__Oreal_J_Mt__Nat__Onat_J,type,
    sigma_1311731663836594126al_nat: sigma_measure_real > sigma_5506949827314733513al_nat > set_re4551664761370959415al_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
    sigma_8384661788165733162l_real: sigma_measure_real > sigma_1811887234549217061l_real > set_re3333496664559395859l_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    sigma_3720449801582733390t_real: sigma_measure_real > sigma_3149601563892791881t_real > set_re2581653195610359479t_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    sigma_707772041119837571at_nat: sigma_measure_real > sigma_5515648953823433982at_nat > set_re3218616107942652140at_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J,type,
    sigma_2975298441655967327t_real: sigma_measure_real > sigma_5310753476256395226t_real > set_re7361189984880485448t_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    sigma_1261203090334623146t_real: sigma_measure_real > sigma_73088456041816485t_real > set_re859989298913791635t_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    sigma_8223622787033862879al_nat: sigma_measure_real > sigma_8500747615449998426al_nat > set_re940251665846180040al_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_001t__Set__Oset_It__Nat__Onat_J,type,
    sigma_5099707562218272132et_nat: sigma_measure_real > sigma_3334325623652945375et_nat > set_real_set_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Real__Oreal_001tf__b,type,
    sigma_523072396149930113real_b: sigma_measure_real > sigma_measure_b > set_real_b ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
    sigma_6407790436254459688at_nat: sigma_3334325623652945375et_nat > sigma_measure_nat > set_set_nat_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Set__Oset_It__Nat__Onat_J_001t__Real__Oreal,type,
    sigma_7357981997995286020t_real: sigma_3334325623652945375et_nat > sigma_measure_real > set_set_nat_real2 ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001_Eo,type,
    sigma_measurable_b_o: sigma_measure_b > sigma_measure_o > set_b_o ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001t__Nat__Onat,type,
    sigma_1308594411581951615_b_nat: sigma_measure_b > sigma_measure_nat > set_b_nat ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001t__Real__Oreal,type,
    sigma_5489272243865111003b_real: sigma_measure_b > sigma_measure_real > set_b_real ).

thf(sy_c_Sigma__Algebra_Omeasurable_001tf__b_001tf__b,type,
    sigma_measurable_b_b: sigma_measure_b > sigma_measure_b > set_b_b ).

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_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    sigma_787286210617551431t_real: sigma_4069289485848494527t_real > set_se8960922493471073746t_real ).

thf(sy_c_Sigma__Algebra_Osets_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    sigma_sets_real_nat: sigma_6586288717683155060al_nat > set_set_real_nat ).

thf(sy_c_Sigma__Algebra_Osets_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    sigma_1129187412287173170et_nat: sigma_7120471488171859498et_nat > set_set_real_set_nat ).

thf(sy_c_Sigma__Algebra_Osets_001_062_It__Real__Oreal_Mtf__b_J,type,
    sigma_sets_real_b: sigma_measure_real_b > set_set_real_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__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_001t__Set__Oset_It__Nat__Onat_J,type,
    sigma_sets_set_nat: sigma_3334325623652945375et_nat > set_set_set_nat ).

thf(sy_c_Sigma__Algebra_Osets_001tf__b,type,
    sigma_sets_b: sigma_measure_b > set_set_b ).

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_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    sigma_7241629334780537822t_real: sigma_4069289485848494527t_real > set_real_nat_real ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    sigma_space_real_nat: sigma_6586288717683155060al_nat > set_real_nat ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    sigma_8533228468570120649et_nat: sigma_7120471488171859498et_nat > set_real_set_nat ).

thf(sy_c_Sigma__Algebra_Ospace_001_062_It__Real__Oreal_Mtf__b_J,type,
    sigma_space_real_b: sigma_measure_real_b > set_real_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__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_001t__Set__Oset_It__Nat__Onat_J,type,
    sigma_space_set_nat: sigma_3334325623652945375et_nat > set_set_nat ).

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

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_Og_001t__Nat__Onat,type,
    standard_g_nat: sigma_measure_nat > real > nat ).

thf(sy_c_Topology__Euclidean__Space_Oeuclidean__space__class_Oeucl__less_001t__Real__Oreal,type,
    topolo2105956845596822908s_real: real > real > $o ).

thf(sy_c_Typedef_Otype__definition_001t__QuasiBorel__Oquasi____borel_Itf__b_J_001t__Product____Type__Oprod_It__Set__Oset_Itf__b_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    type_d4563186451463392057real_b: ( quasi_borel_b > produc4373655991844504306real_b ) > ( produc4373655991844504306real_b > quasi_borel_b ) > set_Pr100679515247169362real_b > $o ).

thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__Real__Oreal_J_Mt__Real__Oreal_J,type,
    member_nat_real_real: ( ( nat > real ) > real ) > set_nat_real_real > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_Mt__Real__Oreal_J,type,
    member6568573216281280328l_real: ( ( real > nat > real ) > real ) > set_re7175981164672885223l_real > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mt__Nat__Onat_J_Mt__Real__Oreal_J,type,
    member_real_nat_real: ( ( real > nat ) > real ) > set_real_nat_real2 > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_Mt__Real__Oreal_J,type,
    member5086141250106945459t_real: ( ( real > set_nat ) > real ) > set_re2661618181930661074t_real > $o ).

thf(sy_c_member_001_062_I_062_It__Real__Oreal_Mtf__b_J_Mt__Real__Oreal_J,type,
    member_real_b_real: ( ( real > b ) > real ) > set_real_b_real > $o ).

thf(sy_c_member_001_062_I_Eo_M_Eo_J,type,
    member_o_o: ( $o > $o ) > set_o_o > $o ).

thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
    member_o_nat: ( $o > nat ) > set_o_nat > $o ).

thf(sy_c_member_001_062_I_Eo_Mt__Real__Oreal_J,type,
    member_o_real: ( $o > real ) > set_o_real > $o ).

thf(sy_c_member_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member_o_set_nat: ( $o > set_nat ) > set_o_set_nat > $o ).

thf(sy_c_member_001_062_I_Eo_Mtf__b_J,type,
    member_o_b: ( $o > b ) > set_o_b > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
    member_nat_o: ( nat > $o ) > set_nat_o > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    member4237364305948650477t_real: ( nat > produc7716430852924023517t_real ) > set_na4391433010951109924t_real > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    member7427358445142253677al_nat: ( nat > produc3741383161447143261al_nat ) > set_na7193866728771580324al_nat > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    member4944736856719700937l_real: ( nat > produc2422161461964618553l_real ) > set_na5185791421324906752l_real > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
    member_nat_real: ( nat > real ) > set_nat_real > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member_nat_set_nat: ( nat > set_nat ) > set_nat_set_nat > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mtf__b_J,type,
    member_nat_b: ( nat > b ) > set_nat_b > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    member_real_nat_real2: ( real > nat > real ) > set_real_nat_real > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    member3012124575732785096t_real: ( real > real > nat > real ) > set_re1338850055584481895t_real > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    member5814558293553255496al_nat: ( real > real > real > nat ) > set_re3347183479607654119al_nat > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    member6395380647429634249real_b: ( real > real > real > b ) > set_real_real_real_b > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    member_real_real_nat: ( real > real > nat ) > set_real_real_nat > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    member6099108769244630579et_nat: ( real > real > set_nat ) > set_re5613859201272075090et_nat > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J,type,
    member_real_real_b: ( real > real > b ) > set_real_real_b > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_M_Eo_J,type,
    member_real_o: ( real > $o ) > set_real_o > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    member_real_nat: ( real > nat ) > set_real_nat > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Real__Oreal_J_Mt__Nat__Onat_J_J,type,
    member1204174549408558720al_nat: ( real > produc6623316681580375884al_nat ) > set_re4551664761370959415al_nat > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Real__Oreal_J_Mt__Real__Oreal_J_J,type,
    member4771552270248568924l_real: ( real > produc7980284238819019688l_real ) > set_re3333496664559395859l_real > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    member8070198322841392896t_real: ( real > produc9032034819326515148t_real ) > set_re2581653195610359479t_real > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member278708040391457717at_nat: ( real > product_prod_nat_nat ) > set_re3218616107942652140at_nat > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    member969486235890772753t_real: ( real > produc7716430852924023517t_real ) > set_re7361189984880485448t_real > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Product____Type__Oprod_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    member3032753491741168348t_real: ( real > produc856825540987909672t_real ) > set_re859989298913791635t_real > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    member4159480375084375953al_nat: ( real > produc3741383161447143261al_nat ) > set_re940251665846180040al_nat > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member_real_real: ( real > real ) > set_real_real > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member_real_set_nat: ( real > set_nat ) > set_real_set_nat > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mtf__b_J,type,
    member_real_b: ( real > b ) > set_real_b > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member_set_nat_nat: ( set_nat > nat ) > set_set_nat_nat > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Real__Oreal_J,type,
    member_set_nat_real: ( set_nat > real ) > set_set_nat_real2 > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mtf__b_J,type,
    member_set_nat_b: ( set_nat > b ) > set_set_nat_b > $o ).

thf(sy_c_member_001_062_Itf__b_M_Eo_J,type,
    member_b_o: ( b > $o ) > set_b_o > $o ).

thf(sy_c_member_001_062_Itf__b_Mt__Nat__Onat_J,type,
    member_b_nat: ( b > nat ) > set_b_nat > $o ).

thf(sy_c_member_001_062_Itf__b_Mt__Real__Oreal_J,type,
    member_b_real: ( b > real ) > set_b_real > $o ).

thf(sy_c_member_001_062_Itf__b_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member_b_set_nat: ( b > set_nat ) > set_b_set_nat > $o ).

thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
    member_b_b: ( b > b ) > set_b_b > $o ).

thf(sy_c_member_001_Eo,type,
    member_o: $o > set_o > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J_Mt__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J_J,type,
    member948614669296314999t_real: produc5618539574968880590t_real > set_Pr1996641493173216942t_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J_Mt__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mt__Nat__Onat_J_J_J_J,type,
    member4954045411463533431al_nat: produc400598280281323214al_nat > set_Pr8267393451869278126al_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_Mt__Set__Oset_I_062_It__Real__Oreal_M_062_It__Real__Oreal_Mtf__b_J_J_J_J,type,
    member2077407680349518043real_b: produc5156796054771612850real_b > set_Pr4282867346099416338real_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J_J,type,
    member4868967264474237409al_nat: produc3490415624965685560al_nat > set_Pr445334604831195416al_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J_J,type,
    member7853754362859631949et_nat: produc5636995695864575140et_nat > set_Pr4307940823654323844et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_Itf__b_J_Mt__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J_J,type,
    member527121533207341851real_b: produc4373655991844504306real_b > set_Pr100679515247169362real_b > $o ).

thf(sy_c_member_001t__Real__Oreal,type,
    member_real: real > set_real > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
    member_set_nat_real2: set_nat_real > set_set_nat_real > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_M_062_It__Nat__Onat_Mt__Real__Oreal_J_J_J,type,
    member890523794726346931t_real: set_real_nat_real > set_se8960922493471073746t_real > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
    member_set_real_nat: set_real_nat > set_set_real_nat > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    member8337256853318788382et_nat: set_real_set_nat > set_set_real_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__b_J_J,type,
    member_set_real_b: set_real_b > set_set_real_b > $o ).

thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
    member_set_o: set_o > set_set_o > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Real__Oreal_J,type,
    member_set_real: set_real > set_set_real > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    member_set_set_nat: set_set_nat > set_set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
    member_set_b: set_b > set_set_b > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_v_Fi,type,
    fi: nat > real > b ).

thf(sy_v_P,type,
    p: real > nat ).

thf(sy_v_X,type,
    x: quasi_borel_b ).

% Relevant facts (1264)
thf(fact_0_assms_I2_J,axiom,
    ! [I: nat] : ( member_real_b @ ( fi @ I ) @ ( qbs_Mx_b @ x ) ) ).

% assms(2)
thf(fact_1_qbs__closed2__dest,axiom,
    ! [X: real > nat > real,X2: quasi_4498916059141695318t_real] :
      ( ( member_real_nat_real2 @ X @ ( qbs_sp1807256676205042352t_real @ X2 ) )
     => ( member3012124575732785096t_real
        @ ^ [R: real] : X
        @ ( qbs_Mx_real_nat_real @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_2_qbs__closed2__dest,axiom,
    ! [X: real > real > b,X2: quasi_4416045583689693371real_b] :
      ( ( member_real_real_b @ X @ ( qbs_sp4769540098391857867real_b @ X2 ) )
     => ( member6395380647429634249real_b
        @ ^ [R: real] : X
        @ ( qbs_Mx_real_real_b @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_3_qbs__closed2__dest,axiom,
    ! [X: real > real > nat,X2: quasi_7301349776962165718al_nat] :
      ( ( member_real_real_nat @ X @ ( qbs_sp4997250815398645552al_nat @ X2 ) )
     => ( member5814558293553255496al_nat
        @ ^ [R: real] : X
        @ ( qbs_Mx_real_real_nat @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_4_qbs__closed2__dest,axiom,
    ! [X: real > set_nat,X2: quasi_1649577361549849025et_nat] :
      ( ( member_real_set_nat @ X @ ( qbs_sp2552676844468483227et_nat @ X2 ) )
     => ( member6099108769244630579et_nat
        @ ^ [R: real] : X
        @ ( qbs_Mx_real_set_nat @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_5_qbs__closed2__dest,axiom,
    ! [X: set_nat,X2: quasi_borel_set_nat] :
      ( ( member_set_nat @ X @ ( qbs_space_set_nat @ X2 ) )
     => ( member_real_set_nat
        @ ^ [R: real] : X
        @ ( qbs_Mx_set_nat @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_6_qbs__closed2__dest,axiom,
    ! [X: nat,X2: quasi_borel_nat] :
      ( ( member_nat @ X @ ( qbs_space_nat @ X2 ) )
     => ( member_real_nat
        @ ^ [R: real] : X
        @ ( qbs_Mx_nat @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_7_qbs__closed2__dest,axiom,
    ! [X: real > nat,X2: quasi_borel_real_nat] :
      ( ( member_real_nat @ X @ ( qbs_space_real_nat @ X2 ) )
     => ( member_real_real_nat
        @ ^ [R: real] : X
        @ ( qbs_Mx_real_nat @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_8_qbs__closed2__dest,axiom,
    ! [X: real > b,X2: quasi_borel_real_b] :
      ( ( member_real_b @ X @ ( qbs_space_real_b @ X2 ) )
     => ( member_real_real_b
        @ ^ [R: real] : X
        @ ( qbs_Mx_real_b @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_9_qbs__closed2__dest,axiom,
    ! [X: nat > real,X2: quasi_borel_nat_real] :
      ( ( member_nat_real @ X @ ( qbs_space_nat_real @ X2 ) )
     => ( member_real_nat_real2
        @ ^ [R: real] : X
        @ ( qbs_Mx_nat_real @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_10_qbs__closed2__dest,axiom,
    ! [X: b,X2: quasi_borel_b] :
      ( ( member_b @ X @ ( qbs_space_b @ X2 ) )
     => ( member_real_b
        @ ^ [R: real] : X
        @ ( qbs_Mx_b @ X2 ) ) ) ).

% qbs_closed2_dest
thf(fact_11_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > nat > real,X2: quasi_4498916059141695318t_real,R2: real] :
      ( ( member3012124575732785096t_real @ Alpha @ ( qbs_Mx_real_nat_real @ X2 ) )
     => ( member_real_nat_real2 @ ( Alpha @ R2 ) @ ( qbs_sp1807256676205042352t_real @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_12_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > real > b,X2: quasi_4416045583689693371real_b,R2: real] :
      ( ( member6395380647429634249real_b @ Alpha @ ( qbs_Mx_real_real_b @ X2 ) )
     => ( member_real_real_b @ ( Alpha @ R2 ) @ ( qbs_sp4769540098391857867real_b @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_13_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > real > nat,X2: quasi_7301349776962165718al_nat,R2: real] :
      ( ( member5814558293553255496al_nat @ Alpha @ ( qbs_Mx_real_real_nat @ X2 ) )
     => ( member_real_real_nat @ ( Alpha @ R2 ) @ ( qbs_sp4997250815398645552al_nat @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_14_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > set_nat,X2: quasi_1649577361549849025et_nat,R2: real] :
      ( ( member6099108769244630579et_nat @ Alpha @ ( qbs_Mx_real_set_nat @ X2 ) )
     => ( member_real_set_nat @ ( Alpha @ R2 ) @ ( qbs_sp2552676844468483227et_nat @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_15_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > set_nat,X2: quasi_borel_set_nat,R2: real] :
      ( ( member_real_set_nat @ Alpha @ ( qbs_Mx_set_nat @ X2 ) )
     => ( member_set_nat @ ( Alpha @ R2 ) @ ( qbs_space_set_nat @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_16_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > nat,X2: quasi_borel_real_nat,R2: real] :
      ( ( member_real_real_nat @ Alpha @ ( qbs_Mx_real_nat @ X2 ) )
     => ( member_real_nat @ ( Alpha @ R2 ) @ ( qbs_space_real_nat @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_17_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > b,X2: quasi_borel_real_b,R2: real] :
      ( ( member_real_real_b @ Alpha @ ( qbs_Mx_real_b @ X2 ) )
     => ( member_real_b @ ( Alpha @ R2 ) @ ( qbs_space_real_b @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_18_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > nat > real,X2: quasi_borel_nat_real,R2: real] :
      ( ( member_real_nat_real2 @ Alpha @ ( qbs_Mx_nat_real @ X2 ) )
     => ( member_nat_real @ ( Alpha @ R2 ) @ ( qbs_space_nat_real @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_19_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > nat,X2: quasi_borel_nat,R2: real] :
      ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X2 ) )
     => ( member_nat @ ( Alpha @ R2 ) @ ( qbs_space_nat @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_20_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > b,X2: quasi_borel_b,R2: real] :
      ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X2 ) )
     => ( member_b @ ( Alpha @ R2 ) @ ( qbs_space_b @ X2 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_21_qbs__closed2I,axiom,
    ! [X2: set_real_nat_real,Mx: set_re1338850055584481895t_real] :
      ( ! [X3: real > nat > real] :
          ( ( member_real_nat_real2 @ X3 @ X2 )
         => ( member3012124575732785096t_real
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_cl6586241553476272112t_real @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_22_qbs__closed2I,axiom,
    ! [X2: set_real_real_b,Mx: set_real_real_real_b] :
      ( ! [X3: real > real > b] :
          ( ( member_real_real_b @ X3 @ X2 )
         => ( member6395380647429634249real_b
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_cl646577709987169675real_b @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_23_qbs__closed2I,axiom,
    ! [X2: set_real_real_nat,Mx: set_re3347183479607654119al_nat] :
      ( ! [X3: real > real > nat] :
          ( ( member_real_real_nat @ X3 @ X2 )
         => ( member5814558293553255496al_nat
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_cl552863655815099504al_nat @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_24_qbs__closed2I,axiom,
    ! [X2: set_real_set_nat,Mx: set_re5613859201272075090et_nat] :
      ( ! [X3: real > set_nat] :
          ( ( member_real_set_nat @ X3 @ X2 )
         => ( member6099108769244630579et_nat
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_cl1079945222394894811et_nat @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_25_qbs__closed2I,axiom,
    ! [X2: set_b,Mx: set_real_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ X2 )
         => ( member_real_b
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_closed2_b @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_26_qbs__closed2I,axiom,
    ! [X2: set_set_nat,Mx: set_real_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ X2 )
         => ( member_real_set_nat
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_closed2_set_nat @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_27_qbs__closed2I,axiom,
    ! [X2: set_nat,Mx: set_real_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ X2 )
         => ( member_real_nat
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_closed2_nat @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_28_qbs__closed2I,axiom,
    ! [X2: set_real_nat,Mx: set_real_real_nat] :
      ( ! [X3: real > nat] :
          ( ( member_real_nat @ X3 @ X2 )
         => ( member_real_real_nat
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_closed2_real_nat @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_29_qbs__closed2I,axiom,
    ! [X2: set_real_b,Mx: set_real_real_b] :
      ( ! [X3: real > b] :
          ( ( member_real_b @ X3 @ X2 )
         => ( member_real_real_b
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_closed2_real_b @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_30_qbs__closed2I,axiom,
    ! [X2: set_nat_real,Mx: set_real_nat_real] :
      ( ! [X3: nat > real] :
          ( ( member_nat_real @ X3 @ X2 )
         => ( member_real_nat_real2
            @ ^ [R: real] : X3
            @ Mx ) )
     => ( qbs_closed2_nat_real @ X2 @ Mx ) ) ).

% qbs_closed2I
thf(fact_31_qbs__closed2__def,axiom,
    ( qbs_closed2_set_nat
    = ( ^ [X4: set_set_nat,Mx2: set_real_set_nat] :
        ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ X4 )
         => ( member_real_set_nat
            @ ^ [R: real] : X5
            @ Mx2 ) ) ) ) ).

% qbs_closed2_def
thf(fact_32_qbs__closed2__def,axiom,
    ( qbs_closed2_real_nat
    = ( ^ [X4: set_real_nat,Mx2: set_real_real_nat] :
        ! [X5: real > nat] :
          ( ( member_real_nat @ X5 @ X4 )
         => ( member_real_real_nat
            @ ^ [R: real] : X5
            @ Mx2 ) ) ) ) ).

% qbs_closed2_def
thf(fact_33_qbs__closed2__def,axiom,
    ( qbs_closed2_real_b
    = ( ^ [X4: set_real_b,Mx2: set_real_real_b] :
        ! [X5: real > b] :
          ( ( member_real_b @ X5 @ X4 )
         => ( member_real_real_b
            @ ^ [R: real] : X5
            @ Mx2 ) ) ) ) ).

% qbs_closed2_def
thf(fact_34_qbs__closed2__def,axiom,
    ( qbs_closed2_nat_real
    = ( ^ [X4: set_nat_real,Mx2: set_real_nat_real] :
        ! [X5: nat > real] :
          ( ( member_nat_real @ X5 @ X4 )
         => ( member_real_nat_real2
            @ ^ [R: real] : X5
            @ Mx2 ) ) ) ) ).

% qbs_closed2_def
thf(fact_35_qbs__closed2__def,axiom,
    ( qbs_closed2_nat
    = ( ^ [X4: set_nat,Mx2: set_real_nat] :
        ! [X5: nat] :
          ( ( member_nat @ X5 @ X4 )
         => ( member_real_nat
            @ ^ [R: real] : X5
            @ Mx2 ) ) ) ) ).

% qbs_closed2_def
thf(fact_36_qbs__closed2__def,axiom,
    ( qbs_closed2_b
    = ( ^ [X4: set_b,Mx2: set_real_b] :
        ! [X5: b] :
          ( ( member_b @ X5 @ X4 )
         => ( member_real_b
            @ ^ [R: real] : X5
            @ Mx2 ) ) ) ) ).

% qbs_closed2_def
thf(fact_37_assms_I1_J,axiom,
    ! [I: nat] : ( member_set_real @ ( vimage_real_nat @ p @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% assms(1)
thf(fact_38_qbs__Mx__def,axiom,
    ( qbs_Mx_nat_real
    = ( ^ [X4: quasi_borel_nat_real] : ( produc7833056394006324t_real @ ( quasi_8146262523719627044t_real @ X4 ) ) ) ) ).

% qbs_Mx_def
thf(fact_39_qbs__Mx__def,axiom,
    ( qbs_Mx_real_b
    = ( ^ [X4: quasi_borel_real_b] : ( produc5800820901218438480real_b @ ( quasi_347859643032404289real_b @ X4 ) ) ) ) ).

% qbs_Mx_def
thf(fact_40_qbs__Mx__def,axiom,
    ( qbs_Mx_real_nat
    = ( ^ [X4: quasi_borel_real_nat] : ( produc4807364092358522932al_nat @ ( quasi_4171214832242746788al_nat @ X4 ) ) ) ) ).

% qbs_Mx_def
thf(fact_41_qbs__Mx__def,axiom,
    ( qbs_Mx_nat
    = ( ^ [X4: quasi_borel_nat] : ( produc8862089267992772894al_nat @ ( quasi_2638801612224710361el_nat @ X4 ) ) ) ) ).

% qbs_Mx_def
thf(fact_42_qbs__Mx__def,axiom,
    ( qbs_Mx_set_nat
    = ( ^ [X4: quasi_borel_set_nat] : ( produc1329678470257883018et_nat @ ( quasi_853997576918650511et_nat @ X4 ) ) ) ) ).

% qbs_Mx_def
thf(fact_43_qbs__Mx__def,axiom,
    ( qbs_Mx_b
    = ( ^ [X4: quasi_borel_b] : ( produc1187263614985732624real_b @ ( quasi_4958298574314430518orel_b @ X4 ) ) ) ) ).

% qbs_Mx_def
thf(fact_44_qbs__Mx__to__X_I1_J,axiom,
    ! [Alpha: real > set_nat,X2: quasi_borel_set_nat] :
      ( ( member_real_set_nat @ Alpha @ ( qbs_Mx_set_nat @ X2 ) )
     => ( member_real_set_nat @ Alpha
        @ ( pi_real_set_nat @ top_top_set_real
          @ ^ [Uu: real] : ( qbs_space_set_nat @ X2 ) ) ) ) ).

% qbs_Mx_to_X(1)
thf(fact_45_qbs__Mx__to__X_I1_J,axiom,
    ! [Alpha: real > real > nat,X2: quasi_borel_real_nat] :
      ( ( member_real_real_nat @ Alpha @ ( qbs_Mx_real_nat @ X2 ) )
     => ( member_real_real_nat @ Alpha
        @ ( pi_real_real_nat @ top_top_set_real
          @ ^ [Uu: real] : ( qbs_space_real_nat @ X2 ) ) ) ) ).

% qbs_Mx_to_X(1)
thf(fact_46_qbs__Mx__to__X_I1_J,axiom,
    ! [Alpha: real > real > b,X2: quasi_borel_real_b] :
      ( ( member_real_real_b @ Alpha @ ( qbs_Mx_real_b @ X2 ) )
     => ( member_real_real_b @ Alpha
        @ ( pi_real_real_b @ top_top_set_real
          @ ^ [Uu: real] : ( qbs_space_real_b @ X2 ) ) ) ) ).

% qbs_Mx_to_X(1)
thf(fact_47_qbs__Mx__to__X_I1_J,axiom,
    ! [Alpha: real > nat > real,X2: quasi_borel_nat_real] :
      ( ( member_real_nat_real2 @ Alpha @ ( qbs_Mx_nat_real @ X2 ) )
     => ( member_real_nat_real2 @ Alpha
        @ ( pi_real_nat_real @ top_top_set_real
          @ ^ [Uu: real] : ( qbs_space_nat_real @ X2 ) ) ) ) ).

% qbs_Mx_to_X(1)
thf(fact_48_qbs__Mx__to__X_I1_J,axiom,
    ! [Alpha: real > nat,X2: quasi_borel_nat] :
      ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X2 ) )
     => ( member_real_nat @ Alpha
        @ ( pi_real_nat @ top_top_set_real
          @ ^ [Uu: real] : ( qbs_space_nat @ X2 ) ) ) ) ).

% qbs_Mx_to_X(1)
thf(fact_49_qbs__Mx__to__X_I1_J,axiom,
    ! [Alpha: real > b,X2: quasi_borel_b] :
      ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X2 ) )
     => ( member_real_b @ Alpha
        @ ( pi_real_b @ top_top_set_real
          @ ^ [Uu: real] : ( qbs_space_b @ X2 ) ) ) ) ).

% qbs_Mx_to_X(1)
thf(fact_50_qbs__decomp,axiom,
    ! [X2: quasi_borel_set_nat] : ( member7853754362859631949et_nat @ ( produc3412633259576239964et_nat @ ( qbs_space_set_nat @ X2 ) @ ( qbs_Mx_set_nat @ X2 ) ) @ ( collec6231571016085039247et_nat @ ( produc1937539024057733095_nat_o @ is_qua1626198732325083535et_nat ) ) ) ).

% qbs_decomp
thf(fact_51_qbs__decomp,axiom,
    ! [X2: quasi_borel_nat] : ( member4868967264474237409al_nat @ ( produc3843710909396041968al_nat @ ( qbs_space_nat @ X2 ) @ ( qbs_Mx_nat @ X2 ) ) @ ( collec8700111531230907683al_nat @ ( produc5377950512376928595_nat_o @ is_quasi_borel_nat ) ) ) ).

% qbs_decomp
thf(fact_52_qbs__decomp,axiom,
    ! [X2: quasi_borel_real_nat] : ( member4954045411463533431al_nat @ ( produc4583020555951415302al_nat @ ( qbs_space_real_nat @ X2 ) @ ( qbs_Mx_real_nat @ X2 ) ) @ ( collec7401705435925069497al_nat @ ( produc5549966297984299197_nat_o @ is_qua3317151984682463396al_nat ) ) ) ).

% qbs_decomp
thf(fact_53_qbs__decomp,axiom,
    ! [X2: quasi_borel_real_b] : ( member2077407680349518043real_b @ ( produc3140906811317515298real_b @ ( qbs_space_real_b @ X2 ) @ ( qbs_Mx_real_b @ X2 ) ) @ ( collec1032871213651684125real_b @ ( produc7687788952222807969al_b_o @ is_qua1212226743203608129real_b ) ) ) ).

% qbs_decomp
thf(fact_54_qbs__decomp,axiom,
    ! [X2: quasi_borel_nat_real] : ( member948614669296314999t_real @ ( produc9006861556841674502t_real @ ( qbs_space_nat_real @ X2 ) @ ( qbs_Mx_nat_real @ X2 ) ) @ ( collec3396274693757851065t_real @ ( produc6450457281045158333real_o @ is_qua7292199676159343652t_real ) ) ) ).

% qbs_decomp
thf(fact_55_qbs__decomp,axiom,
    ! [X2: quasi_borel_b] : ( member527121533207341851real_b @ ( produc5260422477894090978real_b @ ( qbs_space_b @ X2 ) @ ( qbs_Mx_b @ X2 ) ) @ ( collec5860786303674561885real_b @ ( produc9150283905253018337al_b_o @ is_quasi_borel_b ) ) ) ).

% qbs_decomp
thf(fact_56_Rep__quasi__borel,axiom,
    ! [X: quasi_borel_b] : ( member527121533207341851real_b @ ( quasi_4958298574314430518orel_b @ X ) @ ( collec5860786303674561885real_b @ ( produc9150283905253018337al_b_o @ is_quasi_borel_b ) ) ) ).

% Rep_quasi_borel
thf(fact_57_Rep__quasi__borel__cases,axiom,
    ! [Y: produc4373655991844504306real_b] :
      ( ( member527121533207341851real_b @ Y @ ( collec5860786303674561885real_b @ ( produc9150283905253018337al_b_o @ is_quasi_borel_b ) ) )
     => ~ ! [X3: quasi_borel_b] :
            ( Y
           != ( quasi_4958298574314430518orel_b @ X3 ) ) ) ).

% Rep_quasi_borel_cases
thf(fact_58_Rep__quasi__borel__induct,axiom,
    ! [Y: produc4373655991844504306real_b,P: produc4373655991844504306real_b > $o] :
      ( ( member527121533207341851real_b @ Y @ ( collec5860786303674561885real_b @ ( produc9150283905253018337al_b_o @ is_quasi_borel_b ) ) )
     => ( ! [X3: quasi_borel_b] : ( P @ ( quasi_4958298574314430518orel_b @ X3 ) )
       => ( P @ Y ) ) ) ).

% Rep_quasi_borel_induct
thf(fact_59_Rep__quasi__borel__inject,axiom,
    ! [X: quasi_borel_b,Y: quasi_borel_b] :
      ( ( ( quasi_4958298574314430518orel_b @ X )
        = ( quasi_4958298574314430518orel_b @ Y ) )
      = ( X = Y ) ) ).

% Rep_quasi_borel_inject
thf(fact_60_vimage__const,axiom,
    ! [C: b,A: set_b] :
      ( ( ( member_b @ C @ A )
       => ( ( vimage_real_b
            @ ^ [X5: real] : C
            @ A )
          = top_top_set_real ) )
      & ( ~ ( member_b @ C @ A )
       => ( ( vimage_real_b
            @ ^ [X5: real] : C
            @ A )
          = bot_bot_set_real ) ) ) ).

% vimage_const
thf(fact_61_vimage__const,axiom,
    ! [C: nat,A: set_nat] :
      ( ( ( member_nat @ C @ A )
       => ( ( vimage_real_nat
            @ ^ [X5: real] : C
            @ A )
          = top_top_set_real ) )
      & ( ~ ( member_nat @ C @ A )
       => ( ( vimage_real_nat
            @ ^ [X5: real] : C
            @ A )
          = bot_bot_set_real ) ) ) ).

% vimage_const
thf(fact_62_vimage__const,axiom,
    ! [C: nat,A: set_nat] :
      ( ( ( member_nat @ C @ A )
       => ( ( vimage_nat_nat
            @ ^ [X5: nat] : C
            @ A )
          = top_top_set_nat ) )
      & ( ~ ( member_nat @ C @ A )
       => ( ( vimage_nat_nat
            @ ^ [X5: nat] : C
            @ A )
          = bot_bot_set_nat ) ) ) ).

% vimage_const
thf(fact_63_vimage__const,axiom,
    ! [C: b,A: set_b] :
      ( ( ( member_b @ C @ A )
       => ( ( vimage_nat_b
            @ ^ [X5: nat] : C
            @ A )
          = top_top_set_nat ) )
      & ( ~ ( member_b @ C @ A )
       => ( ( vimage_nat_b
            @ ^ [X5: nat] : C
            @ A )
          = bot_bot_set_nat ) ) ) ).

% vimage_const
thf(fact_64_vimage__const,axiom,
    ! [C: nat,A: set_nat] :
      ( ( ( member_nat @ C @ A )
       => ( ( vimage_o_nat
            @ ^ [X5: $o] : C
            @ A )
          = top_top_set_o ) )
      & ( ~ ( member_nat @ C @ A )
       => ( ( vimage_o_nat
            @ ^ [X5: $o] : C
            @ A )
          = bot_bot_set_o ) ) ) ).

% vimage_const
thf(fact_65_vimage__const,axiom,
    ! [C: b,A: set_b] :
      ( ( ( member_b @ C @ A )
       => ( ( vimage_o_b
            @ ^ [X5: $o] : C
            @ A )
          = top_top_set_o ) )
      & ( ~ ( member_b @ C @ A )
       => ( ( vimage_o_b
            @ ^ [X5: $o] : C
            @ A )
          = bot_bot_set_o ) ) ) ).

% vimage_const
thf(fact_66_vimage__const,axiom,
    ! [C: set_nat,A: set_set_nat] :
      ( ( ( member_set_nat @ C @ A )
       => ( ( vimage_real_set_nat
            @ ^ [X5: real] : C
            @ A )
          = top_top_set_real ) )
      & ( ~ ( member_set_nat @ C @ A )
       => ( ( vimage_real_set_nat
            @ ^ [X5: real] : C
            @ A )
          = bot_bot_set_real ) ) ) ).

% vimage_const
thf(fact_67_vimage__const,axiom,
    ! [C: set_nat,A: set_set_nat] :
      ( ( ( member_set_nat @ C @ A )
       => ( ( vimage_nat_set_nat
            @ ^ [X5: nat] : C
            @ A )
          = top_top_set_nat ) )
      & ( ~ ( member_set_nat @ C @ A )
       => ( ( vimage_nat_set_nat
            @ ^ [X5: nat] : C
            @ A )
          = bot_bot_set_nat ) ) ) ).

% vimage_const
thf(fact_68_vimage__const,axiom,
    ! [C: set_nat,A: set_set_nat] :
      ( ( ( member_set_nat @ C @ A )
       => ( ( vimage_o_set_nat
            @ ^ [X5: $o] : C
            @ A )
          = top_top_set_o ) )
      & ( ~ ( member_set_nat @ C @ A )
       => ( ( vimage_o_set_nat
            @ ^ [X5: $o] : C
            @ A )
          = bot_bot_set_o ) ) ) ).

% vimage_const
thf(fact_69_vimage__const,axiom,
    ! [C: real > nat,A: set_real_nat] :
      ( ( ( member_real_nat @ C @ A )
       => ( ( vimage_real_real_nat
            @ ^ [X5: real] : C
            @ A )
          = top_top_set_real ) )
      & ( ~ ( member_real_nat @ C @ A )
       => ( ( vimage_real_real_nat
            @ ^ [X5: real] : C
            @ A )
          = bot_bot_set_real ) ) ) ).

% vimage_const
thf(fact_70_Pi__eq__empty,axiom,
    ! [A: set_real,B: real > set_b] :
      ( ( ( pi_real_b @ A @ B )
        = bot_bot_set_real_b )
      = ( ? [X5: real] :
            ( ( member_real @ X5 @ A )
            & ( ( B @ X5 )
              = bot_bot_set_b ) ) ) ) ).

% Pi_eq_empty
thf(fact_71_Pi__eq__empty,axiom,
    ! [A: set_real,B: real > set_nat] :
      ( ( ( pi_real_nat @ A @ B )
        = bot_bot_set_real_nat )
      = ( ? [X5: real] :
            ( ( member_real @ X5 @ A )
            & ( ( B @ X5 )
              = bot_bot_set_nat ) ) ) ) ).

% Pi_eq_empty
thf(fact_72_Pi__UNIV,axiom,
    ! [A: set_real] :
      ( ( pi_real_b @ A
        @ ^ [Uu: real] : top_top_set_b )
      = top_top_set_real_b ) ).

% Pi_UNIV
thf(fact_73_Pi__UNIV,axiom,
    ! [A: set_real] :
      ( ( pi_real_nat @ A
        @ ^ [Uu: real] : top_top_set_nat )
      = top_top_set_real_nat ) ).

% Pi_UNIV
thf(fact_74_singleton__conv,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ^ [X5: nat] : ( X5 = A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_75_singleton__conv,axiom,
    ! [A2: real] :
      ( ( collect_real
        @ ^ [X5: real] : ( X5 = A2 ) )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_76_singleton__conv,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ^ [X5: $o] : ( X5 = A2 ) )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_77_singleton__conv2,axiom,
    ! [A2: nat] :
      ( ( collect_nat
        @ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
          @ A2 ) )
      = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_78_singleton__conv2,axiom,
    ! [A2: real] :
      ( ( collect_real
        @ ( ^ [Y2: real,Z: real] : ( Y2 = Z )
          @ A2 ) )
      = ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_79_singleton__conv2,axiom,
    ! [A2: $o] :
      ( ( collect_o
        @ ( ^ [Y2: $o,Z: $o] : ( Y2 = Z )
          @ A2 ) )
      = ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_80_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_real
            @ ^ [S: real] : P )
          = top_top_set_real ) )
      & ( ~ P
       => ( ( collect_real
            @ ^ [S: real] : P )
          = bot_bot_set_real ) ) ) ).

% Collect_const
thf(fact_81_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_nat
            @ ^ [S: nat] : P )
          = top_top_set_nat ) )
      & ( ~ P
       => ( ( collect_nat
            @ ^ [S: nat] : P )
          = bot_bot_set_nat ) ) ) ).

% Collect_const
thf(fact_82_Collect__const,axiom,
    ! [P: $o] :
      ( ( P
       => ( ( collect_o
            @ ^ [S: $o] : P )
          = top_top_set_o ) )
      & ( ~ P
       => ( ( collect_o
            @ ^ [S: $o] : P )
          = bot_bot_set_o ) ) ) ).

% Collect_const
thf(fact_83_Pi__split__insert__domain,axiom,
    ! [X: nat > nat,I: nat,I2: set_nat,X2: nat > set_nat] :
      ( ( member_nat_nat @ X @ ( pi_nat_nat @ ( insert_nat @ I @ I2 ) @ X2 ) )
      = ( ( member_nat_nat @ X @ ( pi_nat_nat @ I2 @ X2 ) )
        & ( member_nat @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_84_Pi__split__insert__domain,axiom,
    ! [X: $o > nat,I: $o,I2: set_o,X2: $o > set_nat] :
      ( ( member_o_nat @ X @ ( pi_o_nat @ ( insert_o @ I @ I2 ) @ X2 ) )
      = ( ( member_o_nat @ X @ ( pi_o_nat @ I2 @ X2 ) )
        & ( member_nat @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_85_Pi__split__insert__domain,axiom,
    ! [X: nat > b,I: nat,I2: set_nat,X2: nat > set_b] :
      ( ( member_nat_b @ X @ ( pi_nat_b @ ( insert_nat @ I @ I2 ) @ X2 ) )
      = ( ( member_nat_b @ X @ ( pi_nat_b @ I2 @ X2 ) )
        & ( member_b @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_86_Pi__split__insert__domain,axiom,
    ! [X: $o > b,I: $o,I2: set_o,X2: $o > set_b] :
      ( ( member_o_b @ X @ ( pi_o_b @ ( insert_o @ I @ I2 ) @ X2 ) )
      = ( ( member_o_b @ X @ ( pi_o_b @ I2 @ X2 ) )
        & ( member_b @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_87_Pi__split__insert__domain,axiom,
    ! [X: real > nat,I: real,I2: set_real,X2: real > set_nat] :
      ( ( member_real_nat @ X @ ( pi_real_nat @ ( insert_real @ I @ I2 ) @ X2 ) )
      = ( ( member_real_nat @ X @ ( pi_real_nat @ I2 @ X2 ) )
        & ( member_nat @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_88_Pi__split__insert__domain,axiom,
    ! [X: real > b,I: real,I2: set_real,X2: real > set_b] :
      ( ( member_real_b @ X @ ( pi_real_b @ ( insert_real @ I @ I2 ) @ X2 ) )
      = ( ( member_real_b @ X @ ( pi_real_b @ I2 @ X2 ) )
        & ( member_b @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_89_Pi__split__insert__domain,axiom,
    ! [X: nat > real,I: nat,I2: set_nat,X2: nat > set_real] :
      ( ( member_nat_real @ X @ ( pi_nat_real @ ( insert_nat @ I @ I2 ) @ X2 ) )
      = ( ( member_nat_real @ X @ ( pi_nat_real @ I2 @ X2 ) )
        & ( member_real @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_90_Pi__split__insert__domain,axiom,
    ! [X: nat > set_nat,I: nat,I2: set_nat,X2: nat > set_set_nat] :
      ( ( member_nat_set_nat @ X @ ( pi_nat_set_nat @ ( insert_nat @ I @ I2 ) @ X2 ) )
      = ( ( member_nat_set_nat @ X @ ( pi_nat_set_nat @ I2 @ X2 ) )
        & ( member_set_nat @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_91_Pi__split__insert__domain,axiom,
    ! [X: $o > set_nat,I: $o,I2: set_o,X2: $o > set_set_nat] :
      ( ( member_o_set_nat @ X @ ( pi_o_set_nat @ ( insert_o @ I @ I2 ) @ X2 ) )
      = ( ( member_o_set_nat @ X @ ( pi_o_set_nat @ I2 @ X2 ) )
        & ( member_set_nat @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_92_Pi__split__insert__domain,axiom,
    ! [X: real > set_nat,I: real,I2: set_real,X2: real > set_set_nat] :
      ( ( member_real_set_nat @ X @ ( pi_real_set_nat @ ( insert_real @ I @ I2 ) @ X2 ) )
      = ( ( member_real_set_nat @ X @ ( pi_real_set_nat @ I2 @ X2 ) )
        & ( member_set_nat @ ( X @ I ) @ ( X2 @ I ) ) ) ) ).

% Pi_split_insert_domain
thf(fact_93_vimage__empty,axiom,
    ! [F: nat > nat] :
      ( ( vimage_nat_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% vimage_empty
thf(fact_94_vimage__empty,axiom,
    ! [F: real > nat] :
      ( ( vimage_real_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_real ) ).

% vimage_empty
thf(fact_95_vimage__empty,axiom,
    ! [F: $o > nat] :
      ( ( vimage_o_nat @ F @ bot_bot_set_nat )
      = bot_bot_set_o ) ).

% vimage_empty
thf(fact_96_vimage__empty,axiom,
    ! [F: nat > real] :
      ( ( vimage_nat_real @ F @ bot_bot_set_real )
      = bot_bot_set_nat ) ).

% vimage_empty
thf(fact_97_vimage__empty,axiom,
    ! [F: real > real] :
      ( ( vimage_real_real @ F @ bot_bot_set_real )
      = bot_bot_set_real ) ).

% vimage_empty
thf(fact_98_vimage__empty,axiom,
    ! [F: $o > real] :
      ( ( vimage_o_real @ F @ bot_bot_set_real )
      = bot_bot_set_o ) ).

% vimage_empty
thf(fact_99_vimage__empty,axiom,
    ! [F: nat > $o] :
      ( ( vimage_nat_o @ F @ bot_bot_set_o )
      = bot_bot_set_nat ) ).

% vimage_empty
thf(fact_100_vimage__empty,axiom,
    ! [F: real > $o] :
      ( ( vimage_real_o @ F @ bot_bot_set_o )
      = bot_bot_set_real ) ).

% vimage_empty
thf(fact_101_vimage__empty,axiom,
    ! [F: $o > $o] :
      ( ( vimage_o_o @ F @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% vimage_empty
thf(fact_102_vimage__UNIV,axiom,
    ! [F: real > real] :
      ( ( vimage_real_real @ F @ top_top_set_real )
      = top_top_set_real ) ).

% vimage_UNIV
thf(fact_103_vimage__UNIV,axiom,
    ! [F: nat > real] :
      ( ( vimage_nat_real @ F @ top_top_set_real )
      = top_top_set_nat ) ).

% vimage_UNIV
thf(fact_104_vimage__UNIV,axiom,
    ! [F: $o > real] :
      ( ( vimage_o_real @ F @ top_top_set_real )
      = top_top_set_o ) ).

% vimage_UNIV
thf(fact_105_vimage__UNIV,axiom,
    ! [F: real > nat] :
      ( ( vimage_real_nat @ F @ top_top_set_nat )
      = top_top_set_real ) ).

% vimage_UNIV
thf(fact_106_vimage__UNIV,axiom,
    ! [F: nat > nat] :
      ( ( vimage_nat_nat @ F @ top_top_set_nat )
      = top_top_set_nat ) ).

% vimage_UNIV
thf(fact_107_vimage__UNIV,axiom,
    ! [F: $o > nat] :
      ( ( vimage_o_nat @ F @ top_top_set_nat )
      = top_top_set_o ) ).

% vimage_UNIV
thf(fact_108_vimage__UNIV,axiom,
    ! [F: real > $o] :
      ( ( vimage_real_o @ F @ top_top_set_o )
      = top_top_set_real ) ).

% vimage_UNIV
thf(fact_109_vimage__UNIV,axiom,
    ! [F: nat > $o] :
      ( ( vimage_nat_o @ F @ top_top_set_o )
      = top_top_set_nat ) ).

% vimage_UNIV
thf(fact_110_vimage__UNIV,axiom,
    ! [F: $o > $o] :
      ( ( vimage_o_o @ F @ top_top_set_o )
      = top_top_set_o ) ).

% vimage_UNIV
thf(fact_111_UNIV__I,axiom,
    ! [X: b] : ( member_b @ X @ top_top_set_b ) ).

% UNIV_I
thf(fact_112_UNIV__I,axiom,
    ! [X: real] : ( member_real @ X @ top_top_set_real ) ).

% UNIV_I
thf(fact_113_UNIV__I,axiom,
    ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% UNIV_I
thf(fact_114_UNIV__I,axiom,
    ! [X: $o] : ( member_o @ X @ top_top_set_o ) ).

% UNIV_I
thf(fact_115_UNIV__I,axiom,
    ! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).

% UNIV_I
thf(fact_116_UNIV__I,axiom,
    ! [X: real > nat] : ( member_real_nat @ X @ top_top_set_real_nat ) ).

% UNIV_I
thf(fact_117_UNIV__I,axiom,
    ! [X: real > b] : ( member_real_b @ X @ top_top_set_real_b ) ).

% UNIV_I
thf(fact_118_UNIV__I,axiom,
    ! [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).

% UNIV_I
thf(fact_119_UNIV__I,axiom,
    ! [X: real > set_nat] : ( member_real_set_nat @ X @ top_to245300144855000375et_nat ) ).

% UNIV_I
thf(fact_120_UNIV__I,axiom,
    ! [X: real > nat > real] : ( member_real_nat_real2 @ X @ top_to6902130281023745740t_real ) ).

% UNIV_I
thf(fact_121_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X5: nat] :
            ~ ( P @ X5 ) ) ) ).

% empty_Collect_eq
thf(fact_122_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X5: real] :
            ~ ( P @ X5 ) ) ) ).

% empty_Collect_eq
thf(fact_123_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X5: $o] :
            ~ ( P @ X5 ) ) ) ).

% empty_Collect_eq
thf(fact_124_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X5: nat] :
            ~ ( P @ X5 ) ) ) ).

% Collect_empty_eq
thf(fact_125_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X5: real] :
            ~ ( P @ X5 ) ) ) ).

% Collect_empty_eq
thf(fact_126_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X5: $o] :
            ~ ( P @ X5 ) ) ) ).

% Collect_empty_eq
thf(fact_127_all__not__in__conv,axiom,
    ! [A: set_b] :
      ( ( ! [X5: b] :
            ~ ( member_b @ X5 @ A ) )
      = ( A = bot_bot_set_b ) ) ).

% all_not_in_conv
thf(fact_128_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X5: nat] :
            ~ ( member_nat @ X5 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_129_all__not__in__conv,axiom,
    ! [A: set_real] :
      ( ( ! [X5: real] :
            ~ ( member_real @ X5 @ A ) )
      = ( A = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_130_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ( ! [X5: $o] :
            ~ ( member_o @ X5 @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_131_all__not__in__conv,axiom,
    ! [A: set_set_nat] :
      ( ( ! [X5: set_nat] :
            ~ ( member_set_nat @ X5 @ A ) )
      = ( A = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_132_all__not__in__conv,axiom,
    ! [A: set_real_nat] :
      ( ( ! [X5: real > nat] :
            ~ ( member_real_nat @ X5 @ A ) )
      = ( A = bot_bot_set_real_nat ) ) ).

% all_not_in_conv
thf(fact_133_all__not__in__conv,axiom,
    ! [A: set_real_b] :
      ( ( ! [X5: real > b] :
            ~ ( member_real_b @ X5 @ A ) )
      = ( A = bot_bot_set_real_b ) ) ).

% all_not_in_conv
thf(fact_134_all__not__in__conv,axiom,
    ! [A: set_nat_real] :
      ( ( ! [X5: nat > real] :
            ~ ( member_nat_real @ X5 @ A ) )
      = ( A = bot_bot_set_nat_real ) ) ).

% all_not_in_conv
thf(fact_135_all__not__in__conv,axiom,
    ! [A: set_real_set_nat] :
      ( ( ! [X5: real > set_nat] :
            ~ ( member_real_set_nat @ X5 @ A ) )
      = ( A = bot_bo6814059168456595739et_nat ) ) ).

% all_not_in_conv
thf(fact_136_all__not__in__conv,axiom,
    ! [A: set_real_nat_real] :
      ( ( ! [X5: real > nat > real] :
            ~ ( member_real_nat_real2 @ X5 @ A ) )
      = ( A = bot_bo6533810469807102640t_real ) ) ).

% all_not_in_conv
thf(fact_137_empty__iff,axiom,
    ! [C: b] :
      ~ ( member_b @ C @ bot_bot_set_b ) ).

% empty_iff
thf(fact_138_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_139_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_140_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_141_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_142_empty__iff,axiom,
    ! [C: real > nat] :
      ~ ( member_real_nat @ C @ bot_bot_set_real_nat ) ).

% empty_iff
thf(fact_143_empty__iff,axiom,
    ! [C: real > b] :
      ~ ( member_real_b @ C @ bot_bot_set_real_b ) ).

% empty_iff
thf(fact_144_empty__iff,axiom,
    ! [C: nat > real] :
      ~ ( member_nat_real @ C @ bot_bot_set_nat_real ) ).

% empty_iff
thf(fact_145_empty__iff,axiom,
    ! [C: real > set_nat] :
      ~ ( member_real_set_nat @ C @ bot_bo6814059168456595739et_nat ) ).

% empty_iff
thf(fact_146_empty__iff,axiom,
    ! [C: real > nat > real] :
      ~ ( member_real_nat_real2 @ C @ bot_bo6533810469807102640t_real ) ).

% empty_iff
thf(fact_147_insert__absorb2,axiom,
    ! [X: nat,A: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ X @ A ) )
      = ( insert_nat @ X @ A ) ) ).

% insert_absorb2
thf(fact_148_insert__absorb2,axiom,
    ! [X: real,A: set_real] :
      ( ( insert_real @ X @ ( insert_real @ X @ A ) )
      = ( insert_real @ X @ A ) ) ).

% insert_absorb2
thf(fact_149_insert__absorb2,axiom,
    ! [X: $o,A: set_o] :
      ( ( insert_o @ X @ ( insert_o @ X @ A ) )
      = ( insert_o @ X @ A ) ) ).

% insert_absorb2
thf(fact_150_insert__iff,axiom,
    ! [A2: real,B2: real,A: set_real] :
      ( ( member_real @ A2 @ ( insert_real @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_real @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_151_insert__iff,axiom,
    ! [A2: $o,B2: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_o @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_152_insert__iff,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_153_insert__iff,axiom,
    ! [A2: b,B2: b,A: set_b] :
      ( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_b @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_154_insert__iff,axiom,
    ! [A2: set_nat,B2: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_set_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_155_insert__iff,axiom,
    ! [A2: real > nat,B2: real > nat,A: set_real_nat] :
      ( ( member_real_nat @ A2 @ ( insert_real_nat @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_real_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_156_insert__iff,axiom,
    ! [A2: real > b,B2: real > b,A: set_real_b] :
      ( ( member_real_b @ A2 @ ( insert_real_b @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_real_b @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_157_insert__iff,axiom,
    ! [A2: nat > real,B2: nat > real,A: set_nat_real] :
      ( ( member_nat_real @ A2 @ ( insert_nat_real @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_nat_real @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_158_insert__iff,axiom,
    ! [A2: real > set_nat,B2: real > set_nat,A: set_real_set_nat] :
      ( ( member_real_set_nat @ A2 @ ( insert_real_set_nat @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_real_set_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_159_insert__iff,axiom,
    ! [A2: real > nat > real,B2: real > nat > real,A: set_real_nat_real] :
      ( ( member_real_nat_real2 @ A2 @ ( insert_real_nat_real @ B2 @ A ) )
      = ( ( A2 = B2 )
        | ( member_real_nat_real2 @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_160_insertCI,axiom,
    ! [A2: real,B: set_real,B2: real] :
      ( ( ~ ( member_real @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).

% insertCI
thf(fact_161_insertCI,axiom,
    ! [A2: $o,B: set_o,B2: $o] :
      ( ( ~ ( member_o @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).

% insertCI
thf(fact_162_insertCI,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( ( ~ ( member_nat @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_163_insertCI,axiom,
    ! [A2: b,B: set_b,B2: b] :
      ( ( ~ ( member_b @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).

% insertCI
thf(fact_164_insertCI,axiom,
    ! [A2: set_nat,B: set_set_nat,B2: set_nat] :
      ( ( ~ ( member_set_nat @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_165_insertCI,axiom,
    ! [A2: real > nat,B: set_real_nat,B2: real > nat] :
      ( ( ~ ( member_real_nat @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_real_nat @ A2 @ ( insert_real_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_166_insertCI,axiom,
    ! [A2: real > b,B: set_real_b,B2: real > b] :
      ( ( ~ ( member_real_b @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_real_b @ A2 @ ( insert_real_b @ B2 @ B ) ) ) ).

% insertCI
thf(fact_167_insertCI,axiom,
    ! [A2: nat > real,B: set_nat_real,B2: nat > real] :
      ( ( ~ ( member_nat_real @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_nat_real @ A2 @ ( insert_nat_real @ B2 @ B ) ) ) ).

% insertCI
thf(fact_168_insertCI,axiom,
    ! [A2: real > set_nat,B: set_real_set_nat,B2: real > set_nat] :
      ( ( ~ ( member_real_set_nat @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_real_set_nat @ A2 @ ( insert_real_set_nat @ B2 @ B ) ) ) ).

% insertCI
thf(fact_169_insertCI,axiom,
    ! [A2: real > nat > real,B: set_real_nat_real,B2: real > nat > real] :
      ( ( ~ ( member_real_nat_real2 @ A2 @ B )
       => ( A2 = B2 ) )
     => ( member_real_nat_real2 @ A2 @ ( insert_real_nat_real @ B2 @ B ) ) ) ).

% insertCI
thf(fact_170_vimage__eq,axiom,
    ! [A2: nat,F: nat > nat,B: set_nat] :
      ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
      = ( member_nat @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_171_vimage__eq,axiom,
    ! [A2: nat,F: nat > b,B: set_b] :
      ( ( member_nat @ A2 @ ( vimage_nat_b @ F @ B ) )
      = ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_172_vimage__eq,axiom,
    ! [A2: b,F: b > nat,B: set_nat] :
      ( ( member_b @ A2 @ ( vimage_b_nat @ F @ B ) )
      = ( member_nat @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_173_vimage__eq,axiom,
    ! [A2: b,F: b > b,B: set_b] :
      ( ( member_b @ A2 @ ( vimage_b_b @ F @ B ) )
      = ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_174_vimage__eq,axiom,
    ! [A2: real,F: real > nat,B: set_nat] :
      ( ( member_real @ A2 @ ( vimage_real_nat @ F @ B ) )
      = ( member_nat @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_175_vimage__eq,axiom,
    ! [A2: set_nat,F: set_nat > nat,B: set_nat] :
      ( ( member_set_nat @ A2 @ ( vimage_set_nat_nat @ F @ B ) )
      = ( member_nat @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_176_vimage__eq,axiom,
    ! [A2: set_nat,F: set_nat > b,B: set_b] :
      ( ( member_set_nat @ A2 @ ( vimage_set_nat_b @ F @ B ) )
      = ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_177_vimage__eq,axiom,
    ! [A2: nat,F: nat > set_nat,B: set_set_nat] :
      ( ( member_nat @ A2 @ ( vimage_nat_set_nat @ F @ B ) )
      = ( member_set_nat @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_178_vimage__eq,axiom,
    ! [A2: b,F: b > set_nat,B: set_set_nat] :
      ( ( member_b @ A2 @ ( vimage_b_set_nat @ F @ B ) )
      = ( member_set_nat @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_179_vimage__eq,axiom,
    ! [A2: set_nat,F: set_nat > set_nat,B: set_set_nat] :
      ( ( member_set_nat @ A2 @ ( vimage4765135879290611145et_nat @ F @ B ) )
      = ( member_set_nat @ ( F @ A2 ) @ B ) ) ).

% vimage_eq
thf(fact_180_vimageI,axiom,
    ! [F: nat > nat,A2: nat,B2: nat,B: set_nat] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_nat @ B2 @ B )
       => ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) ) ) ) ).

% vimageI
thf(fact_181_vimageI,axiom,
    ! [F: b > nat,A2: b,B2: nat,B: set_nat] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_nat @ B2 @ B )
       => ( member_b @ A2 @ ( vimage_b_nat @ F @ B ) ) ) ) ).

% vimageI
thf(fact_182_vimageI,axiom,
    ! [F: nat > b,A2: nat,B2: b,B: set_b] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_b @ B2 @ B )
       => ( member_nat @ A2 @ ( vimage_nat_b @ F @ B ) ) ) ) ).

% vimageI
thf(fact_183_vimageI,axiom,
    ! [F: b > b,A2: b,B2: b,B: set_b] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_b @ B2 @ B )
       => ( member_b @ A2 @ ( vimage_b_b @ F @ B ) ) ) ) ).

% vimageI
thf(fact_184_vimageI,axiom,
    ! [F: real > nat,A2: real,B2: nat,B: set_nat] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_nat @ B2 @ B )
       => ( member_real @ A2 @ ( vimage_real_nat @ F @ B ) ) ) ) ).

% vimageI
thf(fact_185_vimageI,axiom,
    ! [F: nat > set_nat,A2: nat,B2: set_nat,B: set_set_nat] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_set_nat @ B2 @ B )
       => ( member_nat @ A2 @ ( vimage_nat_set_nat @ F @ B ) ) ) ) ).

% vimageI
thf(fact_186_vimageI,axiom,
    ! [F: b > set_nat,A2: b,B2: set_nat,B: set_set_nat] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_set_nat @ B2 @ B )
       => ( member_b @ A2 @ ( vimage_b_set_nat @ F @ B ) ) ) ) ).

% vimageI
thf(fact_187_vimageI,axiom,
    ! [F: set_nat > nat,A2: set_nat,B2: nat,B: set_nat] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_nat @ B2 @ B )
       => ( member_set_nat @ A2 @ ( vimage_set_nat_nat @ F @ B ) ) ) ) ).

% vimageI
thf(fact_188_vimageI,axiom,
    ! [F: set_nat > b,A2: set_nat,B2: b,B: set_b] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_b @ B2 @ B )
       => ( member_set_nat @ A2 @ ( vimage_set_nat_b @ F @ B ) ) ) ) ).

% vimageI
thf(fact_189_vimageI,axiom,
    ! [F: set_nat > set_nat,A2: set_nat,B2: set_nat,B: set_set_nat] :
      ( ( ( F @ A2 )
        = B2 )
     => ( ( member_set_nat @ B2 @ B )
       => ( member_set_nat @ A2 @ ( vimage4765135879290611145et_nat @ F @ B ) ) ) ) ).

% vimageI
thf(fact_190_Pi__I,axiom,
    ! [A: set_real,F: real > nat,B: real > set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_real_nat @ F @ ( pi_real_nat @ A @ B ) ) ) ).

% Pi_I
thf(fact_191_Pi__I,axiom,
    ! [A: set_real,F: real > b,B: real > set_b] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_real_b @ F @ ( pi_real_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_192_Pi__I,axiom,
    ! [A: set_nat,F: nat > real,B: nat > set_real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_real @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_real @ F @ ( pi_nat_real @ A @ B ) ) ) ).

% Pi_I
thf(fact_193_Pi__I,axiom,
    ! [A: set_nat,F: nat > nat,B: nat > set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_nat @ F @ ( pi_nat_nat @ A @ B ) ) ) ).

% Pi_I
thf(fact_194_Pi__I,axiom,
    ! [A: set_nat,F: nat > b,B: nat > set_b] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_b @ F @ ( pi_nat_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_195_Pi__I,axiom,
    ! [A: set_b,F: b > nat,B: b > set_nat] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_b_nat @ F @ ( pi_b_nat @ A @ B ) ) ) ).

% Pi_I
thf(fact_196_Pi__I,axiom,
    ! [A: set_b,F: b > b,B: b > set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_b_b @ F @ ( pi_b_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_197_Pi__I,axiom,
    ! [A: set_real,F: real > set_nat,B: real > set_set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_set_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_real_set_nat @ F @ ( pi_real_set_nat @ A @ B ) ) ) ).

% Pi_I
thf(fact_198_Pi__I,axiom,
    ! [A: set_set_nat,F: set_nat > nat,B: set_nat > set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_nat_nat @ F @ ( pi_set_nat_nat @ A @ B ) ) ) ).

% Pi_I
thf(fact_199_Pi__I,axiom,
    ! [A: set_set_nat,F: set_nat > b,B: set_nat > set_b] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_nat_b @ F @ ( pi_set_nat_b @ A @ B ) ) ) ).

% Pi_I
thf(fact_200_mem__Collect__eq,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_201_mem__Collect__eq,axiom,
    ! [A2: real > nat,P: ( real > nat ) > $o] :
      ( ( member_real_nat @ A2 @ ( collect_real_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_202_mem__Collect__eq,axiom,
    ! [A2: real > b,P: ( real > b ) > $o] :
      ( ( member_real_b @ A2 @ ( collect_real_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_203_mem__Collect__eq,axiom,
    ! [A2: nat > real,P: ( nat > real ) > $o] :
      ( ( member_nat_real @ A2 @ ( collect_nat_real @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_204_mem__Collect__eq,axiom,
    ! [A2: b,P: b > $o] :
      ( ( member_b @ A2 @ ( collect_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_205_mem__Collect__eq,axiom,
    ! [A2: real > nat > real,P: ( real > nat > real ) > $o] :
      ( ( member_real_nat_real2 @ A2 @ ( collec336724336074912571t_real @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_206_mem__Collect__eq,axiom,
    ! [A2: real > real > b,P: ( real > real > b ) > $o] :
      ( ( member_real_real_b @ A2 @ ( collect_real_real_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_207_mem__Collect__eq,axiom,
    ! [A2: real > real > nat,P: ( real > real > nat ) > $o] :
      ( ( member_real_real_nat @ A2 @ ( collec3526718475268515771al_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_208_mem__Collect__eq,axiom,
    ! [A2: real > set_nat,P: ( real > set_nat ) > $o] :
      ( ( member_real_set_nat @ A2 @ ( collect_real_set_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_209_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_210_Collect__mem__eq,axiom,
    ! [A: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_211_Collect__mem__eq,axiom,
    ! [A: set_real_nat] :
      ( ( collect_real_nat
        @ ^ [X5: real > nat] : ( member_real_nat @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_212_Collect__mem__eq,axiom,
    ! [A: set_real_b] :
      ( ( collect_real_b
        @ ^ [X5: real > b] : ( member_real_b @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_213_Collect__mem__eq,axiom,
    ! [A: set_nat_real] :
      ( ( collect_nat_real
        @ ^ [X5: nat > real] : ( member_nat_real @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_214_Collect__mem__eq,axiom,
    ! [A: set_b] :
      ( ( collect_b
        @ ^ [X5: b] : ( member_b @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_215_Collect__mem__eq,axiom,
    ! [A: set_real_nat_real] :
      ( ( collec336724336074912571t_real
        @ ^ [X5: real > nat > real] : ( member_real_nat_real2 @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_216_Collect__mem__eq,axiom,
    ! [A: set_real_real_b] :
      ( ( collect_real_real_b
        @ ^ [X5: real > real > b] : ( member_real_real_b @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_217_Collect__mem__eq,axiom,
    ! [A: set_real_real_nat] :
      ( ( collec3526718475268515771al_nat
        @ ^ [X5: real > real > nat] : ( member_real_real_nat @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_218_Collect__mem__eq,axiom,
    ! [A: set_real_set_nat] :
      ( ( collect_real_set_nat
        @ ^ [X5: real > set_nat] : ( member_real_set_nat @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_219_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X5: nat] : ( member_nat @ X5 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_220_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_221_vimage__Collect__eq,axiom,
    ! [F: real > nat,P: nat > $o] :
      ( ( vimage_real_nat @ F @ ( collect_nat @ P ) )
      = ( collect_real
        @ ^ [Y3: real] : ( P @ ( F @ Y3 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_222_vimage__Collect__eq,axiom,
    ! [F: nat > nat,P: nat > $o] :
      ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [Y3: nat] : ( P @ ( F @ Y3 ) ) ) ) ).

% vimage_Collect_eq
thf(fact_223_singletonI,axiom,
    ! [A2: b] : ( member_b @ A2 @ ( insert_b @ A2 @ bot_bot_set_b ) ) ).

% singletonI
thf(fact_224_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_225_singletonI,axiom,
    ! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_226_singletonI,axiom,
    ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_227_singletonI,axiom,
    ! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_228_singletonI,axiom,
    ! [A2: real > nat] : ( member_real_nat @ A2 @ ( insert_real_nat @ A2 @ bot_bot_set_real_nat ) ) ).

% singletonI
thf(fact_229_singletonI,axiom,
    ! [A2: real > b] : ( member_real_b @ A2 @ ( insert_real_b @ A2 @ bot_bot_set_real_b ) ) ).

% singletonI
thf(fact_230_singletonI,axiom,
    ! [A2: nat > real] : ( member_nat_real @ A2 @ ( insert_nat_real @ A2 @ bot_bot_set_nat_real ) ) ).

% singletonI
thf(fact_231_singletonI,axiom,
    ! [A2: real > set_nat] : ( member_real_set_nat @ A2 @ ( insert_real_set_nat @ A2 @ bot_bo6814059168456595739et_nat ) ) ).

% singletonI
thf(fact_232_singletonI,axiom,
    ! [A2: real > nat > real] : ( member_real_nat_real2 @ A2 @ ( insert_real_nat_real @ A2 @ bot_bo6533810469807102640t_real ) ) ).

% singletonI
thf(fact_233_Pi__empty,axiom,
    ! [B: real > set_b] :
      ( ( pi_real_b @ bot_bot_set_real @ B )
      = top_top_set_real_b ) ).

% Pi_empty
thf(fact_234_Pi__empty,axiom,
    ! [B: real > set_nat] :
      ( ( pi_real_nat @ bot_bot_set_real @ B )
      = top_top_set_real_nat ) ).

% Pi_empty
thf(fact_235_funcset__to__empty__iff,axiom,
    ! [A: set_real] :
      ( ( ( A = bot_bot_set_real )
       => ( ( pi_real_b @ A
            @ ^ [Uu: real] : bot_bot_set_b )
          = top_top_set_real_b ) )
      & ( ( A != bot_bot_set_real )
       => ( ( pi_real_b @ A
            @ ^ [Uu: real] : bot_bot_set_b )
          = bot_bot_set_real_b ) ) ) ).

% funcset_to_empty_iff
thf(fact_236_funcset__to__empty__iff,axiom,
    ! [A: set_nat] :
      ( ( ( A = bot_bot_set_nat )
       => ( ( pi_nat_nat @ A
            @ ^ [Uu: nat] : bot_bot_set_nat )
          = top_top_set_nat_nat ) )
      & ( ( A != bot_bot_set_nat )
       => ( ( pi_nat_nat @ A
            @ ^ [Uu: nat] : bot_bot_set_nat )
          = bot_bot_set_nat_nat ) ) ) ).

% funcset_to_empty_iff
thf(fact_237_funcset__to__empty__iff,axiom,
    ! [A: set_real] :
      ( ( ( A = bot_bot_set_real )
       => ( ( pi_real_nat @ A
            @ ^ [Uu: real] : bot_bot_set_nat )
          = top_top_set_real_nat ) )
      & ( ( A != bot_bot_set_real )
       => ( ( pi_real_nat @ A
            @ ^ [Uu: real] : bot_bot_set_nat )
          = bot_bot_set_real_nat ) ) ) ).

% funcset_to_empty_iff
thf(fact_238_funcset__to__empty__iff,axiom,
    ! [A: set_o] :
      ( ( ( A = bot_bot_set_o )
       => ( ( pi_o_nat @ A
            @ ^ [Uu: $o] : bot_bot_set_nat )
          = top_top_set_o_nat ) )
      & ( ( A != bot_bot_set_o )
       => ( ( pi_o_nat @ A
            @ ^ [Uu: $o] : bot_bot_set_nat )
          = bot_bot_set_o_nat ) ) ) ).

% funcset_to_empty_iff
thf(fact_239_funcset__to__empty__iff,axiom,
    ! [A: set_nat] :
      ( ( ( A = bot_bot_set_nat )
       => ( ( pi_nat_real @ A
            @ ^ [Uu: nat] : bot_bot_set_real )
          = top_top_set_nat_real ) )
      & ( ( A != bot_bot_set_nat )
       => ( ( pi_nat_real @ A
            @ ^ [Uu: nat] : bot_bot_set_real )
          = bot_bot_set_nat_real ) ) ) ).

% funcset_to_empty_iff
thf(fact_240_funcset__to__empty__iff,axiom,
    ! [A: set_real] :
      ( ( ( A = bot_bot_set_real )
       => ( ( pi_real_real @ A
            @ ^ [Uu: real] : bot_bot_set_real )
          = top_to2071711978144146653l_real ) )
      & ( ( A != bot_bot_set_real )
       => ( ( pi_real_real @ A
            @ ^ [Uu: real] : bot_bot_set_real )
          = bot_bo6767488733719836353l_real ) ) ) ).

% funcset_to_empty_iff
thf(fact_241_funcset__to__empty__iff,axiom,
    ! [A: set_o] :
      ( ( ( A = bot_bot_set_o )
       => ( ( pi_o_real @ A
            @ ^ [Uu: $o] : bot_bot_set_real )
          = top_top_set_o_real ) )
      & ( ( A != bot_bot_set_o )
       => ( ( pi_o_real @ A
            @ ^ [Uu: $o] : bot_bot_set_real )
          = bot_bot_set_o_real ) ) ) ).

% funcset_to_empty_iff
thf(fact_242_funcset__to__empty__iff,axiom,
    ! [A: set_nat] :
      ( ( ( A = bot_bot_set_nat )
       => ( ( pi_nat_o @ A
            @ ^ [Uu: nat] : bot_bot_set_o )
          = top_top_set_nat_o2 ) )
      & ( ( A != bot_bot_set_nat )
       => ( ( pi_nat_o @ A
            @ ^ [Uu: nat] : bot_bot_set_o )
          = bot_bot_set_nat_o2 ) ) ) ).

% funcset_to_empty_iff
thf(fact_243_funcset__to__empty__iff,axiom,
    ! [A: set_real] :
      ( ( ( A = bot_bot_set_real )
       => ( ( pi_real_o @ A
            @ ^ [Uu: real] : bot_bot_set_o )
          = top_top_set_real_o ) )
      & ( ( A != bot_bot_set_real )
       => ( ( pi_real_o @ A
            @ ^ [Uu: real] : bot_bot_set_o )
          = bot_bot_set_real_o ) ) ) ).

% funcset_to_empty_iff
thf(fact_244_funcset__to__empty__iff,axiom,
    ! [A: set_o] :
      ( ( ( A = bot_bot_set_o )
       => ( ( pi_o_o @ A
            @ ^ [Uu: $o] : bot_bot_set_o )
          = top_top_set_o_o ) )
      & ( ( A != bot_bot_set_o )
       => ( ( pi_o_o @ A
            @ ^ [Uu: $o] : bot_bot_set_o )
          = bot_bot_set_o_o ) ) ) ).

% funcset_to_empty_iff
thf(fact_245_UNIV__witness,axiom,
    ? [X3: b] : ( member_b @ X3 @ top_top_set_b ) ).

% UNIV_witness
thf(fact_246_UNIV__witness,axiom,
    ? [X3: real] : ( member_real @ X3 @ top_top_set_real ) ).

% UNIV_witness
thf(fact_247_UNIV__witness,axiom,
    ? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_248_UNIV__witness,axiom,
    ? [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ).

% UNIV_witness
thf(fact_249_UNIV__witness,axiom,
    ? [X3: set_nat] : ( member_set_nat @ X3 @ top_top_set_set_nat ) ).

% UNIV_witness
thf(fact_250_UNIV__witness,axiom,
    ? [X3: real > nat] : ( member_real_nat @ X3 @ top_top_set_real_nat ) ).

% UNIV_witness
thf(fact_251_UNIV__witness,axiom,
    ? [X3: real > b] : ( member_real_b @ X3 @ top_top_set_real_b ) ).

% UNIV_witness
thf(fact_252_UNIV__witness,axiom,
    ? [X3: nat > real] : ( member_nat_real @ X3 @ top_top_set_nat_real ) ).

% UNIV_witness
thf(fact_253_UNIV__witness,axiom,
    ? [X3: real > set_nat] : ( member_real_set_nat @ X3 @ top_to245300144855000375et_nat ) ).

% UNIV_witness
thf(fact_254_UNIV__witness,axiom,
    ? [X3: real > nat > real] : ( member_real_nat_real2 @ X3 @ top_to6902130281023745740t_real ) ).

% UNIV_witness
thf(fact_255_UNIV__eq__I,axiom,
    ! [A: set_b] :
      ( ! [X3: b] : ( member_b @ X3 @ A )
     => ( top_top_set_b = A ) ) ).

% UNIV_eq_I
thf(fact_256_UNIV__eq__I,axiom,
    ! [A: set_real] :
      ( ! [X3: real] : ( member_real @ X3 @ A )
     => ( top_top_set_real = A ) ) ).

% UNIV_eq_I
thf(fact_257_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] : ( member_nat @ X3 @ A )
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_258_UNIV__eq__I,axiom,
    ! [A: set_o] :
      ( ! [X3: $o] : ( member_o @ X3 @ A )
     => ( top_top_set_o = A ) ) ).

% UNIV_eq_I
thf(fact_259_UNIV__eq__I,axiom,
    ! [A: set_set_nat] :
      ( ! [X3: set_nat] : ( member_set_nat @ X3 @ A )
     => ( top_top_set_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_260_UNIV__eq__I,axiom,
    ! [A: set_real_nat] :
      ( ! [X3: real > nat] : ( member_real_nat @ X3 @ A )
     => ( top_top_set_real_nat = A ) ) ).

% UNIV_eq_I
thf(fact_261_UNIV__eq__I,axiom,
    ! [A: set_real_b] :
      ( ! [X3: real > b] : ( member_real_b @ X3 @ A )
     => ( top_top_set_real_b = A ) ) ).

% UNIV_eq_I
thf(fact_262_UNIV__eq__I,axiom,
    ! [A: set_nat_real] :
      ( ! [X3: nat > real] : ( member_nat_real @ X3 @ A )
     => ( top_top_set_nat_real = A ) ) ).

% UNIV_eq_I
thf(fact_263_UNIV__eq__I,axiom,
    ! [A: set_real_set_nat] :
      ( ! [X3: real > set_nat] : ( member_real_set_nat @ X3 @ A )
     => ( top_to245300144855000375et_nat = A ) ) ).

% UNIV_eq_I
thf(fact_264_UNIV__eq__I,axiom,
    ! [A: set_real_nat_real] :
      ( ! [X3: real > nat > real] : ( member_real_nat_real2 @ X3 @ A )
     => ( top_to6902130281023745740t_real = A ) ) ).

% UNIV_eq_I
thf(fact_265_ex__in__conv,axiom,
    ! [A: set_b] :
      ( ( ? [X5: b] : ( member_b @ X5 @ A ) )
      = ( A != bot_bot_set_b ) ) ).

% ex_in_conv
thf(fact_266_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X5: nat] : ( member_nat @ X5 @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_267_ex__in__conv,axiom,
    ! [A: set_real] :
      ( ( ? [X5: real] : ( member_real @ X5 @ A ) )
      = ( A != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_268_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ( ? [X5: $o] : ( member_o @ X5 @ A ) )
      = ( A != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_269_ex__in__conv,axiom,
    ! [A: set_set_nat] :
      ( ( ? [X5: set_nat] : ( member_set_nat @ X5 @ A ) )
      = ( A != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_270_ex__in__conv,axiom,
    ! [A: set_real_nat] :
      ( ( ? [X5: real > nat] : ( member_real_nat @ X5 @ A ) )
      = ( A != bot_bot_set_real_nat ) ) ).

% ex_in_conv
thf(fact_271_ex__in__conv,axiom,
    ! [A: set_real_b] :
      ( ( ? [X5: real > b] : ( member_real_b @ X5 @ A ) )
      = ( A != bot_bot_set_real_b ) ) ).

% ex_in_conv
thf(fact_272_ex__in__conv,axiom,
    ! [A: set_nat_real] :
      ( ( ? [X5: nat > real] : ( member_nat_real @ X5 @ A ) )
      = ( A != bot_bot_set_nat_real ) ) ).

% ex_in_conv
thf(fact_273_ex__in__conv,axiom,
    ! [A: set_real_set_nat] :
      ( ( ? [X5: real > set_nat] : ( member_real_set_nat @ X5 @ A ) )
      = ( A != bot_bo6814059168456595739et_nat ) ) ).

% ex_in_conv
thf(fact_274_ex__in__conv,axiom,
    ! [A: set_real_nat_real] :
      ( ( ? [X5: real > nat > real] : ( member_real_nat_real2 @ X5 @ A ) )
      = ( A != bot_bo6533810469807102640t_real ) ) ).

% ex_in_conv
thf(fact_275_equals0I,axiom,
    ! [A: set_b] :
      ( ! [Y4: b] :
          ~ ( member_b @ Y4 @ A )
     => ( A = bot_bot_set_b ) ) ).

% equals0I
thf(fact_276_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_277_equals0I,axiom,
    ! [A: set_real] :
      ( ! [Y4: real] :
          ~ ( member_real @ Y4 @ A )
     => ( A = bot_bot_set_real ) ) ).

% equals0I
thf(fact_278_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y4: $o] :
          ~ ( member_o @ Y4 @ A )
     => ( A = bot_bot_set_o ) ) ).

% equals0I
thf(fact_279_equals0I,axiom,
    ! [A: set_set_nat] :
      ( ! [Y4: set_nat] :
          ~ ( member_set_nat @ Y4 @ A )
     => ( A = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_280_equals0I,axiom,
    ! [A: set_real_nat] :
      ( ! [Y4: real > nat] :
          ~ ( member_real_nat @ Y4 @ A )
     => ( A = bot_bot_set_real_nat ) ) ).

% equals0I
thf(fact_281_equals0I,axiom,
    ! [A: set_real_b] :
      ( ! [Y4: real > b] :
          ~ ( member_real_b @ Y4 @ A )
     => ( A = bot_bot_set_real_b ) ) ).

% equals0I
thf(fact_282_equals0I,axiom,
    ! [A: set_nat_real] :
      ( ! [Y4: nat > real] :
          ~ ( member_nat_real @ Y4 @ A )
     => ( A = bot_bot_set_nat_real ) ) ).

% equals0I
thf(fact_283_equals0I,axiom,
    ! [A: set_real_set_nat] :
      ( ! [Y4: real > set_nat] :
          ~ ( member_real_set_nat @ Y4 @ A )
     => ( A = bot_bo6814059168456595739et_nat ) ) ).

% equals0I
thf(fact_284_equals0I,axiom,
    ! [A: set_real_nat_real] :
      ( ! [Y4: real > nat > real] :
          ~ ( member_real_nat_real2 @ Y4 @ A )
     => ( A = bot_bo6533810469807102640t_real ) ) ).

% equals0I
thf(fact_285_equals0D,axiom,
    ! [A: set_b,A2: b] :
      ( ( A = bot_bot_set_b )
     => ~ ( member_b @ A2 @ A ) ) ).

% equals0D
thf(fact_286_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_287_equals0D,axiom,
    ! [A: set_real,A2: real] :
      ( ( A = bot_bot_set_real )
     => ~ ( member_real @ A2 @ A ) ) ).

% equals0D
thf(fact_288_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ ( member_o @ A2 @ A ) ) ).

% equals0D
thf(fact_289_equals0D,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( A = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_290_equals0D,axiom,
    ! [A: set_real_nat,A2: real > nat] :
      ( ( A = bot_bot_set_real_nat )
     => ~ ( member_real_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_291_equals0D,axiom,
    ! [A: set_real_b,A2: real > b] :
      ( ( A = bot_bot_set_real_b )
     => ~ ( member_real_b @ A2 @ A ) ) ).

% equals0D
thf(fact_292_equals0D,axiom,
    ! [A: set_nat_real,A2: nat > real] :
      ( ( A = bot_bot_set_nat_real )
     => ~ ( member_nat_real @ A2 @ A ) ) ).

% equals0D
thf(fact_293_equals0D,axiom,
    ! [A: set_real_set_nat,A2: real > set_nat] :
      ( ( A = bot_bo6814059168456595739et_nat )
     => ~ ( member_real_set_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_294_equals0D,axiom,
    ! [A: set_real_nat_real,A2: real > nat > real] :
      ( ( A = bot_bo6533810469807102640t_real )
     => ~ ( member_real_nat_real2 @ A2 @ A ) ) ).

% equals0D
thf(fact_295_emptyE,axiom,
    ! [A2: b] :
      ~ ( member_b @ A2 @ bot_bot_set_b ) ).

% emptyE
thf(fact_296_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_297_emptyE,axiom,
    ! [A2: real] :
      ~ ( member_real @ A2 @ bot_bot_set_real ) ).

% emptyE
thf(fact_298_emptyE,axiom,
    ! [A2: $o] :
      ~ ( member_o @ A2 @ bot_bot_set_o ) ).

% emptyE
thf(fact_299_emptyE,axiom,
    ! [A2: set_nat] :
      ~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_300_emptyE,axiom,
    ! [A2: real > nat] :
      ~ ( member_real_nat @ A2 @ bot_bot_set_real_nat ) ).

% emptyE
thf(fact_301_emptyE,axiom,
    ! [A2: real > b] :
      ~ ( member_real_b @ A2 @ bot_bot_set_real_b ) ).

% emptyE
thf(fact_302_emptyE,axiom,
    ! [A2: nat > real] :
      ~ ( member_nat_real @ A2 @ bot_bot_set_nat_real ) ).

% emptyE
thf(fact_303_emptyE,axiom,
    ! [A2: real > set_nat] :
      ~ ( member_real_set_nat @ A2 @ bot_bo6814059168456595739et_nat ) ).

% emptyE
thf(fact_304_emptyE,axiom,
    ! [A2: real > nat > real] :
      ~ ( member_real_nat_real2 @ A2 @ bot_bo6533810469807102640t_real ) ).

% emptyE
thf(fact_305_mk__disjoint__insert,axiom,
    ! [A2: real,A: set_real] :
      ( ( member_real @ A2 @ A )
     => ? [B3: set_real] :
          ( ( A
            = ( insert_real @ A2 @ B3 ) )
          & ~ ( member_real @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_306_mk__disjoint__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ? [B3: set_o] :
          ( ( A
            = ( insert_o @ A2 @ B3 ) )
          & ~ ( member_o @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_307_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ? [B3: set_nat] :
          ( ( A
            = ( insert_nat @ A2 @ B3 ) )
          & ~ ( member_nat @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_308_mk__disjoint__insert,axiom,
    ! [A2: b,A: set_b] :
      ( ( member_b @ A2 @ A )
     => ? [B3: set_b] :
          ( ( A
            = ( insert_b @ A2 @ B3 ) )
          & ~ ( member_b @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_309_mk__disjoint__insert,axiom,
    ! [A2: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ A2 @ A )
     => ? [B3: set_set_nat] :
          ( ( A
            = ( insert_set_nat @ A2 @ B3 ) )
          & ~ ( member_set_nat @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_310_mk__disjoint__insert,axiom,
    ! [A2: real > nat,A: set_real_nat] :
      ( ( member_real_nat @ A2 @ A )
     => ? [B3: set_real_nat] :
          ( ( A
            = ( insert_real_nat @ A2 @ B3 ) )
          & ~ ( member_real_nat @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_311_mk__disjoint__insert,axiom,
    ! [A2: real > b,A: set_real_b] :
      ( ( member_real_b @ A2 @ A )
     => ? [B3: set_real_b] :
          ( ( A
            = ( insert_real_b @ A2 @ B3 ) )
          & ~ ( member_real_b @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_312_mk__disjoint__insert,axiom,
    ! [A2: nat > real,A: set_nat_real] :
      ( ( member_nat_real @ A2 @ A )
     => ? [B3: set_nat_real] :
          ( ( A
            = ( insert_nat_real @ A2 @ B3 ) )
          & ~ ( member_nat_real @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_313_mk__disjoint__insert,axiom,
    ! [A2: real > set_nat,A: set_real_set_nat] :
      ( ( member_real_set_nat @ A2 @ A )
     => ? [B3: set_real_set_nat] :
          ( ( A
            = ( insert_real_set_nat @ A2 @ B3 ) )
          & ~ ( member_real_set_nat @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_314_mk__disjoint__insert,axiom,
    ! [A2: real > nat > real,A: set_real_nat_real] :
      ( ( member_real_nat_real2 @ A2 @ A )
     => ? [B3: set_real_nat_real] :
          ( ( A
            = ( insert_real_nat_real @ A2 @ B3 ) )
          & ~ ( member_real_nat_real2 @ A2 @ B3 ) ) ) ).

% mk_disjoint_insert
thf(fact_315_insert__commute,axiom,
    ! [X: nat,Y: nat,A: set_nat] :
      ( ( insert_nat @ X @ ( insert_nat @ Y @ A ) )
      = ( insert_nat @ Y @ ( insert_nat @ X @ A ) ) ) ).

% insert_commute
thf(fact_316_insert__commute,axiom,
    ! [X: real,Y: real,A: set_real] :
      ( ( insert_real @ X @ ( insert_real @ Y @ A ) )
      = ( insert_real @ Y @ ( insert_real @ X @ A ) ) ) ).

% insert_commute
thf(fact_317_insert__commute,axiom,
    ! [X: $o,Y: $o,A: set_o] :
      ( ( insert_o @ X @ ( insert_o @ Y @ A ) )
      = ( insert_o @ Y @ ( insert_o @ X @ A ) ) ) ).

% insert_commute
thf(fact_318_insert__eq__iff,axiom,
    ! [A2: real,A: set_real,B2: real,B: set_real] :
      ( ~ ( member_real @ A2 @ A )
     => ( ~ ( member_real @ B2 @ B )
       => ( ( ( insert_real @ A2 @ A )
            = ( insert_real @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_real] :
                  ( ( A
                    = ( insert_real @ B2 @ C2 ) )
                  & ~ ( member_real @ B2 @ C2 )
                  & ( B
                    = ( insert_real @ A2 @ C2 ) )
                  & ~ ( member_real @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_319_insert__eq__iff,axiom,
    ! [A2: $o,A: set_o,B2: $o,B: set_o] :
      ( ~ ( member_o @ A2 @ A )
     => ( ~ ( member_o @ B2 @ B )
       => ( ( ( insert_o @ A2 @ A )
            = ( insert_o @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 = ~ B2 )
             => ? [C2: set_o] :
                  ( ( A
                    = ( insert_o @ B2 @ C2 ) )
                  & ~ ( member_o @ B2 @ C2 )
                  & ( B
                    = ( insert_o @ A2 @ C2 ) )
                  & ~ ( member_o @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_320_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ~ ( member_nat @ B2 @ B )
       => ( ( ( insert_nat @ A2 @ A )
            = ( insert_nat @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_nat] :
                  ( ( A
                    = ( insert_nat @ B2 @ C2 ) )
                  & ~ ( member_nat @ B2 @ C2 )
                  & ( B
                    = ( insert_nat @ A2 @ C2 ) )
                  & ~ ( member_nat @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_321_insert__eq__iff,axiom,
    ! [A2: b,A: set_b,B2: b,B: set_b] :
      ( ~ ( member_b @ A2 @ A )
     => ( ~ ( member_b @ B2 @ B )
       => ( ( ( insert_b @ A2 @ A )
            = ( insert_b @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_b] :
                  ( ( A
                    = ( insert_b @ B2 @ C2 ) )
                  & ~ ( member_b @ B2 @ C2 )
                  & ( B
                    = ( insert_b @ A2 @ C2 ) )
                  & ~ ( member_b @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_322_insert__eq__iff,axiom,
    ! [A2: set_nat,A: set_set_nat,B2: set_nat,B: set_set_nat] :
      ( ~ ( member_set_nat @ A2 @ A )
     => ( ~ ( member_set_nat @ B2 @ B )
       => ( ( ( insert_set_nat @ A2 @ A )
            = ( insert_set_nat @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_set_nat] :
                  ( ( A
                    = ( insert_set_nat @ B2 @ C2 ) )
                  & ~ ( member_set_nat @ B2 @ C2 )
                  & ( B
                    = ( insert_set_nat @ A2 @ C2 ) )
                  & ~ ( member_set_nat @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_323_insert__eq__iff,axiom,
    ! [A2: real > nat,A: set_real_nat,B2: real > nat,B: set_real_nat] :
      ( ~ ( member_real_nat @ A2 @ A )
     => ( ~ ( member_real_nat @ B2 @ B )
       => ( ( ( insert_real_nat @ A2 @ A )
            = ( insert_real_nat @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_real_nat] :
                  ( ( A
                    = ( insert_real_nat @ B2 @ C2 ) )
                  & ~ ( member_real_nat @ B2 @ C2 )
                  & ( B
                    = ( insert_real_nat @ A2 @ C2 ) )
                  & ~ ( member_real_nat @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_324_insert__eq__iff,axiom,
    ! [A2: real > b,A: set_real_b,B2: real > b,B: set_real_b] :
      ( ~ ( member_real_b @ A2 @ A )
     => ( ~ ( member_real_b @ B2 @ B )
       => ( ( ( insert_real_b @ A2 @ A )
            = ( insert_real_b @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_real_b] :
                  ( ( A
                    = ( insert_real_b @ B2 @ C2 ) )
                  & ~ ( member_real_b @ B2 @ C2 )
                  & ( B
                    = ( insert_real_b @ A2 @ C2 ) )
                  & ~ ( member_real_b @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_325_insert__eq__iff,axiom,
    ! [A2: nat > real,A: set_nat_real,B2: nat > real,B: set_nat_real] :
      ( ~ ( member_nat_real @ A2 @ A )
     => ( ~ ( member_nat_real @ B2 @ B )
       => ( ( ( insert_nat_real @ A2 @ A )
            = ( insert_nat_real @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_nat_real] :
                  ( ( A
                    = ( insert_nat_real @ B2 @ C2 ) )
                  & ~ ( member_nat_real @ B2 @ C2 )
                  & ( B
                    = ( insert_nat_real @ A2 @ C2 ) )
                  & ~ ( member_nat_real @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_326_insert__eq__iff,axiom,
    ! [A2: real > set_nat,A: set_real_set_nat,B2: real > set_nat,B: set_real_set_nat] :
      ( ~ ( member_real_set_nat @ A2 @ A )
     => ( ~ ( member_real_set_nat @ B2 @ B )
       => ( ( ( insert_real_set_nat @ A2 @ A )
            = ( insert_real_set_nat @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_real_set_nat] :
                  ( ( A
                    = ( insert_real_set_nat @ B2 @ C2 ) )
                  & ~ ( member_real_set_nat @ B2 @ C2 )
                  & ( B
                    = ( insert_real_set_nat @ A2 @ C2 ) )
                  & ~ ( member_real_set_nat @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_327_insert__eq__iff,axiom,
    ! [A2: real > nat > real,A: set_real_nat_real,B2: real > nat > real,B: set_real_nat_real] :
      ( ~ ( member_real_nat_real2 @ A2 @ A )
     => ( ~ ( member_real_nat_real2 @ B2 @ B )
       => ( ( ( insert_real_nat_real @ A2 @ A )
            = ( insert_real_nat_real @ B2 @ B ) )
          = ( ( ( A2 = B2 )
             => ( A = B ) )
            & ( ( A2 != B2 )
             => ? [C2: set_real_nat_real] :
                  ( ( A
                    = ( insert_real_nat_real @ B2 @ C2 ) )
                  & ~ ( member_real_nat_real2 @ B2 @ C2 )
                  & ( B
                    = ( insert_real_nat_real @ A2 @ C2 ) )
                  & ~ ( member_real_nat_real2 @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_328_insert__absorb,axiom,
    ! [A2: real,A: set_real] :
      ( ( member_real @ A2 @ A )
     => ( ( insert_real @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_329_insert__absorb,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( insert_o @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_330_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_331_insert__absorb,axiom,
    ! [A2: b,A: set_b] :
      ( ( member_b @ A2 @ A )
     => ( ( insert_b @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_332_insert__absorb,axiom,
    ! [A2: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ A2 @ A )
     => ( ( insert_set_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_333_insert__absorb,axiom,
    ! [A2: real > nat,A: set_real_nat] :
      ( ( member_real_nat @ A2 @ A )
     => ( ( insert_real_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_334_insert__absorb,axiom,
    ! [A2: real > b,A: set_real_b] :
      ( ( member_real_b @ A2 @ A )
     => ( ( insert_real_b @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_335_insert__absorb,axiom,
    ! [A2: nat > real,A: set_nat_real] :
      ( ( member_nat_real @ A2 @ A )
     => ( ( insert_nat_real @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_336_insert__absorb,axiom,
    ! [A2: real > set_nat,A: set_real_set_nat] :
      ( ( member_real_set_nat @ A2 @ A )
     => ( ( insert_real_set_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_337_insert__absorb,axiom,
    ! [A2: real > nat > real,A: set_real_nat_real] :
      ( ( member_real_nat_real2 @ A2 @ A )
     => ( ( insert_real_nat_real @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_338_insert__ident,axiom,
    ! [X: real,A: set_real,B: set_real] :
      ( ~ ( member_real @ X @ A )
     => ( ~ ( member_real @ X @ B )
       => ( ( ( insert_real @ X @ A )
            = ( insert_real @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_339_insert__ident,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ~ ( member_o @ X @ B )
       => ( ( ( insert_o @ X @ A )
            = ( insert_o @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_340_insert__ident,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ~ ( member_nat @ X @ B )
       => ( ( ( insert_nat @ X @ A )
            = ( insert_nat @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_341_insert__ident,axiom,
    ! [X: b,A: set_b,B: set_b] :
      ( ~ ( member_b @ X @ A )
     => ( ~ ( member_b @ X @ B )
       => ( ( ( insert_b @ X @ A )
            = ( insert_b @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_342_insert__ident,axiom,
    ! [X: set_nat,A: set_set_nat,B: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A )
     => ( ~ ( member_set_nat @ X @ B )
       => ( ( ( insert_set_nat @ X @ A )
            = ( insert_set_nat @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_343_insert__ident,axiom,
    ! [X: real > nat,A: set_real_nat,B: set_real_nat] :
      ( ~ ( member_real_nat @ X @ A )
     => ( ~ ( member_real_nat @ X @ B )
       => ( ( ( insert_real_nat @ X @ A )
            = ( insert_real_nat @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_344_insert__ident,axiom,
    ! [X: real > b,A: set_real_b,B: set_real_b] :
      ( ~ ( member_real_b @ X @ A )
     => ( ~ ( member_real_b @ X @ B )
       => ( ( ( insert_real_b @ X @ A )
            = ( insert_real_b @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_345_insert__ident,axiom,
    ! [X: nat > real,A: set_nat_real,B: set_nat_real] :
      ( ~ ( member_nat_real @ X @ A )
     => ( ~ ( member_nat_real @ X @ B )
       => ( ( ( insert_nat_real @ X @ A )
            = ( insert_nat_real @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_346_insert__ident,axiom,
    ! [X: real > set_nat,A: set_real_set_nat,B: set_real_set_nat] :
      ( ~ ( member_real_set_nat @ X @ A )
     => ( ~ ( member_real_set_nat @ X @ B )
       => ( ( ( insert_real_set_nat @ X @ A )
            = ( insert_real_set_nat @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_347_insert__ident,axiom,
    ! [X: real > nat > real,A: set_real_nat_real,B: set_real_nat_real] :
      ( ~ ( member_real_nat_real2 @ X @ A )
     => ( ~ ( member_real_nat_real2 @ X @ B )
       => ( ( ( insert_real_nat_real @ X @ A )
            = ( insert_real_nat_real @ X @ B ) )
          = ( A = B ) ) ) ) ).

% insert_ident
thf(fact_348_Set_Oset__insert,axiom,
    ! [X: real,A: set_real] :
      ( ( member_real @ X @ A )
     => ~ ! [B3: set_real] :
            ( ( A
              = ( insert_real @ X @ B3 ) )
           => ( member_real @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_349_Set_Oset__insert,axiom,
    ! [X: $o,A: set_o] :
      ( ( member_o @ X @ A )
     => ~ ! [B3: set_o] :
            ( ( A
              = ( insert_o @ X @ B3 ) )
           => ( member_o @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_350_Set_Oset__insert,axiom,
    ! [X: nat,A: set_nat] :
      ( ( member_nat @ X @ A )
     => ~ ! [B3: set_nat] :
            ( ( A
              = ( insert_nat @ X @ B3 ) )
           => ( member_nat @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_351_Set_Oset__insert,axiom,
    ! [X: b,A: set_b] :
      ( ( member_b @ X @ A )
     => ~ ! [B3: set_b] :
            ( ( A
              = ( insert_b @ X @ B3 ) )
           => ( member_b @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_352_Set_Oset__insert,axiom,
    ! [X: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ X @ A )
     => ~ ! [B3: set_set_nat] :
            ( ( A
              = ( insert_set_nat @ X @ B3 ) )
           => ( member_set_nat @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_353_Set_Oset__insert,axiom,
    ! [X: real > nat,A: set_real_nat] :
      ( ( member_real_nat @ X @ A )
     => ~ ! [B3: set_real_nat] :
            ( ( A
              = ( insert_real_nat @ X @ B3 ) )
           => ( member_real_nat @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_354_Set_Oset__insert,axiom,
    ! [X: real > b,A: set_real_b] :
      ( ( member_real_b @ X @ A )
     => ~ ! [B3: set_real_b] :
            ( ( A
              = ( insert_real_b @ X @ B3 ) )
           => ( member_real_b @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_355_Set_Oset__insert,axiom,
    ! [X: nat > real,A: set_nat_real] :
      ( ( member_nat_real @ X @ A )
     => ~ ! [B3: set_nat_real] :
            ( ( A
              = ( insert_nat_real @ X @ B3 ) )
           => ( member_nat_real @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_356_Set_Oset__insert,axiom,
    ! [X: real > set_nat,A: set_real_set_nat] :
      ( ( member_real_set_nat @ X @ A )
     => ~ ! [B3: set_real_set_nat] :
            ( ( A
              = ( insert_real_set_nat @ X @ B3 ) )
           => ( member_real_set_nat @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_357_Set_Oset__insert,axiom,
    ! [X: real > nat > real,A: set_real_nat_real] :
      ( ( member_real_nat_real2 @ X @ A )
     => ~ ! [B3: set_real_nat_real] :
            ( ( A
              = ( insert_real_nat_real @ X @ B3 ) )
           => ( member_real_nat_real2 @ X @ B3 ) ) ) ).

% Set.set_insert
thf(fact_358_insertI2,axiom,
    ! [A2: real,B: set_real,B2: real] :
      ( ( member_real @ A2 @ B )
     => ( member_real @ A2 @ ( insert_real @ B2 @ B ) ) ) ).

% insertI2
thf(fact_359_insertI2,axiom,
    ! [A2: $o,B: set_o,B2: $o] :
      ( ( member_o @ A2 @ B )
     => ( member_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).

% insertI2
thf(fact_360_insertI2,axiom,
    ! [A2: nat,B: set_nat,B2: nat] :
      ( ( member_nat @ A2 @ B )
     => ( member_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_361_insertI2,axiom,
    ! [A2: b,B: set_b,B2: b] :
      ( ( member_b @ A2 @ B )
     => ( member_b @ A2 @ ( insert_b @ B2 @ B ) ) ) ).

% insertI2
thf(fact_362_insertI2,axiom,
    ! [A2: set_nat,B: set_set_nat,B2: set_nat] :
      ( ( member_set_nat @ A2 @ B )
     => ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_363_insertI2,axiom,
    ! [A2: real > nat,B: set_real_nat,B2: real > nat] :
      ( ( member_real_nat @ A2 @ B )
     => ( member_real_nat @ A2 @ ( insert_real_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_364_insertI2,axiom,
    ! [A2: real > b,B: set_real_b,B2: real > b] :
      ( ( member_real_b @ A2 @ B )
     => ( member_real_b @ A2 @ ( insert_real_b @ B2 @ B ) ) ) ).

% insertI2
thf(fact_365_insertI2,axiom,
    ! [A2: nat > real,B: set_nat_real,B2: nat > real] :
      ( ( member_nat_real @ A2 @ B )
     => ( member_nat_real @ A2 @ ( insert_nat_real @ B2 @ B ) ) ) ).

% insertI2
thf(fact_366_insertI2,axiom,
    ! [A2: real > set_nat,B: set_real_set_nat,B2: real > set_nat] :
      ( ( member_real_set_nat @ A2 @ B )
     => ( member_real_set_nat @ A2 @ ( insert_real_set_nat @ B2 @ B ) ) ) ).

% insertI2
thf(fact_367_insertI2,axiom,
    ! [A2: real > nat > real,B: set_real_nat_real,B2: real > nat > real] :
      ( ( member_real_nat_real2 @ A2 @ B )
     => ( member_real_nat_real2 @ A2 @ ( insert_real_nat_real @ B2 @ B ) ) ) ).

% insertI2
thf(fact_368_insertI1,axiom,
    ! [A2: real,B: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B ) ) ).

% insertI1
thf(fact_369_insertI1,axiom,
    ! [A2: $o,B: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B ) ) ).

% insertI1
thf(fact_370_insertI1,axiom,
    ! [A2: nat,B: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B ) ) ).

% insertI1
thf(fact_371_insertI1,axiom,
    ! [A2: b,B: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B ) ) ).

% insertI1
thf(fact_372_insertI1,axiom,
    ! [A2: set_nat,B: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B ) ) ).

% insertI1
thf(fact_373_insertI1,axiom,
    ! [A2: real > nat,B: set_real_nat] : ( member_real_nat @ A2 @ ( insert_real_nat @ A2 @ B ) ) ).

% insertI1
thf(fact_374_insertI1,axiom,
    ! [A2: real > b,B: set_real_b] : ( member_real_b @ A2 @ ( insert_real_b @ A2 @ B ) ) ).

% insertI1
thf(fact_375_insertI1,axiom,
    ! [A2: nat > real,B: set_nat_real] : ( member_nat_real @ A2 @ ( insert_nat_real @ A2 @ B ) ) ).

% insertI1
thf(fact_376_insertI1,axiom,
    ! [A2: real > set_nat,B: set_real_set_nat] : ( member_real_set_nat @ A2 @ ( insert_real_set_nat @ A2 @ B ) ) ).

% insertI1
thf(fact_377_insertI1,axiom,
    ! [A2: real > nat > real,B: set_real_nat_real] : ( member_real_nat_real2 @ A2 @ ( insert_real_nat_real @ A2 @ B ) ) ).

% insertI1
thf(fact_378_insertE,axiom,
    ! [A2: real,B2: real,A: set_real] :
      ( ( member_real @ A2 @ ( insert_real @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_real @ A2 @ A ) ) ) ).

% insertE
thf(fact_379_insertE,axiom,
    ! [A2: $o,B2: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B2 @ A ) )
     => ( ( A2 = ~ B2 )
       => ( member_o @ A2 @ A ) ) ) ).

% insertE
thf(fact_380_insertE,axiom,
    ! [A2: nat,B2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_381_insertE,axiom,
    ! [A2: b,B2: b,A: set_b] :
      ( ( member_b @ A2 @ ( insert_b @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_b @ A2 @ A ) ) ) ).

% insertE
thf(fact_382_insertE,axiom,
    ! [A2: set_nat,B2: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ A2 @ ( insert_set_nat @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_set_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_383_insertE,axiom,
    ! [A2: real > nat,B2: real > nat,A: set_real_nat] :
      ( ( member_real_nat @ A2 @ ( insert_real_nat @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_real_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_384_insertE,axiom,
    ! [A2: real > b,B2: real > b,A: set_real_b] :
      ( ( member_real_b @ A2 @ ( insert_real_b @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_real_b @ A2 @ A ) ) ) ).

% insertE
thf(fact_385_insertE,axiom,
    ! [A2: nat > real,B2: nat > real,A: set_nat_real] :
      ( ( member_nat_real @ A2 @ ( insert_nat_real @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_nat_real @ A2 @ A ) ) ) ).

% insertE
thf(fact_386_insertE,axiom,
    ! [A2: real > set_nat,B2: real > set_nat,A: set_real_set_nat] :
      ( ( member_real_set_nat @ A2 @ ( insert_real_set_nat @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_real_set_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_387_insertE,axiom,
    ! [A2: real > nat > real,B2: real > nat > real,A: set_real_nat_real] :
      ( ( member_real_nat_real2 @ A2 @ ( insert_real_nat_real @ B2 @ A ) )
     => ( ( A2 != B2 )
       => ( member_real_nat_real2 @ A2 @ A ) ) ) ).

% insertE
thf(fact_388_vimage__Collect,axiom,
    ! [P: nat > $o,F: real > nat,Q: real > $o] :
      ( ! [X3: real] :
          ( ( P @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage_real_nat @ F @ ( collect_nat @ P ) )
        = ( collect_real @ Q ) ) ) ).

% vimage_Collect
thf(fact_389_vimage__Collect,axiom,
    ! [P: nat > $o,F: nat > nat,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ ( F @ X3 ) )
          = ( Q @ X3 ) )
     => ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
        = ( collect_nat @ Q ) ) ) ).

% vimage_Collect
thf(fact_390_vimageI2,axiom,
    ! [F: nat > nat,A2: nat,A: set_nat] :
      ( ( member_nat @ ( F @ A2 ) @ A )
     => ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) ) ) ).

% vimageI2
thf(fact_391_vimageI2,axiom,
    ! [F: b > nat,A2: b,A: set_nat] :
      ( ( member_nat @ ( F @ A2 ) @ A )
     => ( member_b @ A2 @ ( vimage_b_nat @ F @ A ) ) ) ).

% vimageI2
thf(fact_392_vimageI2,axiom,
    ! [F: nat > b,A2: nat,A: set_b] :
      ( ( member_b @ ( F @ A2 ) @ A )
     => ( member_nat @ A2 @ ( vimage_nat_b @ F @ A ) ) ) ).

% vimageI2
thf(fact_393_vimageI2,axiom,
    ! [F: b > b,A2: b,A: set_b] :
      ( ( member_b @ ( F @ A2 ) @ A )
     => ( member_b @ A2 @ ( vimage_b_b @ F @ A ) ) ) ).

% vimageI2
thf(fact_394_vimageI2,axiom,
    ! [F: real > nat,A2: real,A: set_nat] :
      ( ( member_nat @ ( F @ A2 ) @ A )
     => ( member_real @ A2 @ ( vimage_real_nat @ F @ A ) ) ) ).

% vimageI2
thf(fact_395_vimageI2,axiom,
    ! [F: nat > set_nat,A2: nat,A: set_set_nat] :
      ( ( member_set_nat @ ( F @ A2 ) @ A )
     => ( member_nat @ A2 @ ( vimage_nat_set_nat @ F @ A ) ) ) ).

% vimageI2
thf(fact_396_vimageI2,axiom,
    ! [F: b > set_nat,A2: b,A: set_set_nat] :
      ( ( member_set_nat @ ( F @ A2 ) @ A )
     => ( member_b @ A2 @ ( vimage_b_set_nat @ F @ A ) ) ) ).

% vimageI2
thf(fact_397_vimageI2,axiom,
    ! [F: set_nat > nat,A2: set_nat,A: set_nat] :
      ( ( member_nat @ ( F @ A2 ) @ A )
     => ( member_set_nat @ A2 @ ( vimage_set_nat_nat @ F @ A ) ) ) ).

% vimageI2
thf(fact_398_vimageI2,axiom,
    ! [F: set_nat > b,A2: set_nat,A: set_b] :
      ( ( member_b @ ( F @ A2 ) @ A )
     => ( member_set_nat @ A2 @ ( vimage_set_nat_b @ F @ A ) ) ) ).

% vimageI2
thf(fact_399_vimageI2,axiom,
    ! [F: set_nat > set_nat,A2: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ ( F @ A2 ) @ A )
     => ( member_set_nat @ A2 @ ( vimage4765135879290611145et_nat @ F @ A ) ) ) ).

% vimageI2
thf(fact_400_vimageE,axiom,
    ! [A2: nat,F: nat > nat,B: set_nat] :
      ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
     => ( member_nat @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_401_vimageE,axiom,
    ! [A2: nat,F: nat > b,B: set_b] :
      ( ( member_nat @ A2 @ ( vimage_nat_b @ F @ B ) )
     => ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_402_vimageE,axiom,
    ! [A2: b,F: b > nat,B: set_nat] :
      ( ( member_b @ A2 @ ( vimage_b_nat @ F @ B ) )
     => ( member_nat @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_403_vimageE,axiom,
    ! [A2: b,F: b > b,B: set_b] :
      ( ( member_b @ A2 @ ( vimage_b_b @ F @ B ) )
     => ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_404_vimageE,axiom,
    ! [A2: real,F: real > nat,B: set_nat] :
      ( ( member_real @ A2 @ ( vimage_real_nat @ F @ B ) )
     => ( member_nat @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_405_vimageE,axiom,
    ! [A2: set_nat,F: set_nat > nat,B: set_nat] :
      ( ( member_set_nat @ A2 @ ( vimage_set_nat_nat @ F @ B ) )
     => ( member_nat @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_406_vimageE,axiom,
    ! [A2: set_nat,F: set_nat > b,B: set_b] :
      ( ( member_set_nat @ A2 @ ( vimage_set_nat_b @ F @ B ) )
     => ( member_b @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_407_vimageE,axiom,
    ! [A2: nat,F: nat > set_nat,B: set_set_nat] :
      ( ( member_nat @ A2 @ ( vimage_nat_set_nat @ F @ B ) )
     => ( member_set_nat @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_408_vimageE,axiom,
    ! [A2: b,F: b > set_nat,B: set_set_nat] :
      ( ( member_b @ A2 @ ( vimage_b_set_nat @ F @ B ) )
     => ( member_set_nat @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_409_vimageE,axiom,
    ! [A2: set_nat,F: set_nat > set_nat,B: set_set_nat] :
      ( ( member_set_nat @ A2 @ ( vimage4765135879290611145et_nat @ F @ B ) )
     => ( member_set_nat @ ( F @ A2 ) @ B ) ) ).

% vimageE
thf(fact_410_vimageD,axiom,
    ! [A2: nat,F: nat > nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) )
     => ( member_nat @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_411_vimageD,axiom,
    ! [A2: nat,F: nat > b,A: set_b] :
      ( ( member_nat @ A2 @ ( vimage_nat_b @ F @ A ) )
     => ( member_b @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_412_vimageD,axiom,
    ! [A2: b,F: b > nat,A: set_nat] :
      ( ( member_b @ A2 @ ( vimage_b_nat @ F @ A ) )
     => ( member_nat @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_413_vimageD,axiom,
    ! [A2: b,F: b > b,A: set_b] :
      ( ( member_b @ A2 @ ( vimage_b_b @ F @ A ) )
     => ( member_b @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_414_vimageD,axiom,
    ! [A2: real,F: real > nat,A: set_nat] :
      ( ( member_real @ A2 @ ( vimage_real_nat @ F @ A ) )
     => ( member_nat @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_415_vimageD,axiom,
    ! [A2: set_nat,F: set_nat > nat,A: set_nat] :
      ( ( member_set_nat @ A2 @ ( vimage_set_nat_nat @ F @ A ) )
     => ( member_nat @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_416_vimageD,axiom,
    ! [A2: set_nat,F: set_nat > b,A: set_b] :
      ( ( member_set_nat @ A2 @ ( vimage_set_nat_b @ F @ A ) )
     => ( member_b @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_417_vimageD,axiom,
    ! [A2: nat,F: nat > set_nat,A: set_set_nat] :
      ( ( member_nat @ A2 @ ( vimage_nat_set_nat @ F @ A ) )
     => ( member_set_nat @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_418_vimageD,axiom,
    ! [A2: b,F: b > set_nat,A: set_set_nat] :
      ( ( member_b @ A2 @ ( vimage_b_set_nat @ F @ A ) )
     => ( member_set_nat @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_419_vimageD,axiom,
    ! [A2: set_nat,F: set_nat > set_nat,A: set_set_nat] :
      ( ( member_set_nat @ A2 @ ( vimage4765135879290611145et_nat @ F @ A ) )
     => ( member_set_nat @ ( F @ A2 ) @ A ) ) ).

% vimageD
thf(fact_420_Pi__cong,axiom,
    ! [A: set_real,F: real > nat,G: real > nat,B: real > set_nat] :
      ( ! [W: real] :
          ( ( member_real @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_nat @ F @ ( pi_real_nat @ A @ B ) )
        = ( member_real_nat @ G @ ( pi_real_nat @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_421_Pi__cong,axiom,
    ! [A: set_real,F: real > b,G: real > b,B: real > set_b] :
      ( ! [W: real] :
          ( ( member_real @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_b @ F @ ( pi_real_b @ A @ B ) )
        = ( member_real_b @ G @ ( pi_real_b @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_422_Pi__cong,axiom,
    ! [A: set_real,F: real > nat > real,G: real > nat > real,B: real > set_nat_real] :
      ( ! [W: real] :
          ( ( member_real @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_nat_real2 @ F @ ( pi_real_nat_real @ A @ B ) )
        = ( member_real_nat_real2 @ G @ ( pi_real_nat_real @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_423_Pi__cong,axiom,
    ! [A: set_real,F: real > real > b,G: real > real > b,B: real > set_real_b] :
      ( ! [W: real] :
          ( ( member_real @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_real_b @ F @ ( pi_real_real_b @ A @ B ) )
        = ( member_real_real_b @ G @ ( pi_real_real_b @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_424_Pi__cong,axiom,
    ! [A: set_real,F: real > real > nat,G: real > real > nat,B: real > set_real_nat] :
      ( ! [W: real] :
          ( ( member_real @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_real_nat @ F @ ( pi_real_real_nat @ A @ B ) )
        = ( member_real_real_nat @ G @ ( pi_real_real_nat @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_425_Pi__cong,axiom,
    ! [A: set_real,F: real > set_nat,G: real > set_nat,B: real > set_set_nat] :
      ( ! [W: real] :
          ( ( member_real @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_set_nat @ F @ ( pi_real_set_nat @ A @ B ) )
        = ( member_real_set_nat @ G @ ( pi_real_set_nat @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_426_Pi__cong,axiom,
    ! [A: set_nat,F: nat > real,G: nat > real,B: nat > set_real] :
      ( ! [W: nat] :
          ( ( member_nat @ W @ A )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_nat_real @ F @ ( pi_nat_real @ A @ B ) )
        = ( member_nat_real @ G @ ( pi_nat_real @ A @ B ) ) ) ) ).

% Pi_cong
thf(fact_427_Pi__mem,axiom,
    ! [F: nat > nat,A: set_nat,B: nat > set_nat,X: nat] :
      ( ( member_nat_nat @ F @ ( pi_nat_nat @ A @ B ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_428_Pi__mem,axiom,
    ! [F: nat > b,A: set_nat,B: nat > set_b,X: nat] :
      ( ( member_nat_b @ F @ ( pi_nat_b @ A @ B ) )
     => ( ( member_nat @ X @ A )
       => ( member_b @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_429_Pi__mem,axiom,
    ! [F: b > nat,A: set_b,B: b > set_nat,X: b] :
      ( ( member_b_nat @ F @ ( pi_b_nat @ A @ B ) )
     => ( ( member_b @ X @ A )
       => ( member_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_430_Pi__mem,axiom,
    ! [F: b > b,A: set_b,B: b > set_b,X: b] :
      ( ( member_b_b @ F @ ( pi_b_b @ A @ B ) )
     => ( ( member_b @ X @ A )
       => ( member_b @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_431_Pi__mem,axiom,
    ! [F: real > nat,A: set_real,B: real > set_nat,X: real] :
      ( ( member_real_nat @ F @ ( pi_real_nat @ A @ B ) )
     => ( ( member_real @ X @ A )
       => ( member_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_432_Pi__mem,axiom,
    ! [F: real > b,A: set_real,B: real > set_b,X: real] :
      ( ( member_real_b @ F @ ( pi_real_b @ A @ B ) )
     => ( ( member_real @ X @ A )
       => ( member_b @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_433_Pi__mem,axiom,
    ! [F: nat > real,A: set_nat,B: nat > set_real,X: nat] :
      ( ( member_nat_real @ F @ ( pi_nat_real @ A @ B ) )
     => ( ( member_nat @ X @ A )
       => ( member_real @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_434_Pi__mem,axiom,
    ! [F: set_nat > nat,A: set_set_nat,B: set_nat > set_nat,X: set_nat] :
      ( ( member_set_nat_nat @ F @ ( pi_set_nat_nat @ A @ B ) )
     => ( ( member_set_nat @ X @ A )
       => ( member_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_435_Pi__mem,axiom,
    ! [F: set_nat > b,A: set_set_nat,B: set_nat > set_b,X: set_nat] :
      ( ( member_set_nat_b @ F @ ( pi_set_nat_b @ A @ B ) )
     => ( ( member_set_nat @ X @ A )
       => ( member_b @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_436_Pi__mem,axiom,
    ! [F: nat > set_nat,A: set_nat,B: nat > set_set_nat,X: nat] :
      ( ( member_nat_set_nat @ F @ ( pi_nat_set_nat @ A @ B ) )
     => ( ( member_nat @ X @ A )
       => ( member_set_nat @ ( F @ X ) @ ( B @ X ) ) ) ) ).

% Pi_mem
thf(fact_437_Pi__iff,axiom,
    ! [F: real > nat,I2: set_real,X2: real > set_nat] :
      ( ( member_real_nat @ F @ ( pi_real_nat @ I2 @ X2 ) )
      = ( ! [X5: real] :
            ( ( member_real @ X5 @ I2 )
           => ( member_nat @ ( F @ X5 ) @ ( X2 @ X5 ) ) ) ) ) ).

% Pi_iff
thf(fact_438_Pi__iff,axiom,
    ! [F: real > b,I2: set_real,X2: real > set_b] :
      ( ( member_real_b @ F @ ( pi_real_b @ I2 @ X2 ) )
      = ( ! [X5: real] :
            ( ( member_real @ X5 @ I2 )
           => ( member_b @ ( F @ X5 ) @ ( X2 @ X5 ) ) ) ) ) ).

% Pi_iff
thf(fact_439_Pi__iff,axiom,
    ! [F: nat > real,I2: set_nat,X2: nat > set_real] :
      ( ( member_nat_real @ F @ ( pi_nat_real @ I2 @ X2 ) )
      = ( ! [X5: nat] :
            ( ( member_nat @ X5 @ I2 )
           => ( member_real @ ( F @ X5 ) @ ( X2 @ X5 ) ) ) ) ) ).

% Pi_iff
thf(fact_440_Pi__iff,axiom,
    ! [F: real > nat > real,I2: set_real,X2: real > set_nat_real] :
      ( ( member_real_nat_real2 @ F @ ( pi_real_nat_real @ I2 @ X2 ) )
      = ( ! [X5: real] :
            ( ( member_real @ X5 @ I2 )
           => ( member_nat_real @ ( F @ X5 ) @ ( X2 @ X5 ) ) ) ) ) ).

% Pi_iff
thf(fact_441_Pi__iff,axiom,
    ! [F: real > real > b,I2: set_real,X2: real > set_real_b] :
      ( ( member_real_real_b @ F @ ( pi_real_real_b @ I2 @ X2 ) )
      = ( ! [X5: real] :
            ( ( member_real @ X5 @ I2 )
           => ( member_real_b @ ( F @ X5 ) @ ( X2 @ X5 ) ) ) ) ) ).

% Pi_iff
thf(fact_442_Pi__iff,axiom,
    ! [F: real > real > nat,I2: set_real,X2: real > set_real_nat] :
      ( ( member_real_real_nat @ F @ ( pi_real_real_nat @ I2 @ X2 ) )
      = ( ! [X5: real] :
            ( ( member_real @ X5 @ I2 )
           => ( member_real_nat @ ( F @ X5 ) @ ( X2 @ X5 ) ) ) ) ) ).

% Pi_iff
thf(fact_443_Pi__iff,axiom,
    ! [F: real > set_nat,I2: set_real,X2: real > set_set_nat] :
      ( ( member_real_set_nat @ F @ ( pi_real_set_nat @ I2 @ X2 ) )
      = ( ! [X5: real] :
            ( ( member_real @ X5 @ I2 )
           => ( member_set_nat @ ( F @ X5 ) @ ( X2 @ X5 ) ) ) ) ) ).

% Pi_iff
thf(fact_444_Pi__I_H,axiom,
    ! [A: set_real,F: real > nat,B: real > set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_real_nat @ F @ ( pi_real_nat @ A @ B ) ) ) ).

% Pi_I'
thf(fact_445_Pi__I_H,axiom,
    ! [A: set_real,F: real > b,B: real > set_b] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_real_b @ F @ ( pi_real_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_446_Pi__I_H,axiom,
    ! [A: set_nat,F: nat > real,B: nat > set_real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_real @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_real @ F @ ( pi_nat_real @ A @ B ) ) ) ).

% Pi_I'
thf(fact_447_Pi__I_H,axiom,
    ! [A: set_nat,F: nat > nat,B: nat > set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_nat @ F @ ( pi_nat_nat @ A @ B ) ) ) ).

% Pi_I'
thf(fact_448_Pi__I_H,axiom,
    ! [A: set_nat,F: nat > b,B: nat > set_b] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_nat_b @ F @ ( pi_nat_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_449_Pi__I_H,axiom,
    ! [A: set_b,F: b > nat,B: b > set_nat] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_b_nat @ F @ ( pi_b_nat @ A @ B ) ) ) ).

% Pi_I'
thf(fact_450_Pi__I_H,axiom,
    ! [A: set_b,F: b > b,B: b > set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_b_b @ F @ ( pi_b_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_451_Pi__I_H,axiom,
    ! [A: set_real,F: real > set_nat,B: real > set_set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_set_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_real_set_nat @ F @ ( pi_real_set_nat @ A @ B ) ) ) ).

% Pi_I'
thf(fact_452_Pi__I_H,axiom,
    ! [A: set_set_nat,F: set_nat > nat,B: set_nat > set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_nat_nat @ F @ ( pi_set_nat_nat @ A @ B ) ) ) ).

% Pi_I'
thf(fact_453_Pi__I_H,axiom,
    ! [A: set_set_nat,F: set_nat > b,B: set_nat > set_b] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ ( B @ X3 ) ) )
     => ( member_set_nat_b @ F @ ( pi_set_nat_b @ A @ B ) ) ) ).

% Pi_I'
thf(fact_454_PiE,axiom,
    ! [F: nat > nat,A: set_nat,B: nat > set_nat,X: nat] :
      ( ( member_nat_nat @ F @ ( pi_nat_nat @ A @ B ) )
     => ( ~ ( member_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat @ X @ A ) ) ) ).

% PiE
thf(fact_455_PiE,axiom,
    ! [F: b > nat,A: set_b,B: b > set_nat,X: b] :
      ( ( member_b_nat @ F @ ( pi_b_nat @ A @ B ) )
     => ( ~ ( member_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_b @ X @ A ) ) ) ).

% PiE
thf(fact_456_PiE,axiom,
    ! [F: nat > b,A: set_nat,B: nat > set_b,X: nat] :
      ( ( member_nat_b @ F @ ( pi_nat_b @ A @ B ) )
     => ( ~ ( member_b @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat @ X @ A ) ) ) ).

% PiE
thf(fact_457_PiE,axiom,
    ! [F: b > b,A: set_b,B: b > set_b,X: b] :
      ( ( member_b_b @ F @ ( pi_b_b @ A @ B ) )
     => ( ~ ( member_b @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_b @ X @ A ) ) ) ).

% PiE
thf(fact_458_PiE,axiom,
    ! [F: real > nat,A: set_real,B: real > set_nat,X: real] :
      ( ( member_real_nat @ F @ ( pi_real_nat @ A @ B ) )
     => ( ~ ( member_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_real @ X @ A ) ) ) ).

% PiE
thf(fact_459_PiE,axiom,
    ! [F: real > b,A: set_real,B: real > set_b,X: real] :
      ( ( member_real_b @ F @ ( pi_real_b @ A @ B ) )
     => ( ~ ( member_b @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_real @ X @ A ) ) ) ).

% PiE
thf(fact_460_PiE,axiom,
    ! [F: nat > real,A: set_nat,B: nat > set_real,X: nat] :
      ( ( member_nat_real @ F @ ( pi_nat_real @ A @ B ) )
     => ( ~ ( member_real @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat @ X @ A ) ) ) ).

% PiE
thf(fact_461_PiE,axiom,
    ! [F: nat > set_nat,A: set_nat,B: nat > set_set_nat,X: nat] :
      ( ( member_nat_set_nat @ F @ ( pi_nat_set_nat @ A @ B ) )
     => ( ~ ( member_set_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_nat @ X @ A ) ) ) ).

% PiE
thf(fact_462_PiE,axiom,
    ! [F: b > set_nat,A: set_b,B: b > set_set_nat,X: b] :
      ( ( member_b_set_nat @ F @ ( pi_b_set_nat @ A @ B ) )
     => ( ~ ( member_set_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_b @ X @ A ) ) ) ).

% PiE
thf(fact_463_PiE,axiom,
    ! [F: set_nat > nat,A: set_set_nat,B: set_nat > set_nat,X: set_nat] :
      ( ( member_set_nat_nat @ F @ ( pi_set_nat_nat @ A @ B ) )
     => ( ~ ( member_nat @ ( F @ X ) @ ( B @ X ) )
       => ~ ( member_set_nat @ X @ A ) ) ) ).

% PiE
thf(fact_464_UNIV__def,axiom,
    ( top_top_set_real
    = ( collect_real
      @ ^ [X5: real] : $true ) ) ).

% UNIV_def
thf(fact_465_UNIV__def,axiom,
    ( top_top_set_nat
    = ( collect_nat
      @ ^ [X5: nat] : $true ) ) ).

% UNIV_def
thf(fact_466_UNIV__def,axiom,
    ( top_top_set_o
    = ( collect_o
      @ ^ [X5: $o] : $true ) ) ).

% UNIV_def
thf(fact_467_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X5: nat] : $false ) ) ).

% Set.empty_def
thf(fact_468_Set_Oempty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X5: real] : $false ) ) ).

% Set.empty_def
thf(fact_469_Set_Oempty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X5: $o] : $false ) ) ).

% Set.empty_def
thf(fact_470_insert__Collect,axiom,
    ! [A2: real,P: real > $o] :
      ( ( insert_real @ A2 @ ( collect_real @ P ) )
      = ( collect_real
        @ ^ [U: real] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_471_insert__Collect,axiom,
    ! [A2: $o,P: $o > $o] :
      ( ( insert_o @ A2 @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U: $o] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_472_insert__Collect,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( insert_nat @ A2 @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U: nat] :
            ( ( U != A2 )
           => ( P @ U ) ) ) ) ).

% insert_Collect
thf(fact_473_insert__compr,axiom,
    ( insert_real
    = ( ^ [A3: real,B4: set_real] :
          ( collect_real
          @ ^ [X5: real] :
              ( ( X5 = A3 )
              | ( member_real @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_474_insert__compr,axiom,
    ( insert_o
    = ( ^ [A3: $o,B4: set_o] :
          ( collect_o
          @ ^ [X5: $o] :
              ( ( X5 = A3 )
              | ( member_o @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_475_insert__compr,axiom,
    ( insert_b
    = ( ^ [A3: b,B4: set_b] :
          ( collect_b
          @ ^ [X5: b] :
              ( ( X5 = A3 )
              | ( member_b @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_476_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A3: nat,B4: set_nat] :
          ( collect_nat
          @ ^ [X5: nat] :
              ( ( X5 = A3 )
              | ( member_nat @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_477_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A3: set_nat,B4: set_set_nat] :
          ( collect_set_nat
          @ ^ [X5: set_nat] :
              ( ( X5 = A3 )
              | ( member_set_nat @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_478_insert__compr,axiom,
    ( insert_real_nat
    = ( ^ [A3: real > nat,B4: set_real_nat] :
          ( collect_real_nat
          @ ^ [X5: real > nat] :
              ( ( X5 = A3 )
              | ( member_real_nat @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_479_insert__compr,axiom,
    ( insert_real_b
    = ( ^ [A3: real > b,B4: set_real_b] :
          ( collect_real_b
          @ ^ [X5: real > b] :
              ( ( X5 = A3 )
              | ( member_real_b @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_480_insert__compr,axiom,
    ( insert_nat_real
    = ( ^ [A3: nat > real,B4: set_nat_real] :
          ( collect_nat_real
          @ ^ [X5: nat > real] :
              ( ( X5 = A3 )
              | ( member_nat_real @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_481_insert__compr,axiom,
    ( insert_real_set_nat
    = ( ^ [A3: real > set_nat,B4: set_real_set_nat] :
          ( collect_real_set_nat
          @ ^ [X5: real > set_nat] :
              ( ( X5 = A3 )
              | ( member_real_set_nat @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_482_insert__compr,axiom,
    ( insert_real_nat_real
    = ( ^ [A3: real > nat > real,B4: set_real_nat_real] :
          ( collec336724336074912571t_real
          @ ^ [X5: real > nat > real] :
              ( ( X5 = A3 )
              | ( member_real_nat_real2 @ X5 @ B4 ) ) ) ) ) ).

% insert_compr
thf(fact_483_vimage__def,axiom,
    ( vimage_real_nat
    = ( ^ [F2: real > nat,B4: set_nat] :
          ( collect_real
          @ ^ [X5: real] : ( member_nat @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_484_vimage__def,axiom,
    ( vimage_nat_nat
    = ( ^ [F2: nat > nat,B4: set_nat] :
          ( collect_nat
          @ ^ [X5: nat] : ( member_nat @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_485_vimage__def,axiom,
    ( vimage_nat_b
    = ( ^ [F2: nat > b,B4: set_b] :
          ( collect_nat
          @ ^ [X5: nat] : ( member_b @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_486_vimage__def,axiom,
    ( vimage_nat_set_nat
    = ( ^ [F2: nat > set_nat,B4: set_set_nat] :
          ( collect_nat
          @ ^ [X5: nat] : ( member_set_nat @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_487_vimage__def,axiom,
    ( vimage_nat_real_nat
    = ( ^ [F2: nat > real > nat,B4: set_real_nat] :
          ( collect_nat
          @ ^ [X5: nat] : ( member_real_nat @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_488_vimage__def,axiom,
    ( vimage_nat_real_b
    = ( ^ [F2: nat > real > b,B4: set_real_b] :
          ( collect_nat
          @ ^ [X5: nat] : ( member_real_b @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_489_vimage__def,axiom,
    ( vimage_nat_nat_real
    = ( ^ [F2: nat > nat > real,B4: set_nat_real] :
          ( collect_nat
          @ ^ [X5: nat] : ( member_nat_real @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_490_vimage__def,axiom,
    ( vimage965819199455225310et_nat
    = ( ^ [F2: nat > real > set_nat,B4: set_real_set_nat] :
          ( collect_nat
          @ ^ [X5: nat] : ( member_real_set_nat @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_491_vimage__def,axiom,
    ( vimage264186217122334195t_real
    = ( ^ [F2: nat > real > nat > real,B4: set_real_nat_real] :
          ( collect_nat
          @ ^ [X5: nat] : ( member_real_nat_real2 @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_492_vimage__def,axiom,
    ( vimage3471692816685529672real_b
    = ( ^ [F2: nat > real > real > b,B4: set_real_real_b] :
          ( collect_nat
          @ ^ [X5: nat] : ( member_real_real_b @ ( F2 @ X5 ) @ B4 ) ) ) ) ).

% vimage_def
thf(fact_493_funcset__mem,axiom,
    ! [F: nat > nat,A: set_nat,B: set_nat,X: nat] :
      ( ( member_nat_nat @ F
        @ ( pi_nat_nat @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_494_funcset__mem,axiom,
    ! [F: nat > b,A: set_nat,B: set_b,X: nat] :
      ( ( member_nat_b @ F
        @ ( pi_nat_b @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( member_nat @ X @ A )
       => ( member_b @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_495_funcset__mem,axiom,
    ! [F: b > nat,A: set_b,B: set_nat,X: b] :
      ( ( member_b_nat @ F
        @ ( pi_b_nat @ A
          @ ^ [Uu: b] : B ) )
     => ( ( member_b @ X @ A )
       => ( member_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_496_funcset__mem,axiom,
    ! [F: b > b,A: set_b,B: set_b,X: b] :
      ( ( member_b_b @ F
        @ ( pi_b_b @ A
          @ ^ [Uu: b] : B ) )
     => ( ( member_b @ X @ A )
       => ( member_b @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_497_funcset__mem,axiom,
    ! [F: real > nat,A: set_real,B: set_nat,X: real] :
      ( ( member_real_nat @ F
        @ ( pi_real_nat @ A
          @ ^ [Uu: real] : B ) )
     => ( ( member_real @ X @ A )
       => ( member_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_498_funcset__mem,axiom,
    ! [F: real > b,A: set_real,B: set_b,X: real] :
      ( ( member_real_b @ F
        @ ( pi_real_b @ A
          @ ^ [Uu: real] : B ) )
     => ( ( member_real @ X @ A )
       => ( member_b @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_499_funcset__mem,axiom,
    ! [F: nat > real,A: set_nat,B: set_real,X: nat] :
      ( ( member_nat_real @ F
        @ ( pi_nat_real @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( member_nat @ X @ A )
       => ( member_real @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_500_funcset__mem,axiom,
    ! [F: set_nat > nat,A: set_set_nat,B: set_nat,X: set_nat] :
      ( ( member_set_nat_nat @ F
        @ ( pi_set_nat_nat @ A
          @ ^ [Uu: set_nat] : B ) )
     => ( ( member_set_nat @ X @ A )
       => ( member_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_501_funcset__mem,axiom,
    ! [F: set_nat > b,A: set_set_nat,B: set_b,X: set_nat] :
      ( ( member_set_nat_b @ F
        @ ( pi_set_nat_b @ A
          @ ^ [Uu: set_nat] : B ) )
     => ( ( member_set_nat @ X @ A )
       => ( member_b @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_502_funcset__mem,axiom,
    ! [F: nat > set_nat,A: set_nat,B: set_set_nat,X: nat] :
      ( ( member_nat_set_nat @ F
        @ ( pi_nat_set_nat @ A
          @ ^ [Uu: nat] : B ) )
     => ( ( member_nat @ X @ A )
       => ( member_set_nat @ ( F @ X ) @ B ) ) ) ).

% funcset_mem
thf(fact_503_funcsetI,axiom,
    ! [A: set_real,F: real > nat,B: set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B ) )
     => ( member_real_nat @ F
        @ ( pi_real_nat @ A
          @ ^ [Uu: real] : B ) ) ) ).

% funcsetI
thf(fact_504_funcsetI,axiom,
    ! [A: set_real,F: real > b,B: set_b] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ B ) )
     => ( member_real_b @ F
        @ ( pi_real_b @ A
          @ ^ [Uu: real] : B ) ) ) ).

% funcsetI
thf(fact_505_funcsetI,axiom,
    ! [A: set_nat,F: nat > real,B: set_real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_real @ ( F @ X3 ) @ B ) )
     => ( member_nat_real @ F
        @ ( pi_nat_real @ A
          @ ^ [Uu: nat] : B ) ) ) ).

% funcsetI
thf(fact_506_funcsetI,axiom,
    ! [A: set_nat,F: nat > nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B ) )
     => ( member_nat_nat @ F
        @ ( pi_nat_nat @ A
          @ ^ [Uu: nat] : B ) ) ) ).

% funcsetI
thf(fact_507_funcsetI,axiom,
    ! [A: set_nat,F: nat > b,B: set_b] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ B ) )
     => ( member_nat_b @ F
        @ ( pi_nat_b @ A
          @ ^ [Uu: nat] : B ) ) ) ).

% funcsetI
thf(fact_508_funcsetI,axiom,
    ! [A: set_b,F: b > nat,B: set_nat] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B ) )
     => ( member_b_nat @ F
        @ ( pi_b_nat @ A
          @ ^ [Uu: b] : B ) ) ) ).

% funcsetI
thf(fact_509_funcsetI,axiom,
    ! [A: set_b,F: b > b,B: set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ B ) )
     => ( member_b_b @ F
        @ ( pi_b_b @ A
          @ ^ [Uu: b] : B ) ) ) ).

% funcsetI
thf(fact_510_funcsetI,axiom,
    ! [A: set_real,F: real > set_nat,B: set_set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( member_set_nat @ ( F @ X3 ) @ B ) )
     => ( member_real_set_nat @ F
        @ ( pi_real_set_nat @ A
          @ ^ [Uu: real] : B ) ) ) ).

% funcsetI
thf(fact_511_funcsetI,axiom,
    ! [A: set_set_nat,F: set_nat > nat,B: set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ( member_nat @ ( F @ X3 ) @ B ) )
     => ( member_set_nat_nat @ F
        @ ( pi_set_nat_nat @ A
          @ ^ [Uu: set_nat] : B ) ) ) ).

% funcsetI
thf(fact_512_funcsetI,axiom,
    ! [A: set_set_nat,F: set_nat > b,B: set_b] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ( member_b @ ( F @ X3 ) @ B ) )
     => ( member_set_nat_b @ F
        @ ( pi_set_nat_b @ A
          @ ^ [Uu: set_nat] : B ) ) ) ).

% funcsetI
thf(fact_513_empty__not__UNIV,axiom,
    bot_bot_set_real != top_top_set_real ).

% empty_not_UNIV
thf(fact_514_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_515_empty__not__UNIV,axiom,
    bot_bot_set_o != top_top_set_o ).

% empty_not_UNIV
thf(fact_516_insert__UNIV,axiom,
    ! [X: real] :
      ( ( insert_real @ X @ top_top_set_real )
      = top_top_set_real ) ).

% insert_UNIV
thf(fact_517_insert__UNIV,axiom,
    ! [X: nat] :
      ( ( insert_nat @ X @ top_top_set_nat )
      = top_top_set_nat ) ).

% insert_UNIV
thf(fact_518_insert__UNIV,axiom,
    ! [X: $o] :
      ( ( insert_o @ X @ top_top_set_o )
      = top_top_set_o ) ).

% insert_UNIV
thf(fact_519_singleton__inject,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_520_singleton__inject,axiom,
    ! [A2: real,B2: real] :
      ( ( ( insert_real @ A2 @ bot_bot_set_real )
        = ( insert_real @ B2 @ bot_bot_set_real ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_521_singleton__inject,axiom,
    ! [A2: $o,B2: $o] :
      ( ( ( insert_o @ A2 @ bot_bot_set_o )
        = ( insert_o @ B2 @ bot_bot_set_o ) )
     => ( A2 = B2 ) ) ).

% singleton_inject
thf(fact_522_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ A )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_523_insert__not__empty,axiom,
    ! [A2: real,A: set_real] :
      ( ( insert_real @ A2 @ A )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_524_insert__not__empty,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o @ A2 @ A )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_525_doubleton__eq__iff,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_526_doubleton__eq__iff,axiom,
    ! [A2: real,B2: real,C: real,D: real] :
      ( ( ( insert_real @ A2 @ ( insert_real @ B2 @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_527_doubleton__eq__iff,axiom,
    ! [A2: $o,B2: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A2 = C )
          & ( B2 = D ) )
        | ( ( A2 = D )
          & ( B2 = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_528_singleton__iff,axiom,
    ! [B2: b,A2: b] :
      ( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_529_singleton__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_530_singleton__iff,axiom,
    ! [B2: real,A2: real] :
      ( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_531_singleton__iff,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_532_singleton__iff,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( member_set_nat @ B2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_533_singleton__iff,axiom,
    ! [B2: real > nat,A2: real > nat] :
      ( ( member_real_nat @ B2 @ ( insert_real_nat @ A2 @ bot_bot_set_real_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_534_singleton__iff,axiom,
    ! [B2: real > b,A2: real > b] :
      ( ( member_real_b @ B2 @ ( insert_real_b @ A2 @ bot_bot_set_real_b ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_535_singleton__iff,axiom,
    ! [B2: nat > real,A2: nat > real] :
      ( ( member_nat_real @ B2 @ ( insert_nat_real @ A2 @ bot_bot_set_nat_real ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_536_singleton__iff,axiom,
    ! [B2: real > set_nat,A2: real > set_nat] :
      ( ( member_real_set_nat @ B2 @ ( insert_real_set_nat @ A2 @ bot_bo6814059168456595739et_nat ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_537_singleton__iff,axiom,
    ! [B2: real > nat > real,A2: real > nat > real] :
      ( ( member_real_nat_real2 @ B2 @ ( insert_real_nat_real @ A2 @ bot_bo6533810469807102640t_real ) )
      = ( B2 = A2 ) ) ).

% singleton_iff
thf(fact_538_singletonD,axiom,
    ! [B2: b,A2: b] :
      ( ( member_b @ B2 @ ( insert_b @ A2 @ bot_bot_set_b ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_539_singletonD,axiom,
    ! [B2: nat,A2: nat] :
      ( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_540_singletonD,axiom,
    ! [B2: real,A2: real] :
      ( ( member_real @ B2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_541_singletonD,axiom,
    ! [B2: $o,A2: $o] :
      ( ( member_o @ B2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_542_singletonD,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( member_set_nat @ B2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_543_singletonD,axiom,
    ! [B2: real > nat,A2: real > nat] :
      ( ( member_real_nat @ B2 @ ( insert_real_nat @ A2 @ bot_bot_set_real_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_544_singletonD,axiom,
    ! [B2: real > b,A2: real > b] :
      ( ( member_real_b @ B2 @ ( insert_real_b @ A2 @ bot_bot_set_real_b ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_545_singletonD,axiom,
    ! [B2: nat > real,A2: nat > real] :
      ( ( member_nat_real @ B2 @ ( insert_nat_real @ A2 @ bot_bot_set_nat_real ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_546_singletonD,axiom,
    ! [B2: real > set_nat,A2: real > set_nat] :
      ( ( member_real_set_nat @ B2 @ ( insert_real_set_nat @ A2 @ bot_bo6814059168456595739et_nat ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_547_singletonD,axiom,
    ! [B2: real > nat > real,A2: real > nat > real] :
      ( ( member_real_nat_real2 @ B2 @ ( insert_real_nat_real @ A2 @ bot_bo6533810469807102640t_real ) )
     => ( B2 = A2 ) ) ).

% singletonD
thf(fact_548_snd__conv,axiom,
    ! [X1: set_b,X22: set_real_b] :
      ( ( produc1187263614985732624real_b @ ( produc5260422477894090978real_b @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_549_snd__eqD,axiom,
    ! [X: set_b,Y: set_real_b,A2: set_real_b] :
      ( ( ( produc1187263614985732624real_b @ ( produc5260422477894090978real_b @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_550_Collect__conv__if2,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X5: nat] :
                ( ( A2 = X5 )
                & ( P @ X5 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X5: nat] :
                ( ( A2 = X5 )
                & ( P @ X5 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_551_Collect__conv__if2,axiom,
    ! [P: real > $o,A2: real] :
      ( ( ( P @ A2 )
       => ( ( collect_real
            @ ^ [X5: real] :
                ( ( A2 = X5 )
                & ( P @ X5 ) ) )
          = ( insert_real @ A2 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_real
            @ ^ [X5: real] :
                ( ( A2 = X5 )
                & ( P @ X5 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_552_Collect__conv__if2,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X5: $o] :
                ( ( A2 = X5 )
                & ( P @ X5 ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X5: $o] :
                ( ( A2 = X5 )
                & ( P @ X5 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_553_Collect__conv__if,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X5: nat] :
                ( ( X5 = A2 )
                & ( P @ X5 ) ) )
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_nat
            @ ^ [X5: nat] :
                ( ( X5 = A2 )
                & ( P @ X5 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_554_Collect__conv__if,axiom,
    ! [P: real > $o,A2: real] :
      ( ( ( P @ A2 )
       => ( ( collect_real
            @ ^ [X5: real] :
                ( ( X5 = A2 )
                & ( P @ X5 ) ) )
          = ( insert_real @ A2 @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_real
            @ ^ [X5: real] :
                ( ( X5 = A2 )
                & ( P @ X5 ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_555_Collect__conv__if,axiom,
    ! [P: $o > $o,A2: $o] :
      ( ( ( P @ A2 )
       => ( ( collect_o
            @ ^ [X5: $o] :
                ( ( X5 = A2 )
                & ( P @ X5 ) ) )
          = ( insert_o @ A2 @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A2 )
       => ( ( collect_o
            @ ^ [X5: $o] :
                ( ( X5 = A2 )
                & ( P @ X5 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_556_snd__def,axiom,
    ( produc1187263614985732624real_b
    = ( produc8482724652568146995real_b
      @ ^ [X12: set_b,X23: set_real_b] : X23 ) ) ).

% snd_def
thf(fact_557_vimage__singleton__eq,axiom,
    ! [A2: nat,F: nat > nat,B2: nat] :
      ( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_558_vimage__singleton__eq,axiom,
    ! [A2: b,F: b > nat,B2: nat] :
      ( ( member_b @ A2 @ ( vimage_b_nat @ F @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_559_vimage__singleton__eq,axiom,
    ! [A2: real,F: real > nat,B2: nat] :
      ( ( member_real @ A2 @ ( vimage_real_nat @ F @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_560_vimage__singleton__eq,axiom,
    ! [A2: nat,F: nat > real,B2: real] :
      ( ( member_nat @ A2 @ ( vimage_nat_real @ F @ ( insert_real @ B2 @ bot_bot_set_real ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_561_vimage__singleton__eq,axiom,
    ! [A2: b,F: b > real,B2: real] :
      ( ( member_b @ A2 @ ( vimage_b_real @ F @ ( insert_real @ B2 @ bot_bot_set_real ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_562_vimage__singleton__eq,axiom,
    ! [A2: nat,F: nat > $o,B2: $o] :
      ( ( member_nat @ A2 @ ( vimage_nat_o @ F @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_563_vimage__singleton__eq,axiom,
    ! [A2: b,F: b > $o,B2: $o] :
      ( ( member_b @ A2 @ ( vimage_b_o @ F @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_564_vimage__singleton__eq,axiom,
    ! [A2: set_nat,F: set_nat > nat,B2: nat] :
      ( ( member_set_nat @ A2 @ ( vimage_set_nat_nat @ F @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_565_vimage__singleton__eq,axiom,
    ! [A2: set_nat,F: set_nat > real,B2: real] :
      ( ( member_set_nat @ A2 @ ( vimage_set_nat_real @ F @ ( insert_real @ B2 @ bot_bot_set_real ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_566_vimage__singleton__eq,axiom,
    ! [A2: set_nat,F: set_nat > $o,B2: $o] :
      ( ( member_set_nat @ A2 @ ( vimage_set_nat_o @ F @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
      = ( ( F @ A2 )
        = B2 ) ) ).

% vimage_singleton_eq
thf(fact_567_snd__bot,axiom,
    ( ( produc1187263614985732624real_b @ bot_bo932696493702709598real_b )
    = bot_bot_set_real_b ) ).

% snd_bot
thf(fact_568_snd__top,axiom,
    ( ( produc1187263614985732624real_b @ top_to1114480880377561154real_b )
    = top_top_set_real_b ) ).

% snd_top
thf(fact_569_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_nat] : ( member_set_nat @ bot_bot_set_nat @ ( sigma_sets_nat @ M ) ) ).

% sets.empty_sets
thf(fact_570_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_real] : ( member_set_real @ bot_bot_set_real @ ( sigma_sets_real @ M ) ) ).

% sets.empty_sets
thf(fact_571_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_o] : ( member_set_o @ bot_bot_set_o @ ( sigma_sets_o @ M ) ) ).

% sets.empty_sets
thf(fact_572_qbs__closed3__def,axiom,
    ( qbs_closed3_nat
    = ( ^ [Mx2: set_real_nat] :
        ! [P2: real > nat,Fi: nat > real > nat] :
          ( ! [I3: nat] : ( member_set_real @ ( vimage_real_nat @ P2 @ ( insert_nat @ I3 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I3: nat] : ( member_real_nat @ ( Fi @ I3 ) @ Mx2 )
           => ( member_real_nat
              @ ^ [R: real] : ( Fi @ ( P2 @ R ) @ R )
              @ Mx2 ) ) ) ) ) ).

% qbs_closed3_def
thf(fact_573_qbs__closed3__def,axiom,
    ( qbs_closed3_b
    = ( ^ [Mx2: set_real_b] :
        ! [P2: real > nat,Fi: nat > real > b] :
          ( ! [I3: nat] : ( member_set_real @ ( vimage_real_nat @ P2 @ ( insert_nat @ I3 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I3: nat] : ( member_real_b @ ( Fi @ I3 ) @ Mx2 )
           => ( member_real_b
              @ ^ [R: real] : ( Fi @ ( P2 @ R ) @ R )
              @ Mx2 ) ) ) ) ) ).

% qbs_closed3_def
thf(fact_574_qbs__closed3__def,axiom,
    ( qbs_closed3_nat_real
    = ( ^ [Mx2: set_real_nat_real] :
        ! [P2: real > nat,Fi: nat > real > nat > real] :
          ( ! [I3: nat] : ( member_set_real @ ( vimage_real_nat @ P2 @ ( insert_nat @ I3 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I3: nat] : ( member_real_nat_real2 @ ( Fi @ I3 ) @ Mx2 )
           => ( member_real_nat_real2
              @ ^ [R: real] : ( Fi @ ( P2 @ R ) @ R )
              @ Mx2 ) ) ) ) ) ).

% qbs_closed3_def
thf(fact_575_qbs__closed3__def,axiom,
    ( qbs_closed3_real_b
    = ( ^ [Mx2: set_real_real_b] :
        ! [P2: real > nat,Fi: nat > real > real > b] :
          ( ! [I3: nat] : ( member_set_real @ ( vimage_real_nat @ P2 @ ( insert_nat @ I3 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I3: nat] : ( member_real_real_b @ ( Fi @ I3 ) @ Mx2 )
           => ( member_real_real_b
              @ ^ [R: real] : ( Fi @ ( P2 @ R ) @ R )
              @ Mx2 ) ) ) ) ) ).

% qbs_closed3_def
thf(fact_576_qbs__closed3__def,axiom,
    ( qbs_closed3_real_nat
    = ( ^ [Mx2: set_real_real_nat] :
        ! [P2: real > nat,Fi: nat > real > real > nat] :
          ( ! [I3: nat] : ( member_set_real @ ( vimage_real_nat @ P2 @ ( insert_nat @ I3 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I3: nat] : ( member_real_real_nat @ ( Fi @ I3 ) @ Mx2 )
           => ( member_real_real_nat
              @ ^ [R: real] : ( Fi @ ( P2 @ R ) @ R )
              @ Mx2 ) ) ) ) ) ).

% qbs_closed3_def
thf(fact_577_qbs__closed3__def,axiom,
    ( qbs_closed3_set_nat
    = ( ^ [Mx2: set_real_set_nat] :
        ! [P2: real > nat,Fi: nat > real > set_nat] :
          ( ! [I3: nat] : ( member_set_real @ ( vimage_real_nat @ P2 @ ( insert_nat @ I3 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I3: nat] : ( member_real_set_nat @ ( Fi @ I3 ) @ Mx2 )
           => ( member_real_set_nat
              @ ^ [R: real] : ( Fi @ ( P2 @ R ) @ R )
              @ Mx2 ) ) ) ) ) ).

% qbs_closed3_def
thf(fact_578_qbs__closed3I,axiom,
    ! [Mx: set_real_nat] :
      ( ! [P3: real > nat,Fi2: nat > real > nat] :
          ( ! [I4: nat] : ( member_set_real @ ( vimage_real_nat @ P3 @ ( insert_nat @ I4 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I4: nat] : ( member_real_nat @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_nat
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_nat @ Mx ) ) ).

% qbs_closed3I
thf(fact_579_qbs__closed3I,axiom,
    ! [Mx: set_real_b] :
      ( ! [P3: real > nat,Fi2: nat > real > b] :
          ( ! [I4: nat] : ( member_set_real @ ( vimage_real_nat @ P3 @ ( insert_nat @ I4 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I4: nat] : ( member_real_b @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_b
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_b @ Mx ) ) ).

% qbs_closed3I
thf(fact_580_qbs__closed3I,axiom,
    ! [Mx: set_real_nat_real] :
      ( ! [P3: real > nat,Fi2: nat > real > nat > real] :
          ( ! [I4: nat] : ( member_set_real @ ( vimage_real_nat @ P3 @ ( insert_nat @ I4 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I4: nat] : ( member_real_nat_real2 @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_nat_real2
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_nat_real @ Mx ) ) ).

% qbs_closed3I
thf(fact_581_qbs__closed3I,axiom,
    ! [Mx: set_real_real_b] :
      ( ! [P3: real > nat,Fi2: nat > real > real > b] :
          ( ! [I4: nat] : ( member_set_real @ ( vimage_real_nat @ P3 @ ( insert_nat @ I4 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I4: nat] : ( member_real_real_b @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_real_b
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_real_b @ Mx ) ) ).

% qbs_closed3I
thf(fact_582_qbs__closed3I,axiom,
    ! [Mx: set_real_real_nat] :
      ( ! [P3: real > nat,Fi2: nat > real > real > nat] :
          ( ! [I4: nat] : ( member_set_real @ ( vimage_real_nat @ P3 @ ( insert_nat @ I4 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I4: nat] : ( member_real_real_nat @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_real_nat
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_real_nat @ Mx ) ) ).

% qbs_closed3I
thf(fact_583_qbs__closed3I,axiom,
    ! [Mx: set_real_set_nat] :
      ( ! [P3: real > nat,Fi2: nat > real > set_nat] :
          ( ! [I4: nat] : ( member_set_real @ ( vimage_real_nat @ P3 @ ( insert_nat @ I4 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
         => ( ! [I4: nat] : ( member_real_set_nat @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_set_nat
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_set_nat @ Mx ) ) ).

% qbs_closed3I
thf(fact_584_borel__singleton,axiom,
    ! [A: set_real,X: real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( member_set_real @ ( insert_real @ X @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% borel_singleton
thf(fact_585_borel__singleton,axiom,
    ! [A: set_nat,X: nat] :
      ( ( member_set_nat @ A @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) )
     => ( member_set_nat @ ( insert_nat @ X @ A ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% borel_singleton
thf(fact_586_borel__singleton,axiom,
    ! [A: set_o,X: $o] :
      ( ( member_set_o @ A @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) )
     => ( member_set_o @ ( insert_o @ X @ A ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% borel_singleton
thf(fact_587_space__in__borel,axiom,
    member_set_real @ top_top_set_real @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).

% space_in_borel
thf(fact_588_space__in__borel,axiom,
    member_set_nat @ top_top_set_nat @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ).

% space_in_borel
thf(fact_589_space__in__borel,axiom,
    member_set_o @ top_top_set_o @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ).

% space_in_borel
thf(fact_590_sets_Oinsert__in__sets,axiom,
    ! [X: nat,M: sigma_measure_nat,A: set_nat] :
      ( ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat @ ( insert_nat @ X @ A ) @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_591_sets_Oinsert__in__sets,axiom,
    ! [X: real,M: sigma_measure_real,A: set_real] :
      ( ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
       => ( member_set_real @ ( insert_real @ X @ A ) @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_592_sets_Oinsert__in__sets,axiom,
    ! [X: $o,M: sigma_measure_o,A: set_o] :
      ( ( member_set_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
       => ( member_set_o @ ( insert_o @ X @ A ) @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_593_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [X: real] :
      ( top_top_set_real
     != ( insert_real @ X @ bot_bot_set_real ) ) ).

% perfect_space_class.UNIV_not_singleton
thf(fact_594_Abs__quasi__borel__inverse,axiom,
    ! [Y: produc4373655991844504306real_b] :
      ( ( member527121533207341851real_b @ Y @ ( collec5860786303674561885real_b @ ( produc9150283905253018337al_b_o @ is_quasi_borel_b ) ) )
     => ( ( quasi_4958298574314430518orel_b @ ( quasi_2002468295286565185orel_b @ Y ) )
        = Y ) ) ).

% Abs_quasi_borel_inverse
thf(fact_595_top__empty__eq,axiom,
    ( top_top_b_o
    = ( ^ [X5: b] : ( member_b @ X5 @ top_top_set_b ) ) ) ).

% top_empty_eq
thf(fact_596_top__empty__eq,axiom,
    ( top_top_real_o
    = ( ^ [X5: real] : ( member_real @ X5 @ top_top_set_real ) ) ) ).

% top_empty_eq
thf(fact_597_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_598_top__empty__eq,axiom,
    ( top_top_o_o
    = ( ^ [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ) ) ).

% top_empty_eq
thf(fact_599_top__empty__eq,axiom,
    ( top_top_set_nat_o
    = ( ^ [X5: set_nat] : ( member_set_nat @ X5 @ top_top_set_set_nat ) ) ) ).

% top_empty_eq
thf(fact_600_top__empty__eq,axiom,
    ( top_top_real_nat_o
    = ( ^ [X5: real > nat] : ( member_real_nat @ X5 @ top_top_set_real_nat ) ) ) ).

% top_empty_eq
thf(fact_601_top__empty__eq,axiom,
    ( top_top_real_b_o
    = ( ^ [X5: real > b] : ( member_real_b @ X5 @ top_top_set_real_b ) ) ) ).

% top_empty_eq
thf(fact_602_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_603_top__empty__eq,axiom,
    ( top_to150443550350409742_nat_o
    = ( ^ [X5: real > set_nat] : ( member_real_set_nat @ X5 @ top_to245300144855000375et_nat ) ) ) ).

% top_empty_eq
thf(fact_604_top__empty__eq,axiom,
    ( top_to1109506300641978553real_o
    = ( ^ [X5: real > nat > real] : ( member_real_nat_real2 @ X5 @ top_to6902130281023745740t_real ) ) ) ).

% top_empty_eq
thf(fact_605_top__set__def,axiom,
    ( top_top_set_real
    = ( collect_real @ top_top_real_o ) ) ).

% top_set_def
thf(fact_606_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_607_top__set__def,axiom,
    ( top_top_set_o
    = ( collect_o @ top_top_o_o ) ) ).

% top_set_def
thf(fact_608_bot__empty__eq,axiom,
    ( bot_bot_b_o
    = ( ^ [X5: b] : ( member_b @ X5 @ bot_bot_set_b ) ) ) ).

% bot_empty_eq
thf(fact_609_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X5: nat] : ( member_nat @ X5 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_610_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X5: real] : ( member_real @ X5 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_611_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X5: $o] : ( member_o @ X5 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_612_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X5: set_nat] : ( member_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_613_bot__empty__eq,axiom,
    ( bot_bot_real_nat_o
    = ( ^ [X5: real > nat] : ( member_real_nat @ X5 @ bot_bot_set_real_nat ) ) ) ).

% bot_empty_eq
thf(fact_614_bot__empty__eq,axiom,
    ( bot_bot_real_b_o
    = ( ^ [X5: real > b] : ( member_real_b @ X5 @ bot_bot_set_real_b ) ) ) ).

% bot_empty_eq
thf(fact_615_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_616_bot__empty__eq,axiom,
    ( bot_bo2654456731858362666_nat_o
    = ( ^ [X5: real > set_nat] : ( member_real_set_nat @ X5 @ bot_bo6814059168456595739et_nat ) ) ) ).

% bot_empty_eq
thf(fact_617_bot__empty__eq,axiom,
    ( bot_bo839430502184047061real_o
    = ( ^ [X5: real > nat > real] : ( member_real_nat_real2 @ X5 @ bot_bo6533810469807102640t_real ) ) ) ).

% bot_empty_eq
thf(fact_618_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_619_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_620_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_621_Rep__quasi__borel__inverse,axiom,
    ! [X: quasi_borel_b] :
      ( ( quasi_2002468295286565185orel_b @ ( quasi_4958298574314430518orel_b @ X ) )
      = X ) ).

% Rep_quasi_borel_inverse
thf(fact_622_top__prod__def,axiom,
    ( top_to1822899090350206709t_real
    = ( produc6377492392003646173t_real @ top_top_set_real @ top_top_set_real ) ) ).

% top_prod_def
thf(fact_623_top__prod__def,axiom,
    ( top_to8233803056352276889et_nat
    = ( produc4432056366243864833et_nat @ top_top_set_real @ top_top_set_nat ) ) ).

% top_prod_def
thf(fact_624_top__prod__def,axiom,
    ( top_to5627709924142012091_set_o
    = ( produc9171573333813886013_set_o @ top_top_set_real @ top_top_set_o ) ) ).

% top_prod_def
thf(fact_625_top__prod__def,axiom,
    ( top_to8609433042786791961t_real
    = ( produc3494877982625406337t_real @ top_top_set_nat @ top_top_set_real ) ) ).

% top_prod_def
thf(fact_626_top__prod__def,axiom,
    ( top_to7593079806418470589et_nat
    = ( produc4532415448927165861et_nat @ top_top_set_nat @ top_top_set_nat ) ) ).

% top_prod_def
thf(fact_627_top__prod__def,axiom,
    ( top_to4137661810880245911_set_o
    = ( produc2750217728217304729_set_o @ top_top_set_nat @ top_top_set_o ) ) ).

% top_prod_def
thf(fact_628_top__prod__def,axiom,
    ( top_to8585549949225041977t_real
    = ( produc2850126452351882627t_real @ top_top_set_o @ top_top_set_real ) ) ).

% top_prod_def
thf(fact_629_top__prod__def,axiom,
    ( top_to2343176657905263837et_nat
    = ( produc2212626075605038759et_nat @ top_top_set_o @ top_top_set_nat ) ) ).

% top_prod_def
thf(fact_630_top__prod__def,axiom,
    ( top_to8356300885640301175_set_o
    = ( produc5838405689764958487_set_o @ top_top_set_o @ top_top_set_o ) ) ).

% top_prod_def
thf(fact_631_bot__prod__def,axiom,
    ( bot_bo3047382831089536473et_nat
    = ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) ) ).

% bot_prod_def
thf(fact_632_bot__prod__def,axiom,
    ( bot_bo4718971705043381045t_real
    = ( produc3494877982625406337t_real @ bot_bot_set_nat @ bot_bot_set_real ) ) ).

% bot_prod_def
thf(fact_633_bot__prod__def,axiom,
    ( bot_bo6859089648663734907_set_o
    = ( produc2750217728217304729_set_o @ bot_bot_set_nat @ bot_bot_set_o ) ) ).

% bot_prod_def
thf(fact_634_bot__prod__def,axiom,
    ( bot_bo4343341718608865973et_nat
    = ( produc4432056366243864833et_nat @ bot_bot_set_real @ bot_bot_set_nat ) ) ).

% bot_prod_def
thf(fact_635_bot__prod__def,axiom,
    ( bot_bo987450378295171601t_real
    = ( produc6377492392003646173t_real @ bot_bot_set_real @ bot_bot_set_real ) ) ).

% bot_prod_def
thf(fact_636_bot__prod__def,axiom,
    ( bot_bo3099358262589906591_set_o
    = ( produc9171573333813886013_set_o @ bot_bot_set_real @ bot_bot_set_o ) ) ).

% bot_prod_def
thf(fact_637_bot__prod__def,axiom,
    ( bot_bo5064604495688752833et_nat
    = ( produc2212626075605038759et_nat @ bot_bot_set_o @ bot_bot_set_nat ) ) ).

% bot_prod_def
thf(fact_638_bot__prod__def,axiom,
    ( bot_bo6057198287672936477t_real
    = ( produc2850126452351882627t_real @ bot_bot_set_o @ bot_bot_set_real ) ) ).

% bot_prod_def
thf(fact_639_bot__prod__def,axiom,
    ( bot_bo5661300308423610259_set_o
    = ( produc5838405689764958487_set_o @ bot_bot_set_o @ bot_bot_set_o ) ) ).

% bot_prod_def
thf(fact_640_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_641_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_642_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_643_ball__insert,axiom,
    ! [A2: nat,B: set_nat,P: nat > $o] :
      ( ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( insert_nat @ A2 @ B ) )
           => ( P @ X5 ) ) )
      = ( ( P @ A2 )
        & ! [X5: nat] :
            ( ( member_nat @ X5 @ B )
           => ( P @ X5 ) ) ) ) ).

% ball_insert
thf(fact_644_ball__insert,axiom,
    ! [A2: real,B: set_real,P: real > $o] :
      ( ( ! [X5: real] :
            ( ( member_real @ X5 @ ( insert_real @ A2 @ B ) )
           => ( P @ X5 ) ) )
      = ( ( P @ A2 )
        & ! [X5: real] :
            ( ( member_real @ X5 @ B )
           => ( P @ X5 ) ) ) ) ).

% ball_insert
thf(fact_645_ball__insert,axiom,
    ! [A2: $o,B: set_o,P: $o > $o] :
      ( ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( insert_o @ A2 @ B ) )
           => ( P @ X5 ) ) )
      = ( ( P @ A2 )
        & ! [X5: $o] :
            ( ( member_o @ X5 @ B )
           => ( P @ X5 ) ) ) ) ).

% ball_insert
thf(fact_646_iso__tuple__UNIV__I,axiom,
    ! [X: b] : ( member_b @ X @ top_top_set_b ) ).

% iso_tuple_UNIV_I
thf(fact_647_iso__tuple__UNIV__I,axiom,
    ! [X: real] : ( member_real @ X @ top_top_set_real ) ).

% iso_tuple_UNIV_I
thf(fact_648_iso__tuple__UNIV__I,axiom,
    ! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_649_iso__tuple__UNIV__I,axiom,
    ! [X: $o] : ( member_o @ X @ top_top_set_o ) ).

% iso_tuple_UNIV_I
thf(fact_650_iso__tuple__UNIV__I,axiom,
    ! [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_651_iso__tuple__UNIV__I,axiom,
    ! [X: real > nat] : ( member_real_nat @ X @ top_top_set_real_nat ) ).

% iso_tuple_UNIV_I
thf(fact_652_iso__tuple__UNIV__I,axiom,
    ! [X: real > b] : ( member_real_b @ X @ top_top_set_real_b ) ).

% iso_tuple_UNIV_I
thf(fact_653_iso__tuple__UNIV__I,axiom,
    ! [X: nat > real] : ( member_nat_real @ X @ top_top_set_nat_real ) ).

% iso_tuple_UNIV_I
thf(fact_654_iso__tuple__UNIV__I,axiom,
    ! [X: real > set_nat] : ( member_real_set_nat @ X @ top_to245300144855000375et_nat ) ).

% iso_tuple_UNIV_I
thf(fact_655_iso__tuple__UNIV__I,axiom,
    ! [X: real > nat > real] : ( member_real_nat_real2 @ X @ top_to6902130281023745740t_real ) ).

% iso_tuple_UNIV_I
thf(fact_656_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_657_is__singletonI,axiom,
    ! [X: real] : ( is_singleton_real @ ( insert_real @ X @ bot_bot_set_real ) ) ).

% is_singletonI
thf(fact_658_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_659_sndI,axiom,
    ! [X: produc4373655991844504306real_b,Y: set_b,Z2: set_real_b] :
      ( ( X
        = ( produc5260422477894090978real_b @ Y @ Z2 ) )
     => ( ( produc1187263614985732624real_b @ X )
        = Z2 ) ) ).

% sndI
thf(fact_660_is__singletonI_H,axiom,
    ! [A: set_b] :
      ( ( A != bot_bot_set_b )
     => ( ! [X3: b,Y4: b] :
            ( ( member_b @ X3 @ A )
           => ( ( member_b @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_b @ A ) ) ) ).

% is_singletonI'
thf(fact_661_is__singletonI_H,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X3: nat,Y4: nat] :
            ( ( member_nat @ X3 @ A )
           => ( ( member_nat @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_662_is__singletonI_H,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
     => ( ! [X3: real,Y4: real] :
            ( ( member_real @ X3 @ A )
           => ( ( member_real @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_real @ A ) ) ) ).

% is_singletonI'
thf(fact_663_is__singletonI_H,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
     => ( ! [X3: $o,Y4: $o] :
            ( ( member_o @ X3 @ A )
           => ( ( member_o @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_o @ A ) ) ) ).

% is_singletonI'
thf(fact_664_is__singletonI_H,axiom,
    ! [A: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => ( ! [X3: set_nat,Y4: set_nat] :
            ( ( member_set_nat @ X3 @ A )
           => ( ( member_set_nat @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_set_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_665_is__singletonI_H,axiom,
    ! [A: set_real_nat] :
      ( ( A != bot_bot_set_real_nat )
     => ( ! [X3: real > nat,Y4: real > nat] :
            ( ( member_real_nat @ X3 @ A )
           => ( ( member_real_nat @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_sin9065187617944275111al_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_666_is__singletonI_H,axiom,
    ! [A: set_real_b] :
      ( ( A != bot_bot_set_real_b )
     => ( ! [X3: real > b,Y4: real > b] :
            ( ( member_real_b @ X3 @ A )
           => ( ( member_real_b @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_real_b @ A ) ) ) ).

% is_singletonI'
thf(fact_667_is__singletonI_H,axiom,
    ! [A: set_nat_real] :
      ( ( A != bot_bot_set_nat_real )
     => ( ! [X3: nat > real,Y4: nat > real] :
            ( ( member_nat_real @ X3 @ A )
           => ( ( member_nat_real @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_sin3816863272566379559t_real @ A ) ) ) ).

% is_singletonI'
thf(fact_668_is__singletonI_H,axiom,
    ! [A: set_real_set_nat] :
      ( ( A != bot_bo6814059168456595739et_nat )
     => ( ! [X3: real > set_nat,Y4: real > set_nat] :
            ( ( member_real_set_nat @ X3 @ A )
           => ( ( member_real_set_nat @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_sin5719733656810162525et_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_669_is__singletonI_H,axiom,
    ! [A: set_real_nat_real] :
      ( ( A != bot_bo6533810469807102640t_real )
     => ( ! [X3: real > nat > real,Y4: real > nat > real] :
            ( ( member_real_nat_real2 @ X3 @ A )
           => ( ( member_real_nat_real2 @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_sin4536856481422437234t_real @ A ) ) ) ).

% is_singletonI'
thf(fact_670_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A4: set_nat] :
        ? [X5: nat] :
          ( A4
          = ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_671_is__singleton__def,axiom,
    ( is_singleton_real
    = ( ^ [A4: set_real] :
        ? [X5: real] :
          ( A4
          = ( insert_real @ X5 @ bot_bot_set_real ) ) ) ) ).

% is_singleton_def
thf(fact_672_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A4: set_o] :
        ? [X5: $o] :
          ( A4
          = ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_673_is__singletonE,axiom,
    ! [A: set_nat] :
      ( ( is_singleton_nat @ A )
     => ~ ! [X3: nat] :
            ( A
           != ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_674_is__singletonE,axiom,
    ! [A: set_real] :
      ( ( is_singleton_real @ A )
     => ~ ! [X3: real] :
            ( A
           != ( insert_real @ X3 @ bot_bot_set_real ) ) ) ).

% is_singletonE
thf(fact_675_is__singletonE,axiom,
    ! [A: set_o] :
      ( ( is_singleton_o @ A )
     => ~ ! [X3: $o] :
            ( A
           != ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_676_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_677_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_678_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_679_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_680_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_681_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_682_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_683_Collect__empty__eq__bot,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( P = bot_bot_real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_684_Collect__empty__eq__bot,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( P = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_685_eq__snd__iff,axiom,
    ! [B2: set_real_b,P4: produc4373655991844504306real_b] :
      ( ( B2
        = ( produc1187263614985732624real_b @ P4 ) )
      = ( ? [A3: set_b] :
            ( P4
            = ( produc5260422477894090978real_b @ A3 @ B2 ) ) ) ) ).

% eq_snd_iff
thf(fact_686_type__definition__quasi__borel,axiom,
    type_d4563186451463392057real_b @ quasi_4958298574314430518orel_b @ quasi_2002468295286565185orel_b @ ( collec5860786303674561885real_b @ ( produc9150283905253018337al_b_o @ is_quasi_borel_b ) ) ).

% type_definition_quasi_borel
thf(fact_687_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A4: set_nat] :
          ( A4
          = ( insert_nat @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_688_is__singleton__the__elem,axiom,
    ( is_singleton_real
    = ( ^ [A4: set_real] :
          ( A4
          = ( insert_real @ ( the_elem_real @ A4 ) @ bot_bot_set_real ) ) ) ) ).

% is_singleton_the_elem
thf(fact_689_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A4: set_o] :
          ( A4
          = ( insert_o @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_690_separate__measurable,axiom,
    ! [P: real > nat] :
      ( ! [I5: nat] : ( member_set_real @ ( vimage_real_nat @ P @ ( insert_nat @ I5 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) ) ) ).

% separate_measurable
thf(fact_691_measurable__separate,axiom,
    ! [P: real > nat,I: nat] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( member_set_real @ ( vimage_real_nat @ P @ ( insert_nat @ I @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% measurable_separate
thf(fact_692_is__quasi__borel__intro,axiom,
    ! [Mx: set_real_nat,X2: set_nat] :
      ( ( ord_le6098800555920186673al_nat @ Mx
        @ ( pi_real_nat @ top_top_set_real
          @ ^ [Uu: real] : X2 ) )
     => ( ( qbs_closed1_nat @ Mx )
       => ( ( qbs_closed2_nat @ X2 @ Mx )
         => ( ( qbs_closed3_nat @ Mx )
           => ( is_quasi_borel_nat @ X2 @ Mx ) ) ) ) ) ).

% is_quasi_borel_intro
thf(fact_693_is__quasi__borel__intro,axiom,
    ! [Mx: set_real_real_nat,X2: set_real_nat] :
      ( ( ord_le491656556176229628al_nat @ Mx
        @ ( pi_real_real_nat @ top_top_set_real
          @ ^ [Uu: real] : X2 ) )
     => ( ( qbs_closed1_real_nat @ Mx )
       => ( ( qbs_closed2_real_nat @ X2 @ Mx )
         => ( ( qbs_closed3_real_nat @ Mx )
           => ( is_qua3317151984682463396al_nat @ X2 @ Mx ) ) ) ) ) ).

% is_quasi_borel_intro
thf(fact_694_is__quasi__borel__intro,axiom,
    ! [Mx: set_real_real_b,X2: set_real_b] :
      ( ( ord_le227027178093677845real_b @ Mx
        @ ( pi_real_real_b @ top_top_set_real
          @ ^ [Uu: real] : X2 ) )
     => ( ( qbs_closed1_real_b @ Mx )
       => ( ( qbs_closed2_real_b @ X2 @ Mx )
         => ( ( qbs_closed3_real_b @ Mx )
           => ( is_qua1212226743203608129real_b @ X2 @ Mx ) ) ) ) ) ).

% is_quasi_borel_intro
thf(fact_695_is__quasi__borel__intro,axiom,
    ! [Mx: set_real_nat_real,X2: set_nat_real] :
      ( ( ord_le6912594875210535036t_real @ Mx
        @ ( pi_real_nat_real @ top_top_set_real
          @ ^ [Uu: real] : X2 ) )
     => ( ( qbs_closed1_nat_real @ Mx )
       => ( ( qbs_closed2_nat_real @ X2 @ Mx )
         => ( ( qbs_closed3_nat_real @ Mx )
           => ( is_qua7292199676159343652t_real @ X2 @ Mx ) ) ) ) ) ).

% is_quasi_borel_intro
thf(fact_696_is__quasi__borel__intro,axiom,
    ! [Mx: set_real_b,X2: set_b] :
      ( ( ord_le5814440863667440394real_b @ Mx
        @ ( pi_real_b @ top_top_set_real
          @ ^ [Uu: real] : X2 ) )
     => ( ( qbs_closed1_b @ Mx )
       => ( ( qbs_closed2_b @ X2 @ Mx )
         => ( ( qbs_closed3_b @ Mx )
           => ( is_quasi_borel_b @ X2 @ Mx ) ) ) ) ) ).

% is_quasi_borel_intro
thf(fact_697_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_698_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_699_subsetI,axiom,
    ! [A: set_set_nat,B: set_set_nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A )
         => ( member_set_nat @ X3 @ B ) )
     => ( ord_le6893508408891458716et_nat @ A @ B ) ) ).

% subsetI
thf(fact_700_subsetI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A )
         => ( member_nat @ X3 @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% subsetI
thf(fact_701_subsetI,axiom,
    ! [A: set_real_nat,B: set_real_nat] :
      ( ! [X3: real > nat] :
          ( ( member_real_nat @ X3 @ A )
         => ( member_real_nat @ X3 @ B ) )
     => ( ord_le6098800555920186673al_nat @ A @ B ) ) ).

% subsetI
thf(fact_702_subsetI,axiom,
    ! [A: set_real_b,B: set_real_b] :
      ( ! [X3: real > b] :
          ( ( member_real_b @ X3 @ A )
         => ( member_real_b @ X3 @ B ) )
     => ( ord_le5814440863667440394real_b @ A @ B ) ) ).

% subsetI
thf(fact_703_subsetI,axiom,
    ! [A: set_nat_real,B: set_nat_real] :
      ( ! [X3: nat > real] :
          ( ( member_nat_real @ X3 @ A )
         => ( member_nat_real @ X3 @ B ) )
     => ( ord_le2908806416726583473t_real @ A @ B ) ) ).

% subsetI
thf(fact_704_subsetI,axiom,
    ! [A: set_b,B: set_b] :
      ( ! [X3: b] :
          ( ( member_b @ X3 @ A )
         => ( member_b @ X3 @ B ) )
     => ( ord_less_eq_set_b @ A @ B ) ) ).

% subsetI
thf(fact_705_subsetI,axiom,
    ! [A: set_real_nat_real,B: set_real_nat_real] :
      ( ! [X3: real > nat > real] :
          ( ( member_real_nat_real2 @ X3 @ A )
         => ( member_real_nat_real2 @ X3 @ B ) )
     => ( ord_le6912594875210535036t_real @ A @ B ) ) ).

% subsetI
thf(fact_706_subsetI,axiom,
    ! [A: set_real_real_b,B: set_real_real_b] :
      ( ! [X3: real > real > b] :
          ( ( member_real_real_b @ X3 @ A )
         => ( member_real_real_b @ X3 @ B ) )
     => ( ord_le227027178093677845real_b @ A @ B ) ) ).

% subsetI
thf(fact_707_subsetI,axiom,
    ! [A: set_real_real_nat,B: set_real_real_nat] :
      ( ! [X3: real > real > nat] :
          ( ( member_real_real_nat @ X3 @ A )
         => ( member_real_real_nat @ X3 @ B ) )
     => ( ord_le491656556176229628al_nat @ A @ B ) ) ).

% subsetI
thf(fact_708_subsetI,axiom,
    ! [A: set_real_set_nat,B: set_real_set_nat] :
      ( ! [X3: real > set_nat] :
          ( ( member_real_set_nat @ X3 @ A )
         => ( member_real_set_nat @ X3 @ B ) )
     => ( ord_le7035988643939837671et_nat @ A @ B ) ) ).

% subsetI
thf(fact_709_Pair__le,axiom,
    ! [A2: nat,B2: nat,C: nat,D: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( product_Pair_nat_nat @ C @ D ) )
      = ( ( ord_less_eq_nat @ A2 @ C )
        & ( ord_less_eq_nat @ B2 @ D ) ) ) ).

% Pair_le
thf(fact_710_subset__empty,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_711_subset__empty,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% subset_empty
thf(fact_712_subset__empty,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_713_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_714_empty__subsetI,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% empty_subsetI
thf(fact_715_empty__subsetI,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% empty_subsetI
thf(fact_716_insert__subset,axiom,
    ! [X: real,A: set_real,B: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X @ A ) @ B )
      = ( ( member_real @ X @ B )
        & ( ord_less_eq_set_real @ A @ B ) ) ) ).

% insert_subset
thf(fact_717_insert__subset,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X @ A ) @ B )
      = ( ( member_o @ X @ B )
        & ( ord_less_eq_set_o @ A @ B ) ) ) ).

% insert_subset
thf(fact_718_insert__subset,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B )
      = ( ( member_nat @ X @ B )
        & ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_719_insert__subset,axiom,
    ! [X: b,A: set_b,B: set_b] :
      ( ( ord_less_eq_set_b @ ( insert_b @ X @ A ) @ B )
      = ( ( member_b @ X @ B )
        & ( ord_less_eq_set_b @ A @ B ) ) ) ).

% insert_subset
thf(fact_720_insert__subset,axiom,
    ! [X: set_nat,A: set_set_nat,B: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A ) @ B )
      = ( ( member_set_nat @ X @ B )
        & ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_721_insert__subset,axiom,
    ! [X: real > nat,A: set_real_nat,B: set_real_nat] :
      ( ( ord_le6098800555920186673al_nat @ ( insert_real_nat @ X @ A ) @ B )
      = ( ( member_real_nat @ X @ B )
        & ( ord_le6098800555920186673al_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_722_insert__subset,axiom,
    ! [X: real > b,A: set_real_b,B: set_real_b] :
      ( ( ord_le5814440863667440394real_b @ ( insert_real_b @ X @ A ) @ B )
      = ( ( member_real_b @ X @ B )
        & ( ord_le5814440863667440394real_b @ A @ B ) ) ) ).

% insert_subset
thf(fact_723_insert__subset,axiom,
    ! [X: nat > real,A: set_nat_real,B: set_nat_real] :
      ( ( ord_le2908806416726583473t_real @ ( insert_nat_real @ X @ A ) @ B )
      = ( ( member_nat_real @ X @ B )
        & ( ord_le2908806416726583473t_real @ A @ B ) ) ) ).

% insert_subset
thf(fact_724_insert__subset,axiom,
    ! [X: real > set_nat,A: set_real_set_nat,B: set_real_set_nat] :
      ( ( ord_le7035988643939837671et_nat @ ( insert_real_set_nat @ X @ A ) @ B )
      = ( ( member_real_set_nat @ X @ B )
        & ( ord_le7035988643939837671et_nat @ A @ B ) ) ) ).

% insert_subset
thf(fact_725_insert__subset,axiom,
    ! [X: real > nat > real,A: set_real_nat_real,B: set_real_nat_real] :
      ( ( ord_le6912594875210535036t_real @ ( insert_real_nat_real @ X @ A ) @ B )
      = ( ( member_real_nat_real2 @ X @ B )
        & ( ord_le6912594875210535036t_real @ A @ B ) ) ) ).

% insert_subset
thf(fact_726_singleton__insert__inj__eq,axiom,
    ! [B2: nat,A2: nat,A: set_nat] :
      ( ( ( insert_nat @ B2 @ bot_bot_set_nat )
        = ( insert_nat @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_727_singleton__insert__inj__eq,axiom,
    ! [B2: real,A2: real,A: set_real] :
      ( ( ( insert_real @ B2 @ bot_bot_set_real )
        = ( insert_real @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_728_singleton__insert__inj__eq,axiom,
    ! [B2: $o,A2: $o,A: set_o] :
      ( ( ( insert_o @ B2 @ bot_bot_set_o )
        = ( insert_o @ A2 @ A ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_729_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B2: nat] :
      ( ( ( insert_nat @ A2 @ A )
        = ( insert_nat @ B2 @ bot_bot_set_nat ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_730_singleton__insert__inj__eq_H,axiom,
    ! [A2: real,A: set_real,B2: real] :
      ( ( ( insert_real @ A2 @ A )
        = ( insert_real @ B2 @ bot_bot_set_real ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_731_singleton__insert__inj__eq_H,axiom,
    ! [A2: $o,A: set_o,B2: $o] :
      ( ( ( insert_o @ A2 @ A )
        = ( insert_o @ B2 @ bot_bot_set_o ) )
      = ( ( A2 = B2 )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_732_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_733_the__elem__eq,axiom,
    ! [X: real] :
      ( ( the_elem_real @ ( insert_real @ X @ bot_bot_set_real ) )
      = X ) ).

% the_elem_eq
thf(fact_734_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_735_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_736_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_737_ord__le__eq__subst,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ( F @ B2 )
          = C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_738_ord__eq__le__subst,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( A2
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_739_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_740_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_741_order__subst2,axiom,
    ! [A2: nat,B2: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_742_order__subst1,axiom,
    ! [A2: nat,F: nat > nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ! [X3: nat,Y4: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_743_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A3: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A3 @ B5 )
          & ( ord_less_eq_nat @ B5 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_744_antisym,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% antisym
thf(fact_745_dual__order_Otrans,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ C @ B2 )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_746_dual__order_Oantisym,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B2 )
       => ( A2 = B2 ) ) ) ).

% dual_order.antisym
thf(fact_747_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [A3: nat,B5: nat] :
          ( ( ord_less_eq_nat @ B5 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B5 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_748_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A5: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A5 @ B6 )
         => ( P @ A5 @ B6 ) )
     => ( ! [A5: nat,B6: nat] :
            ( ( P @ B6 @ A5 )
           => ( P @ A5 @ B6 ) )
       => ( P @ A2 @ B2 ) ) ) ).

% linorder_wlog
thf(fact_749_order__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_750_order_Otrans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_751_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_752_ord__le__eq__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B2 )
     => ( ( B2 = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_753_ord__eq__le__trans,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( A2 = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_754_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
    = ( ^ [X5: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y3 )
          & ( ord_less_eq_nat @ Y3 @ X5 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_755_le__cases3,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_756_nle__le,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A2 )
        & ( B2 != A2 ) ) ) ).

% nle_le
thf(fact_757_Collect__subset,axiom,
    ! [A: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_758_Collect__subset,axiom,
    ! [A: set_real_nat,P: ( real > nat ) > $o] :
      ( ord_le6098800555920186673al_nat
      @ ( collect_real_nat
        @ ^ [X5: real > nat] :
            ( ( member_real_nat @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_759_Collect__subset,axiom,
    ! [A: set_real_b,P: ( real > b ) > $o] :
      ( ord_le5814440863667440394real_b
      @ ( collect_real_b
        @ ^ [X5: real > b] :
            ( ( member_real_b @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_760_Collect__subset,axiom,
    ! [A: set_nat_real,P: ( nat > real ) > $o] :
      ( ord_le2908806416726583473t_real
      @ ( collect_nat_real
        @ ^ [X5: nat > real] :
            ( ( member_nat_real @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_761_Collect__subset,axiom,
    ! [A: set_b,P: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X5: b] :
            ( ( member_b @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_762_Collect__subset,axiom,
    ! [A: set_real_nat_real,P: ( real > nat > real ) > $o] :
      ( ord_le6912594875210535036t_real
      @ ( collec336724336074912571t_real
        @ ^ [X5: real > nat > real] :
            ( ( member_real_nat_real2 @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_763_Collect__subset,axiom,
    ! [A: set_real_real_b,P: ( real > real > b ) > $o] :
      ( ord_le227027178093677845real_b
      @ ( collect_real_real_b
        @ ^ [X5: real > real > b] :
            ( ( member_real_real_b @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_764_Collect__subset,axiom,
    ! [A: set_real_real_nat,P: ( real > real > nat ) > $o] :
      ( ord_le491656556176229628al_nat
      @ ( collec3526718475268515771al_nat
        @ ^ [X5: real > real > nat] :
            ( ( member_real_real_nat @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_765_Collect__subset,axiom,
    ! [A: set_real_set_nat,P: ( real > set_nat ) > $o] :
      ( ord_le7035988643939837671et_nat
      @ ( collect_real_set_nat
        @ ^ [X5: real > set_nat] :
            ( ( member_real_set_nat @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_766_Collect__subset,axiom,
    ! [A: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ A )
            & ( P @ X5 ) ) )
      @ A ) ).

% Collect_subset
thf(fact_767_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X5: nat] :
            ( ( P @ X5 )
           => ( Q @ X5 ) ) ) ) ).

% Collect_mono_iff
thf(fact_768_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
         => ( Q @ X3 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_769_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
        ! [T: set_nat] :
          ( ( member_set_nat @ T @ A4 )
         => ( member_set_nat @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_770_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
        ! [T: nat] :
          ( ( member_nat @ T @ A4 )
         => ( member_nat @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_771_subset__iff,axiom,
    ( ord_le6098800555920186673al_nat
    = ( ^ [A4: set_real_nat,B4: set_real_nat] :
        ! [T: real > nat] :
          ( ( member_real_nat @ T @ A4 )
         => ( member_real_nat @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_772_subset__iff,axiom,
    ( ord_le5814440863667440394real_b
    = ( ^ [A4: set_real_b,B4: set_real_b] :
        ! [T: real > b] :
          ( ( member_real_b @ T @ A4 )
         => ( member_real_b @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_773_subset__iff,axiom,
    ( ord_le2908806416726583473t_real
    = ( ^ [A4: set_nat_real,B4: set_nat_real] :
        ! [T: nat > real] :
          ( ( member_nat_real @ T @ A4 )
         => ( member_nat_real @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_774_subset__iff,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B4: set_b] :
        ! [T: b] :
          ( ( member_b @ T @ A4 )
         => ( member_b @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_775_subset__iff,axiom,
    ( ord_le6912594875210535036t_real
    = ( ^ [A4: set_real_nat_real,B4: set_real_nat_real] :
        ! [T: real > nat > real] :
          ( ( member_real_nat_real2 @ T @ A4 )
         => ( member_real_nat_real2 @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_776_subset__iff,axiom,
    ( ord_le227027178093677845real_b
    = ( ^ [A4: set_real_real_b,B4: set_real_real_b] :
        ! [T: real > real > b] :
          ( ( member_real_real_b @ T @ A4 )
         => ( member_real_real_b @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_777_subset__iff,axiom,
    ( ord_le491656556176229628al_nat
    = ( ^ [A4: set_real_real_nat,B4: set_real_real_nat] :
        ! [T: real > real > nat] :
          ( ( member_real_real_nat @ T @ A4 )
         => ( member_real_real_nat @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_778_subset__iff,axiom,
    ( ord_le7035988643939837671et_nat
    = ( ^ [A4: set_real_set_nat,B4: set_real_set_nat] :
        ! [T: real > set_nat] :
          ( ( member_real_set_nat @ T @ A4 )
         => ( member_real_set_nat @ T @ B4 ) ) ) ) ).

% subset_iff
thf(fact_779_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
        ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ A4 )
         => ( member_set_nat @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_780_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
        ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( member_nat @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_781_subset__eq,axiom,
    ( ord_le6098800555920186673al_nat
    = ( ^ [A4: set_real_nat,B4: set_real_nat] :
        ! [X5: real > nat] :
          ( ( member_real_nat @ X5 @ A4 )
         => ( member_real_nat @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_782_subset__eq,axiom,
    ( ord_le5814440863667440394real_b
    = ( ^ [A4: set_real_b,B4: set_real_b] :
        ! [X5: real > b] :
          ( ( member_real_b @ X5 @ A4 )
         => ( member_real_b @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_783_subset__eq,axiom,
    ( ord_le2908806416726583473t_real
    = ( ^ [A4: set_nat_real,B4: set_nat_real] :
        ! [X5: nat > real] :
          ( ( member_nat_real @ X5 @ A4 )
         => ( member_nat_real @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_784_subset__eq,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B4: set_b] :
        ! [X5: b] :
          ( ( member_b @ X5 @ A4 )
         => ( member_b @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_785_subset__eq,axiom,
    ( ord_le6912594875210535036t_real
    = ( ^ [A4: set_real_nat_real,B4: set_real_nat_real] :
        ! [X5: real > nat > real] :
          ( ( member_real_nat_real2 @ X5 @ A4 )
         => ( member_real_nat_real2 @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_786_subset__eq,axiom,
    ( ord_le227027178093677845real_b
    = ( ^ [A4: set_real_real_b,B4: set_real_real_b] :
        ! [X5: real > real > b] :
          ( ( member_real_real_b @ X5 @ A4 )
         => ( member_real_real_b @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_787_subset__eq,axiom,
    ( ord_le491656556176229628al_nat
    = ( ^ [A4: set_real_real_nat,B4: set_real_real_nat] :
        ! [X5: real > real > nat] :
          ( ( member_real_real_nat @ X5 @ A4 )
         => ( member_real_real_nat @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_788_subset__eq,axiom,
    ( ord_le7035988643939837671et_nat
    = ( ^ [A4: set_real_set_nat,B4: set_real_set_nat] :
        ! [X5: real > set_nat] :
          ( ( member_real_set_nat @ X5 @ A4 )
         => ( member_real_set_nat @ X5 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_789_subsetD,axiom,
    ! [A: set_set_nat,B: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( member_set_nat @ C @ A )
       => ( member_set_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_790_subsetD,axiom,
    ! [A: set_nat,B: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_791_subsetD,axiom,
    ! [A: set_real_nat,B: set_real_nat,C: real > nat] :
      ( ( ord_le6098800555920186673al_nat @ A @ B )
     => ( ( member_real_nat @ C @ A )
       => ( member_real_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_792_subsetD,axiom,
    ! [A: set_real_b,B: set_real_b,C: real > b] :
      ( ( ord_le5814440863667440394real_b @ A @ B )
     => ( ( member_real_b @ C @ A )
       => ( member_real_b @ C @ B ) ) ) ).

% subsetD
thf(fact_793_subsetD,axiom,
    ! [A: set_nat_real,B: set_nat_real,C: nat > real] :
      ( ( ord_le2908806416726583473t_real @ A @ B )
     => ( ( member_nat_real @ C @ A )
       => ( member_nat_real @ C @ B ) ) ) ).

% subsetD
thf(fact_794_subsetD,axiom,
    ! [A: set_b,B: set_b,C: b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( member_b @ C @ A )
       => ( member_b @ C @ B ) ) ) ).

% subsetD
thf(fact_795_subsetD,axiom,
    ! [A: set_real_nat_real,B: set_real_nat_real,C: real > nat > real] :
      ( ( ord_le6912594875210535036t_real @ A @ B )
     => ( ( member_real_nat_real2 @ C @ A )
       => ( member_real_nat_real2 @ C @ B ) ) ) ).

% subsetD
thf(fact_796_subsetD,axiom,
    ! [A: set_real_real_b,B: set_real_real_b,C: real > real > b] :
      ( ( ord_le227027178093677845real_b @ A @ B )
     => ( ( member_real_real_b @ C @ A )
       => ( member_real_real_b @ C @ B ) ) ) ).

% subsetD
thf(fact_797_subsetD,axiom,
    ! [A: set_real_real_nat,B: set_real_real_nat,C: real > real > nat] :
      ( ( ord_le491656556176229628al_nat @ A @ B )
     => ( ( member_real_real_nat @ C @ A )
       => ( member_real_real_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_798_subsetD,axiom,
    ! [A: set_real_set_nat,B: set_real_set_nat,C: real > set_nat] :
      ( ( ord_le7035988643939837671et_nat @ A @ B )
     => ( ( member_real_set_nat @ C @ A )
       => ( member_real_set_nat @ C @ B ) ) ) ).

% subsetD
thf(fact_799_in__mono,axiom,
    ! [A: set_set_nat,B: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B )
     => ( ( member_set_nat @ X @ A )
       => ( member_set_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_800_in__mono,axiom,
    ! [A: set_nat,B: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_801_in__mono,axiom,
    ! [A: set_real_nat,B: set_real_nat,X: real > nat] :
      ( ( ord_le6098800555920186673al_nat @ A @ B )
     => ( ( member_real_nat @ X @ A )
       => ( member_real_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_802_in__mono,axiom,
    ! [A: set_real_b,B: set_real_b,X: real > b] :
      ( ( ord_le5814440863667440394real_b @ A @ B )
     => ( ( member_real_b @ X @ A )
       => ( member_real_b @ X @ B ) ) ) ).

% in_mono
thf(fact_803_in__mono,axiom,
    ! [A: set_nat_real,B: set_nat_real,X: nat > real] :
      ( ( ord_le2908806416726583473t_real @ A @ B )
     => ( ( member_nat_real @ X @ A )
       => ( member_nat_real @ X @ B ) ) ) ).

% in_mono
thf(fact_804_in__mono,axiom,
    ! [A: set_b,B: set_b,X: b] :
      ( ( ord_less_eq_set_b @ A @ B )
     => ( ( member_b @ X @ A )
       => ( member_b @ X @ B ) ) ) ).

% in_mono
thf(fact_805_in__mono,axiom,
    ! [A: set_real_nat_real,B: set_real_nat_real,X: real > nat > real] :
      ( ( ord_le6912594875210535036t_real @ A @ B )
     => ( ( member_real_nat_real2 @ X @ A )
       => ( member_real_nat_real2 @ X @ B ) ) ) ).

% in_mono
thf(fact_806_in__mono,axiom,
    ! [A: set_real_real_b,B: set_real_real_b,X: real > real > b] :
      ( ( ord_le227027178093677845real_b @ A @ B )
     => ( ( member_real_real_b @ X @ A )
       => ( member_real_real_b @ X @ B ) ) ) ).

% in_mono
thf(fact_807_in__mono,axiom,
    ! [A: set_real_real_nat,B: set_real_real_nat,X: real > real > nat] :
      ( ( ord_le491656556176229628al_nat @ A @ B )
     => ( ( member_real_real_nat @ X @ A )
       => ( member_real_real_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_808_in__mono,axiom,
    ! [A: set_real_set_nat,B: set_real_set_nat,X: real > set_nat] :
      ( ( ord_le7035988643939837671et_nat @ A @ B )
     => ( ( member_real_set_nat @ X @ A )
       => ( member_real_set_nat @ X @ B ) ) ) ).

% in_mono
thf(fact_809_Pi__mono,axiom,
    ! [A: set_real,B: real > set_b,C3: real > set_b] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( ord_less_eq_set_b @ ( B @ X3 ) @ ( C3 @ X3 ) ) )
     => ( ord_le5814440863667440394real_b @ ( pi_real_b @ A @ B ) @ ( pi_real_b @ A @ C3 ) ) ) ).

% Pi_mono
thf(fact_810_Pi__mono,axiom,
    ! [A: set_real,B: real > set_nat,C3: real > set_nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A )
         => ( ord_less_eq_set_nat @ ( B @ X3 ) @ ( C3 @ X3 ) ) )
     => ( ord_le6098800555920186673al_nat @ ( pi_real_nat @ A @ B ) @ ( pi_real_nat @ A @ C3 ) ) ) ).

% Pi_mono
thf(fact_811_Pi__anti__mono,axiom,
    ! [A6: set_real,A: set_real,B: real > set_b] :
      ( ( ord_less_eq_set_real @ A6 @ A )
     => ( ord_le5814440863667440394real_b @ ( pi_real_b @ A @ B ) @ ( pi_real_b @ A6 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_812_Pi__anti__mono,axiom,
    ! [A6: set_real,A: set_real,B: real > set_nat] :
      ( ( ord_less_eq_set_real @ A6 @ A )
     => ( ord_le6098800555920186673al_nat @ ( pi_real_nat @ A @ B ) @ ( pi_real_nat @ A6 @ B ) ) ) ).

% Pi_anti_mono
thf(fact_813_snd__mono,axiom,
    ! [X: produc4373655991844504306real_b,Y: produc4373655991844504306real_b] :
      ( ( ord_le209527602574031506real_b @ X @ Y )
     => ( ord_le5814440863667440394real_b @ ( produc1187263614985732624real_b @ X ) @ ( produc1187263614985732624real_b @ Y ) ) ) ).

% snd_mono
thf(fact_814_measurable__compose,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,G: real > b,L: sigma_measure_b] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( ( member_real_b @ G @ ( sigma_523072396149930113real_b @ N @ L ) )
       => ( member_real_b
          @ ^ [X5: real] : ( G @ ( F @ X5 ) )
          @ ( sigma_523072396149930113real_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_815_measurable__compose,axiom,
    ! [F: real > b,M: sigma_measure_real,N: sigma_measure_b,G: b > b,L: sigma_measure_b] :
      ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ N ) )
     => ( ( member_b_b @ G @ ( sigma_measurable_b_b @ N @ L ) )
       => ( member_real_b
          @ ^ [X5: real] : ( G @ ( F @ X5 ) )
          @ ( sigma_523072396149930113real_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_816_measurable__compose,axiom,
    ! [F: real > b,M: sigma_measure_real,N: sigma_measure_b,G: b > nat,L: sigma_measure_nat] :
      ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ N ) )
     => ( ( member_b_nat @ G @ ( sigma_1308594411581951615_b_nat @ N @ L ) )
       => ( member_real_nat
          @ ^ [X5: real] : ( G @ ( F @ X5 ) )
          @ ( sigma_6315060578831106510al_nat @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_817_measurable__compose,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,G: real > nat,L: sigma_measure_nat] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ N @ L ) )
       => ( member_real_nat
          @ ^ [X5: real] : ( G @ ( F @ X5 ) )
          @ ( sigma_6315060578831106510al_nat @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_818_measurable__compose,axiom,
    ! [F: nat > nat,M: sigma_measure_nat,N: sigma_measure_nat,G: nat > real,L: sigma_measure_real] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ N ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ N @ L ) )
       => ( member_nat_real
          @ ^ [X5: nat] : ( G @ ( F @ X5 ) )
          @ ( sigma_1747752005702207822t_real @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_819_measurable__compose,axiom,
    ! [F: real > nat,M: sigma_measure_real,N: sigma_measure_nat,G: nat > b,L: sigma_measure_b] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ N ) )
     => ( ( member_nat_b @ G @ ( sigma_4105081583803843549_nat_b @ N @ L ) )
       => ( member_real_b
          @ ^ [X5: real] : ( G @ ( F @ X5 ) )
          @ ( sigma_523072396149930113real_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_820_measurable__compose,axiom,
    ! [F: real > nat,M: sigma_measure_real,N: sigma_measure_nat,G: nat > nat,L: sigma_measure_nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ N ) )
     => ( ( member_nat_nat @ G @ ( sigma_4350458207664084850at_nat @ N @ L ) )
       => ( member_real_nat
          @ ^ [X5: real] : ( G @ ( F @ X5 ) )
          @ ( sigma_6315060578831106510al_nat @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_821_measurable__compose,axiom,
    ! [F: real > nat,M: sigma_measure_real,N: sigma_measure_nat,G: nat > real,L: sigma_measure_real] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ N ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ N @ L ) )
       => ( member_real_real
          @ ^ [X5: real] : ( G @ ( F @ X5 ) )
          @ ( sigma_5267869275261027754l_real @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_822_measurable__compose,axiom,
    ! [F: nat > real,M: sigma_measure_nat,N: sigma_measure_real,G: real > b,L: sigma_measure_b] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ N ) )
     => ( ( member_real_b @ G @ ( sigma_523072396149930113real_b @ N @ L ) )
       => ( member_nat_b
          @ ^ [X5: nat] : ( G @ ( F @ X5 ) )
          @ ( sigma_4105081583803843549_nat_b @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_823_measurable__compose,axiom,
    ! [F: nat > real,M: sigma_measure_nat,N: sigma_measure_real,G: real > real,L: sigma_measure_real] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ N ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ N @ L ) )
       => ( member_nat_real
          @ ^ [X5: nat] : ( G @ ( F @ X5 ) )
          @ ( sigma_1747752005702207822t_real @ M @ L ) ) ) ) ).

% measurable_compose
thf(fact_824_measurable__compose__rev,axiom,
    ! [F: b > b,L: sigma_measure_b,N: sigma_measure_b,G: real > b,M: sigma_measure_real] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ L @ N ) )
     => ( ( member_real_b @ G @ ( sigma_523072396149930113real_b @ M @ L ) )
       => ( member_real_b
          @ ^ [X5: real] : ( F @ ( G @ X5 ) )
          @ ( sigma_523072396149930113real_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_825_measurable__compose__rev,axiom,
    ! [F: real > b,L: sigma_measure_real,N: sigma_measure_b,G: real > real,M: sigma_measure_real] :
      ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ L @ N ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ L ) )
       => ( member_real_b
          @ ^ [X5: real] : ( F @ ( G @ X5 ) )
          @ ( sigma_523072396149930113real_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_826_measurable__compose__rev,axiom,
    ! [F: b > nat,L: sigma_measure_b,N: sigma_measure_nat,G: real > b,M: sigma_measure_real] :
      ( ( member_b_nat @ F @ ( sigma_1308594411581951615_b_nat @ L @ N ) )
     => ( ( member_real_b @ G @ ( sigma_523072396149930113real_b @ M @ L ) )
       => ( member_real_nat
          @ ^ [X5: real] : ( F @ ( G @ X5 ) )
          @ ( sigma_6315060578831106510al_nat @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_827_measurable__compose__rev,axiom,
    ! [F: nat > b,L: sigma_measure_nat,N: sigma_measure_b,G: real > nat,M: sigma_measure_real] :
      ( ( member_nat_b @ F @ ( sigma_4105081583803843549_nat_b @ L @ N ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ L ) )
       => ( member_real_b
          @ ^ [X5: real] : ( F @ ( G @ X5 ) )
          @ ( sigma_523072396149930113real_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_828_measurable__compose__rev,axiom,
    ! [F: nat > nat,L: sigma_measure_nat,N: sigma_measure_nat,G: real > nat,M: sigma_measure_real] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ L @ N ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ L ) )
       => ( member_real_nat
          @ ^ [X5: real] : ( F @ ( G @ X5 ) )
          @ ( sigma_6315060578831106510al_nat @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_829_measurable__compose__rev,axiom,
    ! [F: real > b,L: sigma_measure_real,N: sigma_measure_b,G: nat > real,M: sigma_measure_nat] :
      ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ L @ N ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ L ) )
       => ( member_nat_b
          @ ^ [X5: nat] : ( F @ ( G @ X5 ) )
          @ ( sigma_4105081583803843549_nat_b @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_830_measurable__compose__rev,axiom,
    ! [F: real > real,L: sigma_measure_real,N: sigma_measure_real,G: nat > real,M: sigma_measure_nat] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ L @ N ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ L ) )
       => ( member_nat_real
          @ ^ [X5: nat] : ( F @ ( G @ X5 ) )
          @ ( sigma_1747752005702207822t_real @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_831_measurable__compose__rev,axiom,
    ! [F: real > nat,L: sigma_measure_real,N: sigma_measure_nat,G: real > real,M: sigma_measure_real] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ L @ N ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ L ) )
       => ( member_real_nat
          @ ^ [X5: real] : ( F @ ( G @ X5 ) )
          @ ( sigma_6315060578831106510al_nat @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_832_measurable__compose__rev,axiom,
    ! [F: real > nat,L: sigma_measure_real,N: sigma_measure_nat,G: nat > real,M: sigma_measure_nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ L @ N ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ L ) )
       => ( member_nat_nat
          @ ^ [X5: nat] : ( F @ ( G @ X5 ) )
          @ ( sigma_4350458207664084850at_nat @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_833_measurable__compose__rev,axiom,
    ! [F: nat > real,L: sigma_measure_nat,N: sigma_measure_real,G: nat > nat,M: sigma_measure_nat] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ L @ N ) )
     => ( ( member_nat_nat @ G @ ( sigma_4350458207664084850at_nat @ M @ L ) )
       => ( member_nat_real
          @ ^ [X5: nat] : ( F @ ( G @ X5 ) )
          @ ( sigma_1747752005702207822t_real @ M @ N ) ) ) ) ).

% measurable_compose_rev
thf(fact_834_Pair__mono,axiom,
    ! [X: nat,X6: nat,Y: nat,Y5: nat] :
      ( ( ord_less_eq_nat @ X @ X6 )
     => ( ( ord_less_eq_nat @ Y @ Y5 )
       => ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X6 @ Y5 ) ) ) ) ).

% Pair_mono
thf(fact_835_borel__measurable__Pair,axiom,
    ! [F: nat > real,M: sigma_measure_nat,G: nat > real] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
       => ( member4944736856719700937l_real
          @ ^ [X5: nat] : ( produc4511245868158468465l_real @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_5893925349391517463l_real @ M @ borel_9155112475215227991l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_836_borel__measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > nat] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member4159480375084375953al_nat
          @ ^ [X5: real] : ( produc3181502643871035669al_nat @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_8223622787033862879al_nat @ M @ borel_8300761668267423611al_nat ) ) ) ) ).

% borel_measurable_Pair
thf(fact_837_borel__measurable__Pair,axiom,
    ! [F: nat > real,M: sigma_measure_nat,G: nat > nat] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_nat_nat @ G @ ( sigma_4350458207664084850at_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member7427358445142253677al_nat
          @ ^ [X5: nat] : ( produc3181502643871035669al_nat @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_4775202647280382523al_nat @ M @ borel_8300761668267423611al_nat ) ) ) ) ).

% borel_measurable_Pair
thf(fact_838_borel__measurable__Pair,axiom,
    ! [F: nat > nat,M: sigma_measure_nat,G: nat > real] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
       => ( member4237364305948650477t_real
          @ ^ [X5: nat] : ( produc7837566107596912789t_real @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_8750250338757262779t_real @ M @ borel_3052437322889528059t_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_839_borel__measurable__Pair,axiom,
    ! [F: real > nat,M: sigma_measure_real,G: real > real] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member969486235890772753t_real
          @ ^ [X5: real] : ( produc7837566107596912789t_real @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_2975298441655967327t_real @ M @ borel_3052437322889528059t_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_840_borel__measurable__Pair,axiom,
    ! [F: real > nat,M: sigma_measure_real,G: real > nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member278708040391457717at_nat
          @ ^ [X5: real] : ( product_Pair_nat_nat @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_707772041119837571at_nat @ M @ borel_2901874158916493343at_nat ) ) ) ) ).

% borel_measurable_Pair
thf(fact_841_borel__measurable__Pair,axiom,
    ! [F: real > nat > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_nat_real2 @ F @ ( sigma_783869947231497753t_real @ M @ borel_5725509229735958141t_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member4771552270248568924l_real
          @ ^ [X5: real] : ( produc7188456909510567776l_real @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_8384661788165733162l_real @ M @ borel_5246341344715896774l_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_842_borel__measurable__Pair,axiom,
    ! [F: real > nat > real,M: sigma_measure_real,G: real > nat] :
      ( ( member_real_nat_real2 @ F @ ( sigma_783869947231497753t_real @ M @ borel_5725509229735958141t_real ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member1204174549408558720al_nat
          @ ^ [X5: real] : ( produc157057340832456324al_nat @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_1311731663836594126al_nat @ M @ borel_8282316692848364138al_nat ) ) ) ) ).

% borel_measurable_Pair
thf(fact_843_borel__measurable__Pair,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > nat > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_nat_real2 @ G @ ( sigma_783869947231497753t_real @ M @ borel_5725509229735958141t_real ) )
       => ( member3032753491741168348t_real
          @ ^ [X5: real] : ( produc8719833358983803744t_real @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_1261203090334623146t_real @ M @ borel_7346254683739562566t_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_844_borel__measurable__Pair,axiom,
    ! [F: real > nat,M: sigma_measure_real,G: real > nat > real] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_nat_real2 @ G @ ( sigma_783869947231497753t_real @ M @ borel_5725509229735958141t_real ) )
       => ( member8070198322841392896t_real
          @ ^ [X5: real] : ( produc2678254361277162628t_real @ ( F @ X5 ) @ ( G @ X5 ) )
          @ ( sigma_3720449801582733390t_real @ M @ borel_1467662793739727594t_real ) ) ) ) ).

% borel_measurable_Pair
thf(fact_845_measurable__cong__sets,axiom,
    ! [M: sigma_measure_real,M2: sigma_measure_real,N: sigma_measure_real,N2: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M2 ) )
     => ( ( ( sigma_sets_real @ N )
          = ( sigma_sets_real @ N2 ) )
       => ( ( sigma_5267869275261027754l_real @ M @ N )
          = ( sigma_5267869275261027754l_real @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_846_measurable__cong__sets,axiom,
    ! [M: sigma_measure_real,M2: sigma_measure_real,N: sigma_measure_o,N2: sigma_measure_o] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M2 ) )
     => ( ( ( sigma_sets_o @ N )
          = ( sigma_sets_o @ N2 ) )
       => ( ( sigma_3939073009482781210real_o @ M @ N )
          = ( sigma_3939073009482781210real_o @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_847_measurable__cong__sets,axiom,
    ! [M: sigma_measure_nat,M2: sigma_measure_nat,N: sigma_measure_nat,N2: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ M2 ) )
     => ( ( ( sigma_sets_nat @ N )
          = ( sigma_sets_nat @ N2 ) )
       => ( ( sigma_4350458207664084850at_nat @ M @ N )
          = ( sigma_4350458207664084850at_nat @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_848_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_849_measurable__cong__sets,axiom,
    ! [M: sigma_measure_o,M2: sigma_measure_o,N: sigma_measure_real,N2: sigma_measure_real] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ M2 ) )
     => ( ( ( sigma_sets_real @ N )
          = ( sigma_sets_real @ N2 ) )
       => ( ( sigma_2430008634441611636o_real @ M @ N )
          = ( sigma_2430008634441611636o_real @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_850_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_851_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_852_measurable__cong__sets,axiom,
    ! [M: sigma_measure_real,M2: sigma_measure_real,N: sigma_measure_nat,N2: sigma_measure_nat] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M2 ) )
     => ( ( ( sigma_sets_nat @ N )
          = ( sigma_sets_nat @ N2 ) )
       => ( ( sigma_6315060578831106510al_nat @ M @ N )
          = ( sigma_6315060578831106510al_nat @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_853_measurable__cong__sets,axiom,
    ! [M: sigma_measure_nat,M2: sigma_measure_nat,N: sigma_measure_real,N2: sigma_measure_real] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ M2 ) )
     => ( ( ( sigma_sets_real @ N )
          = ( sigma_sets_real @ N2 ) )
       => ( ( sigma_1747752005702207822t_real @ M @ N )
          = ( sigma_1747752005702207822t_real @ M2 @ N2 ) ) ) ) ).

% measurable_cong_sets
thf(fact_854_top_Oextremum__uniqueI,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
     => ( A2 = top_top_set_real ) ) ).

% top.extremum_uniqueI
thf(fact_855_top_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
     => ( A2 = top_top_set_nat ) ) ).

% top.extremum_uniqueI
thf(fact_856_top_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
     => ( A2 = top_top_set_o ) ) ).

% top.extremum_uniqueI
thf(fact_857_top_Oextremum__unique,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ top_top_set_real @ A2 )
      = ( A2 = top_top_set_real ) ) ).

% top.extremum_unique
thf(fact_858_top_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
      = ( A2 = top_top_set_nat ) ) ).

% top.extremum_unique
thf(fact_859_top_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ top_top_set_o @ A2 )
      = ( A2 = top_top_set_o ) ) ).

% top.extremum_unique
thf(fact_860_top__greatest,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).

% top_greatest
thf(fact_861_top__greatest,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% top_greatest
thf(fact_862_top__greatest,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).

% top_greatest
thf(fact_863_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
     => ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_864_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
     => ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_865_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
     => ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_866_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_867_bot_Oextremum__unique,axiom,
    ! [A2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
      = ( A2 = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_868_bot_Oextremum__unique,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
      = ( A2 = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_869_bot_Oextremum__unique,axiom,
    ! [A2: set_o] :
      ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
      = ( A2 = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_870_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_871_bot_Oextremum,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% bot.extremum
thf(fact_872_bot_Oextremum,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% bot.extremum
thf(fact_873_bot_Oextremum,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% bot.extremum
thf(fact_874_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_875_subset__UNIV,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).

% subset_UNIV
thf(fact_876_subset__UNIV,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_877_subset__UNIV,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).

% subset_UNIV
thf(fact_878_Set_Oinsert__mono,axiom,
    ! [C3: set_nat,D2: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ C3 @ D2 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C3 ) @ ( insert_nat @ A2 @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_879_Set_Oinsert__mono,axiom,
    ! [C3: set_real,D2: set_real,A2: real] :
      ( ( ord_less_eq_set_real @ C3 @ D2 )
     => ( ord_less_eq_set_real @ ( insert_real @ A2 @ C3 ) @ ( insert_real @ A2 @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_880_Set_Oinsert__mono,axiom,
    ! [C3: set_o,D2: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ C3 @ D2 )
     => ( ord_less_eq_set_o @ ( insert_o @ A2 @ C3 ) @ ( insert_o @ A2 @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_881_subset__insert,axiom,
    ! [X: real,A: set_real,B: set_real] :
      ( ~ ( member_real @ X @ A )
     => ( ( ord_less_eq_set_real @ A @ ( insert_real @ X @ B ) )
        = ( ord_less_eq_set_real @ A @ B ) ) ) ).

% subset_insert
thf(fact_882_subset__insert,axiom,
    ! [X: $o,A: set_o,B: set_o] :
      ( ~ ( member_o @ X @ A )
     => ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ B ) )
        = ( ord_less_eq_set_o @ A @ B ) ) ) ).

% subset_insert
thf(fact_883_subset__insert,axiom,
    ! [X: nat,A: set_nat,B: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B ) )
        = ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% subset_insert
thf(fact_884_subset__insert,axiom,
    ! [X: b,A: set_b,B: set_b] :
      ( ~ ( member_b @ X @ A )
     => ( ( ord_less_eq_set_b @ A @ ( insert_b @ X @ B ) )
        = ( ord_less_eq_set_b @ A @ B ) ) ) ).

% subset_insert
thf(fact_885_subset__insert,axiom,
    ! [X: set_nat,A: set_set_nat,B: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A )
     => ( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X @ B ) )
        = ( ord_le6893508408891458716et_nat @ A @ B ) ) ) ).

% subset_insert
thf(fact_886_subset__insert,axiom,
    ! [X: real > nat,A: set_real_nat,B: set_real_nat] :
      ( ~ ( member_real_nat @ X @ A )
     => ( ( ord_le6098800555920186673al_nat @ A @ ( insert_real_nat @ X @ B ) )
        = ( ord_le6098800555920186673al_nat @ A @ B ) ) ) ).

% subset_insert
thf(fact_887_subset__insert,axiom,
    ! [X: real > b,A: set_real_b,B: set_real_b] :
      ( ~ ( member_real_b @ X @ A )
     => ( ( ord_le5814440863667440394real_b @ A @ ( insert_real_b @ X @ B ) )
        = ( ord_le5814440863667440394real_b @ A @ B ) ) ) ).

% subset_insert
thf(fact_888_subset__insert,axiom,
    ! [X: nat > real,A: set_nat_real,B: set_nat_real] :
      ( ~ ( member_nat_real @ X @ A )
     => ( ( ord_le2908806416726583473t_real @ A @ ( insert_nat_real @ X @ B ) )
        = ( ord_le2908806416726583473t_real @ A @ B ) ) ) ).

% subset_insert
thf(fact_889_subset__insert,axiom,
    ! [X: real > set_nat,A: set_real_set_nat,B: set_real_set_nat] :
      ( ~ ( member_real_set_nat @ X @ A )
     => ( ( ord_le7035988643939837671et_nat @ A @ ( insert_real_set_nat @ X @ B ) )
        = ( ord_le7035988643939837671et_nat @ A @ B ) ) ) ).

% subset_insert
thf(fact_890_subset__insert,axiom,
    ! [X: real > nat > real,A: set_real_nat_real,B: set_real_nat_real] :
      ( ~ ( member_real_nat_real2 @ X @ A )
     => ( ( ord_le6912594875210535036t_real @ A @ ( insert_real_nat_real @ X @ B ) )
        = ( ord_le6912594875210535036t_real @ A @ B ) ) ) ).

% subset_insert
thf(fact_891_subset__insertI,axiom,
    ! [B: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A2 @ B ) ) ).

% subset_insertI
thf(fact_892_subset__insertI,axiom,
    ! [B: set_real,A2: real] : ( ord_less_eq_set_real @ B @ ( insert_real @ A2 @ B ) ) ).

% subset_insertI
thf(fact_893_subset__insertI,axiom,
    ! [B: set_o,A2: $o] : ( ord_less_eq_set_o @ B @ ( insert_o @ A2 @ B ) ) ).

% subset_insertI
thf(fact_894_subset__insertI2,axiom,
    ! [A: set_nat,B: set_nat,B2: nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_895_subset__insertI2,axiom,
    ! [A: set_real,B: set_real,B2: real] :
      ( ( ord_less_eq_set_real @ A @ B )
     => ( ord_less_eq_set_real @ A @ ( insert_real @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_896_subset__insertI2,axiom,
    ! [A: set_o,B: set_o,B2: $o] :
      ( ( ord_less_eq_set_o @ A @ B )
     => ( ord_less_eq_set_o @ A @ ( insert_o @ B2 @ B ) ) ) ).

% subset_insertI2
thf(fact_897_vimage__mono,axiom,
    ! [A: set_nat,B: set_nat,F: real > nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ord_less_eq_set_real @ ( vimage_real_nat @ F @ A ) @ ( vimage_real_nat @ F @ B ) ) ) ).

% vimage_mono
thf(fact_898_subset__vimage__iff,axiom,
    ! [A: set_real,F: real > nat,B: set_nat] :
      ( ( ord_less_eq_set_real @ A @ ( vimage_real_nat @ F @ B ) )
      = ( ! [X5: real] :
            ( ( member_real @ X5 @ A )
           => ( member_nat @ ( F @ X5 ) @ B ) ) ) ) ).

% subset_vimage_iff
thf(fact_899_measurable__ident__sets,axiom,
    ! [M: sigma_measure_real,M2: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M2 ) )
     => ( member_real_real
        @ ^ [X5: real] : X5
        @ ( sigma_5267869275261027754l_real @ M @ M2 ) ) ) ).

% measurable_ident_sets
thf(fact_900_measurable__ident__sets,axiom,
    ! [M: sigma_measure_nat,M2: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ M2 ) )
     => ( member_nat_nat
        @ ^ [X5: nat] : X5
        @ ( sigma_4350458207664084850at_nat @ M @ M2 ) ) ) ).

% measurable_ident_sets
thf(fact_901_measurable__ident__sets,axiom,
    ! [M: sigma_measure_o,M2: sigma_measure_o] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ M2 ) )
     => ( member_o_o
        @ ^ [X5: $o] : X5
        @ ( sigma_measurable_o_o @ M @ M2 ) ) ) ).

% measurable_ident_sets
thf(fact_902_borel__measurable__const,axiom,
    ! [C: nat > real,M: sigma_measure_real] :
      ( member_real_nat_real2
      @ ^ [X5: real] : C
      @ ( sigma_783869947231497753t_real @ M @ borel_5725509229735958141t_real ) ) ).

% borel_measurable_const
thf(fact_903_borel__measurable__const,axiom,
    ! [C: real > nat,M: sigma_measure_real] :
      ( member_real_real_nat
      @ ^ [X5: real] : C
      @ ( sigma_6032194292609393305al_nat @ M @ borel_1750461538259077885al_nat ) ) ).

% borel_measurable_const
thf(fact_904_borel__measurable__const,axiom,
    ! [C: real,M: sigma_measure_nat] :
      ( member_nat_real
      @ ^ [X5: nat] : C
      @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) ) ).

% borel_measurable_const
thf(fact_905_borel__measurable__const,axiom,
    ! [C: nat,M: sigma_measure_real] :
      ( member_real_nat
      @ ^ [X5: real] : C
      @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) ) ).

% borel_measurable_const
thf(fact_906_subset__singletonD,axiom,
    ! [A: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A = bot_bot_set_nat )
        | ( A
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_907_subset__singletonD,axiom,
    ! [A: set_real,X: real] :
      ( ( ord_less_eq_set_real @ A @ ( insert_real @ X @ bot_bot_set_real ) )
     => ( ( A = bot_bot_set_real )
        | ( A
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_908_subset__singletonD,axiom,
    ! [A: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A @ ( insert_o @ X @ bot_bot_set_o ) )
     => ( ( A = bot_bot_set_o )
        | ( A
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_909_subset__singleton__iff,axiom,
    ! [X2: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ X2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( ( X2 = bot_bot_set_nat )
        | ( X2
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_910_subset__singleton__iff,axiom,
    ! [X2: set_real,A2: real] :
      ( ( ord_less_eq_set_real @ X2 @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( ( X2 = bot_bot_set_real )
        | ( X2
          = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_911_subset__singleton__iff,axiom,
    ! [X2: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ X2 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( ( X2 = bot_bot_set_o )
        | ( X2
          = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_912_measurable__sets__borel,axiom,
    ! [F: real > b,M: sigma_measure_b,A: set_b] :
      ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ borel_5078946678739801102l_real @ M ) )
     => ( ( member_set_b @ A @ ( sigma_sets_b @ M ) )
       => ( member_set_real @ ( vimage_real_b @ F @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).

% measurable_sets_borel
thf(fact_913_measurable__sets__borel,axiom,
    ! [F: real > real,M: sigma_measure_real,A: set_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ M ) )
     => ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
       => ( member_set_real @ ( vimage_real_real @ F @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).

% measurable_sets_borel
thf(fact_914_measurable__sets__borel,axiom,
    ! [F: real > $o,M: sigma_measure_o,A: set_o] :
      ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ M ) )
     => ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
       => ( member_set_real @ ( vimage_real_o @ F @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).

% measurable_sets_borel
thf(fact_915_measurable__sets__borel,axiom,
    ! [F: real > nat,M: sigma_measure_nat,A: set_nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ M ) )
     => ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
       => ( member_set_real @ ( vimage_real_nat @ F @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).

% measurable_sets_borel
thf(fact_916_measurable__sets__borel,axiom,
    ! [F: nat > nat,M: sigma_measure_nat,A: set_nat] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ borel_8449730974584783410el_nat @ M ) )
     => ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ) ).

% measurable_sets_borel
thf(fact_917_measurable__sets__borel,axiom,
    ! [F: nat > $o,M: sigma_measure_o,A: set_o] :
      ( ( member_nat_o @ F @ ( sigma_5101835498682829686_nat_o @ borel_8449730974584783410el_nat @ M ) )
     => ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
       => ( member_set_nat @ ( vimage_nat_o @ F @ A ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ) ).

% measurable_sets_borel
thf(fact_918_measurable__sets__borel,axiom,
    ! [F: nat > real,M: sigma_measure_real,A: set_real] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ M ) )
     => ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
       => ( member_set_nat @ ( vimage_nat_real @ F @ A ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ) ).

% measurable_sets_borel
thf(fact_919_measurable__sets__borel,axiom,
    ! [F: $o > real,M: sigma_measure_real,A: set_real] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ M ) )
     => ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
       => ( member_set_o @ ( vimage_o_real @ F @ A ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ) ).

% measurable_sets_borel
thf(fact_920_measurable__sets__borel,axiom,
    ! [F: $o > nat,M: sigma_measure_nat,A: set_nat] :
      ( ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ borel_5500255247093592246orel_o @ M ) )
     => ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
       => ( member_set_o @ ( vimage_o_nat @ F @ A ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ) ).

% measurable_sets_borel
thf(fact_921_measurable__sets__borel,axiom,
    ! [F: $o > $o,M: sigma_measure_o,A: set_o] :
      ( ( member_o_o @ F @ ( sigma_measurable_o_o @ borel_5500255247093592246orel_o @ M ) )
     => ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
       => ( member_set_o @ ( vimage_o_o @ F @ A ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ) ).

% measurable_sets_borel
thf(fact_922_qbs__closed3I_H,axiom,
    ! [Mx: set_real_nat] :
      ( ! [P3: real > nat,Fi2: nat > real > nat] :
          ( ( member_real_nat @ P3 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
         => ( ! [I4: nat] : ( member_real_nat @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_nat
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_nat @ Mx ) ) ).

% qbs_closed3I'
thf(fact_923_qbs__closed3I_H,axiom,
    ! [Mx: set_real_b] :
      ( ! [P3: real > nat,Fi2: nat > real > b] :
          ( ( member_real_nat @ P3 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
         => ( ! [I4: nat] : ( member_real_b @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_b
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_b @ Mx ) ) ).

% qbs_closed3I'
thf(fact_924_qbs__closed3I_H,axiom,
    ! [Mx: set_real_nat_real] :
      ( ! [P3: real > nat,Fi2: nat > real > nat > real] :
          ( ( member_real_nat @ P3 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
         => ( ! [I4: nat] : ( member_real_nat_real2 @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_nat_real2
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_nat_real @ Mx ) ) ).

% qbs_closed3I'
thf(fact_925_qbs__closed3I_H,axiom,
    ! [Mx: set_real_real_b] :
      ( ! [P3: real > nat,Fi2: nat > real > real > b] :
          ( ( member_real_nat @ P3 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
         => ( ! [I4: nat] : ( member_real_real_b @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_real_b
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_real_b @ Mx ) ) ).

% qbs_closed3I'
thf(fact_926_qbs__closed3I_H,axiom,
    ! [Mx: set_real_real_nat] :
      ( ! [P3: real > nat,Fi2: nat > real > real > nat] :
          ( ( member_real_nat @ P3 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
         => ( ! [I4: nat] : ( member_real_real_nat @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_real_nat
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_real_nat @ Mx ) ) ).

% qbs_closed3I'
thf(fact_927_qbs__closed3I_H,axiom,
    ! [Mx: set_real_set_nat] :
      ( ! [P3: real > nat,Fi2: nat > real > set_nat] :
          ( ( member_real_nat @ P3 @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
         => ( ! [I4: nat] : ( member_real_set_nat @ ( Fi2 @ I4 ) @ Mx )
           => ( member_real_set_nat
              @ ^ [R: real] : ( Fi2 @ ( P3 @ R ) @ R )
              @ Mx ) ) )
     => ( qbs_closed3_set_nat @ Mx ) ) ).

% qbs_closed3I'
thf(fact_928_is__quasi__borel__def,axiom,
    ( is_quasi_borel_nat
    = ( ^ [X4: set_nat,Mx2: set_real_nat] :
          ( ( ord_le6098800555920186673al_nat @ Mx2
            @ ( pi_real_nat @ top_top_set_real
              @ ^ [Uu: real] : X4 ) )
          & ( qbs_closed1_nat @ Mx2 )
          & ( qbs_closed2_nat @ X4 @ Mx2 )
          & ( qbs_closed3_nat @ Mx2 ) ) ) ) ).

% is_quasi_borel_def
thf(fact_929_is__quasi__borel__def,axiom,
    ( is_qua3317151984682463396al_nat
    = ( ^ [X4: set_real_nat,Mx2: set_real_real_nat] :
          ( ( ord_le491656556176229628al_nat @ Mx2
            @ ( pi_real_real_nat @ top_top_set_real
              @ ^ [Uu: real] : X4 ) )
          & ( qbs_closed1_real_nat @ Mx2 )
          & ( qbs_closed2_real_nat @ X4 @ Mx2 )
          & ( qbs_closed3_real_nat @ Mx2 ) ) ) ) ).

% is_quasi_borel_def
thf(fact_930_is__quasi__borel__def,axiom,
    ( is_qua1212226743203608129real_b
    = ( ^ [X4: set_real_b,Mx2: set_real_real_b] :
          ( ( ord_le227027178093677845real_b @ Mx2
            @ ( pi_real_real_b @ top_top_set_real
              @ ^ [Uu: real] : X4 ) )
          & ( qbs_closed1_real_b @ Mx2 )
          & ( qbs_closed2_real_b @ X4 @ Mx2 )
          & ( qbs_closed3_real_b @ Mx2 ) ) ) ) ).

% is_quasi_borel_def
thf(fact_931_is__quasi__borel__def,axiom,
    ( is_qua7292199676159343652t_real
    = ( ^ [X4: set_nat_real,Mx2: set_real_nat_real] :
          ( ( ord_le6912594875210535036t_real @ Mx2
            @ ( pi_real_nat_real @ top_top_set_real
              @ ^ [Uu: real] : X4 ) )
          & ( qbs_closed1_nat_real @ Mx2 )
          & ( qbs_closed2_nat_real @ X4 @ Mx2 )
          & ( qbs_closed3_nat_real @ Mx2 ) ) ) ) ).

% is_quasi_borel_def
thf(fact_932_is__quasi__borel__def,axiom,
    ( is_quasi_borel_b
    = ( ^ [X4: set_b,Mx2: set_real_b] :
          ( ( ord_le5814440863667440394real_b @ Mx2
            @ ( pi_real_b @ top_top_set_real
              @ ^ [Uu: real] : X4 ) )
          & ( qbs_closed1_b @ Mx2 )
          & ( qbs_closed2_b @ X4 @ Mx2 )
          & ( qbs_closed3_b @ Mx2 ) ) ) ) ).

% is_quasi_borel_def
thf(fact_933_is__borel__def,axiom,
    ( borel_7664774927250867717t_real
    = ( ^ [F2: real > nat > real,M3: sigma_measure_real] : ( member_real_nat_real2 @ F2 @ ( sigma_783869947231497753t_real @ M3 @ borel_5725509229735958141t_real ) ) ) ) ).

% is_borel_def
thf(fact_934_is__borel__def,axiom,
    ( borel_3689727235773987461al_nat
    = ( ^ [F2: real > real > nat,M3: sigma_measure_real] : ( member_real_real_nat @ F2 @ ( sigma_6032194292609393305al_nat @ M3 @ borel_1750461538259077885al_nat ) ) ) ) ).

% is_borel_def
thf(fact_935_is__borel__def,axiom,
    ( borel_9213571707143006522t_real
    = ( ^ [F2: nat > real,M3: sigma_measure_nat] : ( member_nat_real @ F2 @ ( sigma_1747752005702207822t_real @ M3 @ borel_5078946678739801102l_real ) ) ) ) ).

% is_borel_def
thf(fact_936_is__borel__def,axiom,
    ( borel_4557508243417129402al_nat
    = ( ^ [F2: real > nat,M3: sigma_measure_real] : ( member_real_nat @ F2 @ ( sigma_6315060578831106510al_nat @ M3 @ borel_8449730974584783410el_nat ) ) ) ) ).

% is_borel_def
thf(fact_937_measurable__coordinatewise__then__product,axiom,
    ! [F: real > nat > real,M: sigma_measure_real] :
      ( ! [I5: nat] :
          ( member_real_real
          @ ^ [X5: real] : ( F @ X5 @ I5 )
          @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( member_real_nat_real2 @ F @ ( sigma_783869947231497753t_real @ M @ borel_5725509229735958141t_real ) ) ) ).

% measurable_coordinatewise_then_product
thf(fact_938_measurable__product__then__coordinatewise,axiom,
    ! [F: real > nat > real,M: sigma_measure_real,I: nat] :
      ( ( member_real_nat_real2 @ F @ ( sigma_783869947231497753t_real @ M @ borel_5725509229735958141t_real ) )
     => ( member_real_real
        @ ^ [X5: real] : ( F @ X5 @ I )
        @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% measurable_product_then_coordinatewise
thf(fact_939_measurable__product__then__coordinatewise,axiom,
    ! [F: real > real > nat,M: sigma_measure_real,I: real] :
      ( ( member_real_real_nat @ F @ ( sigma_6032194292609393305al_nat @ M @ borel_1750461538259077885al_nat ) )
     => ( member_real_nat
        @ ^ [X5: real] : ( F @ X5 @ I )
        @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% measurable_product_then_coordinatewise
thf(fact_940_increasingD,axiom,
    ! [M: set_set_nat,F: set_nat > nat,X: set_nat,Y: set_nat] :
      ( ( measur1302623347068717141at_nat @ M @ F )
     => ( ( ord_less_eq_set_nat @ X @ Y )
       => ( ( member_set_nat @ X @ M )
         => ( ( member_set_nat @ Y @ M )
           => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ).

% increasingD
thf(fact_941_insert__subsetI,axiom,
    ! [X: real,A: set_real,X2: set_real] :
      ( ( member_real @ X @ A )
     => ( ( ord_less_eq_set_real @ X2 @ A )
       => ( ord_less_eq_set_real @ ( insert_real @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_942_insert__subsetI,axiom,
    ! [X: $o,A: set_o,X2: set_o] :
      ( ( member_o @ X @ A )
     => ( ( ord_less_eq_set_o @ X2 @ A )
       => ( ord_less_eq_set_o @ ( insert_o @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_943_insert__subsetI,axiom,
    ! [X: nat,A: set_nat,X2: set_nat] :
      ( ( member_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ X2 @ A )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_944_insert__subsetI,axiom,
    ! [X: b,A: set_b,X2: set_b] :
      ( ( member_b @ X @ A )
     => ( ( ord_less_eq_set_b @ X2 @ A )
       => ( ord_less_eq_set_b @ ( insert_b @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_945_insert__subsetI,axiom,
    ! [X: set_nat,A: set_set_nat,X2: set_set_nat] :
      ( ( member_set_nat @ X @ A )
     => ( ( ord_le6893508408891458716et_nat @ X2 @ A )
       => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_946_insert__subsetI,axiom,
    ! [X: real > nat,A: set_real_nat,X2: set_real_nat] :
      ( ( member_real_nat @ X @ A )
     => ( ( ord_le6098800555920186673al_nat @ X2 @ A )
       => ( ord_le6098800555920186673al_nat @ ( insert_real_nat @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_947_insert__subsetI,axiom,
    ! [X: real > b,A: set_real_b,X2: set_real_b] :
      ( ( member_real_b @ X @ A )
     => ( ( ord_le5814440863667440394real_b @ X2 @ A )
       => ( ord_le5814440863667440394real_b @ ( insert_real_b @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_948_insert__subsetI,axiom,
    ! [X: nat > real,A: set_nat_real,X2: set_nat_real] :
      ( ( member_nat_real @ X @ A )
     => ( ( ord_le2908806416726583473t_real @ X2 @ A )
       => ( ord_le2908806416726583473t_real @ ( insert_nat_real @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_949_insert__subsetI,axiom,
    ! [X: real > set_nat,A: set_real_set_nat,X2: set_real_set_nat] :
      ( ( member_real_set_nat @ X @ A )
     => ( ( ord_le7035988643939837671et_nat @ X2 @ A )
       => ( ord_le7035988643939837671et_nat @ ( insert_real_set_nat @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_950_insert__subsetI,axiom,
    ! [X: real > nat > real,A: set_real_nat_real,X2: set_real_nat_real] :
      ( ( member_real_nat_real2 @ X @ A )
     => ( ( ord_le6912594875210535036t_real @ X2 @ A )
       => ( ord_le6912594875210535036t_real @ ( insert_real_nat_real @ X @ X2 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_951_subset__emptyI,axiom,
    ! [A: set_b] :
      ( ! [X3: b] :
          ~ ( member_b @ X3 @ A )
     => ( ord_less_eq_set_b @ A @ bot_bot_set_b ) ) ).

% subset_emptyI
thf(fact_952_subset__emptyI,axiom,
    ! [A: set_nat] :
      ( ! [X3: nat] :
          ~ ( member_nat @ X3 @ A )
     => ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_953_subset__emptyI,axiom,
    ! [A: set_real] :
      ( ! [X3: real] :
          ~ ( member_real @ X3 @ A )
     => ( ord_less_eq_set_real @ A @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_954_subset__emptyI,axiom,
    ! [A: set_o] :
      ( ! [X3: $o] :
          ~ ( member_o @ X3 @ A )
     => ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_955_subset__emptyI,axiom,
    ! [A: set_set_nat] :
      ( ! [X3: set_nat] :
          ~ ( member_set_nat @ X3 @ A )
     => ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_956_subset__emptyI,axiom,
    ! [A: set_real_nat] :
      ( ! [X3: real > nat] :
          ~ ( member_real_nat @ X3 @ A )
     => ( ord_le6098800555920186673al_nat @ A @ bot_bot_set_real_nat ) ) ).

% subset_emptyI
thf(fact_957_subset__emptyI,axiom,
    ! [A: set_real_b] :
      ( ! [X3: real > b] :
          ~ ( member_real_b @ X3 @ A )
     => ( ord_le5814440863667440394real_b @ A @ bot_bot_set_real_b ) ) ).

% subset_emptyI
thf(fact_958_subset__emptyI,axiom,
    ! [A: set_nat_real] :
      ( ! [X3: nat > real] :
          ~ ( member_nat_real @ X3 @ A )
     => ( ord_le2908806416726583473t_real @ A @ bot_bot_set_nat_real ) ) ).

% subset_emptyI
thf(fact_959_subset__emptyI,axiom,
    ! [A: set_real_set_nat] :
      ( ! [X3: real > set_nat] :
          ~ ( member_real_set_nat @ X3 @ A )
     => ( ord_le7035988643939837671et_nat @ A @ bot_bo6814059168456595739et_nat ) ) ).

% subset_emptyI
thf(fact_960_subset__emptyI,axiom,
    ! [A: set_real_nat_real] :
      ( ! [X3: real > nat > real] :
          ~ ( member_real_nat_real2 @ X3 @ A )
     => ( ord_le6912594875210535036t_real @ A @ bot_bo6533810469807102640t_real ) ) ).

% subset_emptyI
thf(fact_961_sets__eq__iff__bounded,axiom,
    ! [A: sigma_measure_real,B: sigma_measure_real,C3: sigma_measure_real] :
      ( ( ord_le487379304121309861e_real @ A @ B )
     => ( ( ord_le487379304121309861e_real @ B @ C3 )
       => ( ( ( sigma_sets_real @ A )
            = ( sigma_sets_real @ C3 ) )
         => ( ( sigma_sets_real @ B )
            = ( sigma_sets_real @ A ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_962_sets__eq__iff__bounded,axiom,
    ! [A: sigma_measure_nat,B: sigma_measure_nat,C3: sigma_measure_nat] :
      ( ( ord_le2862109966718184649re_nat @ A @ B )
     => ( ( ord_le2862109966718184649re_nat @ B @ C3 )
       => ( ( ( sigma_sets_nat @ A )
            = ( sigma_sets_nat @ C3 ) )
         => ( ( sigma_sets_nat @ B )
            = ( sigma_sets_nat @ A ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_963_sets__eq__iff__bounded,axiom,
    ! [A: sigma_measure_o,B: sigma_measure_o,C3: sigma_measure_o] :
      ( ( ord_le478349814012620405sure_o @ A @ B )
     => ( ( ord_le478349814012620405sure_o @ B @ C3 )
       => ( ( ( sigma_sets_o @ A )
            = ( sigma_sets_o @ C3 ) )
         => ( ( sigma_sets_o @ B )
            = ( sigma_sets_o @ A ) ) ) ) ) ).

% sets_eq_iff_bounded
thf(fact_964_pred__subset__eq,axiom,
    ! [R3: set_set_nat,S2: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ R3 )
        @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ S2 ) )
      = ( ord_le6893508408891458716et_nat @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_965_pred__subset__eq,axiom,
    ! [R3: set_nat,S2: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X5: nat] : ( member_nat @ X5 @ R3 )
        @ ^ [X5: nat] : ( member_nat @ X5 @ S2 ) )
      = ( ord_less_eq_set_nat @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_966_pred__subset__eq,axiom,
    ! [R3: set_real_nat,S2: set_real_nat] :
      ( ( ord_le5694569607330361492_nat_o
        @ ^ [X5: real > nat] : ( member_real_nat @ X5 @ R3 )
        @ ^ [X5: real > nat] : ( member_real_nat @ X5 @ S2 ) )
      = ( ord_le6098800555920186673al_nat @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_967_pred__subset__eq,axiom,
    ! [R3: set_real_b,S2: set_real_b] :
      ( ( ord_less_eq_real_b_o
        @ ^ [X5: real > b] : ( member_real_b @ X5 @ R3 )
        @ ^ [X5: real > b] : ( member_real_b @ X5 @ S2 ) )
      = ( ord_le5814440863667440394real_b @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_968_pred__subset__eq,axiom,
    ! [R3: set_nat_real,S2: set_nat_real] :
      ( ( ord_le7676461544873280788real_o
        @ ^ [X5: nat > real] : ( member_nat_real @ X5 @ R3 )
        @ ^ [X5: nat > real] : ( member_nat_real @ X5 @ S2 ) )
      = ( ord_le2908806416726583473t_real @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_969_pred__subset__eq,axiom,
    ! [R3: set_b,S2: set_b] :
      ( ( ord_less_eq_b_o
        @ ^ [X5: b] : ( member_b @ X5 @ R3 )
        @ ^ [X5: b] : ( member_b @ X5 @ S2 ) )
      = ( ord_less_eq_set_b @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_970_pred__subset__eq,axiom,
    ! [R3: set_real_nat_real,S2: set_real_nat_real] :
      ( ( ord_le2056620204158255881real_o
        @ ^ [X5: real > nat > real] : ( member_real_nat_real2 @ X5 @ R3 )
        @ ^ [X5: real > nat > real] : ( member_real_nat_real2 @ X5 @ S2 ) )
      = ( ord_le6912594875210535036t_real @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_971_pred__subset__eq,axiom,
    ! [R3: set_real_real_b,S2: set_real_real_b] :
      ( ( ord_le7427477666976093192al_b_o
        @ ^ [X5: real > real > b] : ( member_real_real_b @ X5 @ R3 )
        @ ^ [X5: real > real > b] : ( member_real_real_b @ X5 @ S2 ) )
      = ( ord_le227027178093677845real_b @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_972_pred__subset__eq,axiom,
    ! [R3: set_real_real_nat,S2: set_real_real_nat] :
      ( ( ord_le5498898627171184265_nat_o
        @ ^ [X5: real > real > nat] : ( member_real_real_nat @ X5 @ R3 )
        @ ^ [X5: real > real > nat] : ( member_real_real_nat @ X5 @ S2 ) )
      = ( ord_le491656556176229628al_nat @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_973_pred__subset__eq,axiom,
    ! [R3: set_real_set_nat,S2: set_real_set_nat] :
      ( ( ord_le5255281653000370782_nat_o
        @ ^ [X5: real > set_nat] : ( member_real_set_nat @ X5 @ R3 )
        @ ^ [X5: real > set_nat] : ( member_real_set_nat @ X5 @ S2 ) )
      = ( ord_le7035988643939837671et_nat @ R3 @ S2 ) ) ).

% pred_subset_eq
thf(fact_974_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A4: set_set_nat,B4: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A4 )
          @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_975_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A4: set_nat,B4: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X5: nat] : ( member_nat @ X5 @ A4 )
          @ ^ [X5: nat] : ( member_nat @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_976_less__eq__set__def,axiom,
    ( ord_le6098800555920186673al_nat
    = ( ^ [A4: set_real_nat,B4: set_real_nat] :
          ( ord_le5694569607330361492_nat_o
          @ ^ [X5: real > nat] : ( member_real_nat @ X5 @ A4 )
          @ ^ [X5: real > nat] : ( member_real_nat @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_977_less__eq__set__def,axiom,
    ( ord_le5814440863667440394real_b
    = ( ^ [A4: set_real_b,B4: set_real_b] :
          ( ord_less_eq_real_b_o
          @ ^ [X5: real > b] : ( member_real_b @ X5 @ A4 )
          @ ^ [X5: real > b] : ( member_real_b @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_978_less__eq__set__def,axiom,
    ( ord_le2908806416726583473t_real
    = ( ^ [A4: set_nat_real,B4: set_nat_real] :
          ( ord_le7676461544873280788real_o
          @ ^ [X5: nat > real] : ( member_nat_real @ X5 @ A4 )
          @ ^ [X5: nat > real] : ( member_nat_real @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_979_less__eq__set__def,axiom,
    ( ord_less_eq_set_b
    = ( ^ [A4: set_b,B4: set_b] :
          ( ord_less_eq_b_o
          @ ^ [X5: b] : ( member_b @ X5 @ A4 )
          @ ^ [X5: b] : ( member_b @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_980_less__eq__set__def,axiom,
    ( ord_le6912594875210535036t_real
    = ( ^ [A4: set_real_nat_real,B4: set_real_nat_real] :
          ( ord_le2056620204158255881real_o
          @ ^ [X5: real > nat > real] : ( member_real_nat_real2 @ X5 @ A4 )
          @ ^ [X5: real > nat > real] : ( member_real_nat_real2 @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_981_less__eq__set__def,axiom,
    ( ord_le227027178093677845real_b
    = ( ^ [A4: set_real_real_b,B4: set_real_real_b] :
          ( ord_le7427477666976093192al_b_o
          @ ^ [X5: real > real > b] : ( member_real_real_b @ X5 @ A4 )
          @ ^ [X5: real > real > b] : ( member_real_real_b @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_982_less__eq__set__def,axiom,
    ( ord_le491656556176229628al_nat
    = ( ^ [A4: set_real_real_nat,B4: set_real_real_nat] :
          ( ord_le5498898627171184265_nat_o
          @ ^ [X5: real > real > nat] : ( member_real_real_nat @ X5 @ A4 )
          @ ^ [X5: real > real > nat] : ( member_real_real_nat @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_983_less__eq__set__def,axiom,
    ( ord_le7035988643939837671et_nat
    = ( ^ [A4: set_real_set_nat,B4: set_real_set_nat] :
          ( ord_le5255281653000370782_nat_o
          @ ^ [X5: real > set_nat] : ( member_real_set_nat @ X5 @ A4 )
          @ ^ [X5: real > set_nat] : ( member_real_set_nat @ X5 @ B4 ) ) ) ) ).

% less_eq_set_def
thf(fact_984_Pi__cong__sets,axiom,
    ! [I2: set_real,J: set_real,M: real > set_b,N: real > set_b] :
      ( ( I2 = J )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ I2 )
           => ( ( M @ X3 )
              = ( N @ X3 ) ) )
       => ( ( pi_real_b @ I2 @ M )
          = ( pi_real_b @ J @ N ) ) ) ) ).

% Pi_cong_sets
thf(fact_985_Pi__cong__sets,axiom,
    ! [I2: set_real,J: set_real,M: real > set_nat,N: real > set_nat] :
      ( ( I2 = J )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ I2 )
           => ( ( M @ X3 )
              = ( N @ X3 ) ) )
       => ( ( pi_real_nat @ I2 @ M )
          = ( pi_real_nat @ J @ N ) ) ) ) ).

% Pi_cong_sets
thf(fact_986_conj__subset__def,axiom,
    ! [A: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( P @ X5 )
              & ( Q @ X5 ) ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_987_prop__restrict,axiom,
    ! [X: set_nat,Z3: set_set_nat,X2: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X @ Z3 )
     => ( ( ord_le6893508408891458716et_nat @ Z3
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_988_prop__restrict,axiom,
    ! [X: real > nat,Z3: set_real_nat,X2: set_real_nat,P: ( real > nat ) > $o] :
      ( ( member_real_nat @ X @ Z3 )
     => ( ( ord_le6098800555920186673al_nat @ Z3
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_989_prop__restrict,axiom,
    ! [X: real > b,Z3: set_real_b,X2: set_real_b,P: ( real > b ) > $o] :
      ( ( member_real_b @ X @ Z3 )
     => ( ( ord_le5814440863667440394real_b @ Z3
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_990_prop__restrict,axiom,
    ! [X: nat > real,Z3: set_nat_real,X2: set_nat_real,P: ( nat > real ) > $o] :
      ( ( member_nat_real @ X @ Z3 )
     => ( ( ord_le2908806416726583473t_real @ Z3
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_991_prop__restrict,axiom,
    ! [X: b,Z3: set_b,X2: set_b,P: b > $o] :
      ( ( member_b @ X @ Z3 )
     => ( ( ord_less_eq_set_b @ Z3
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_992_prop__restrict,axiom,
    ! [X: real > nat > real,Z3: set_real_nat_real,X2: set_real_nat_real,P: ( real > nat > real ) > $o] :
      ( ( member_real_nat_real2 @ X @ Z3 )
     => ( ( ord_le6912594875210535036t_real @ Z3
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_993_prop__restrict,axiom,
    ! [X: real > real > b,Z3: set_real_real_b,X2: set_real_real_b,P: ( real > real > b ) > $o] :
      ( ( member_real_real_b @ X @ Z3 )
     => ( ( ord_le227027178093677845real_b @ Z3
          @ ( collect_real_real_b
            @ ^ [X5: real > real > b] :
                ( ( member_real_real_b @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_994_prop__restrict,axiom,
    ! [X: real > real > nat,Z3: set_real_real_nat,X2: set_real_real_nat,P: ( real > real > nat ) > $o] :
      ( ( member_real_real_nat @ X @ Z3 )
     => ( ( ord_le491656556176229628al_nat @ Z3
          @ ( collec3526718475268515771al_nat
            @ ^ [X5: real > real > nat] :
                ( ( member_real_real_nat @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_995_prop__restrict,axiom,
    ! [X: real > set_nat,Z3: set_real_set_nat,X2: set_real_set_nat,P: ( real > set_nat ) > $o] :
      ( ( member_real_set_nat @ X @ Z3 )
     => ( ( ord_le7035988643939837671et_nat @ Z3
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_996_prop__restrict,axiom,
    ! [X: nat,Z3: set_nat,X2: set_nat,P: nat > $o] :
      ( ( member_nat @ X @ Z3 )
     => ( ( ord_less_eq_set_nat @ Z3
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ X2 )
                & ( P @ X5 ) ) ) )
       => ( P @ X ) ) ) ).

% prop_restrict
thf(fact_997_Collect__restrict,axiom,
    ! [X2: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X5: set_nat] :
            ( ( member_set_nat @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_998_Collect__restrict,axiom,
    ! [X2: set_real_nat,P: ( real > nat ) > $o] :
      ( ord_le6098800555920186673al_nat
      @ ( collect_real_nat
        @ ^ [X5: real > nat] :
            ( ( member_real_nat @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_999_Collect__restrict,axiom,
    ! [X2: set_real_b,P: ( real > b ) > $o] :
      ( ord_le5814440863667440394real_b
      @ ( collect_real_b
        @ ^ [X5: real > b] :
            ( ( member_real_b @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_1000_Collect__restrict,axiom,
    ! [X2: set_nat_real,P: ( nat > real ) > $o] :
      ( ord_le2908806416726583473t_real
      @ ( collect_nat_real
        @ ^ [X5: nat > real] :
            ( ( member_nat_real @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_1001_Collect__restrict,axiom,
    ! [X2: set_b,P: b > $o] :
      ( ord_less_eq_set_b
      @ ( collect_b
        @ ^ [X5: b] :
            ( ( member_b @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_1002_Collect__restrict,axiom,
    ! [X2: set_real_nat_real,P: ( real > nat > real ) > $o] :
      ( ord_le6912594875210535036t_real
      @ ( collec336724336074912571t_real
        @ ^ [X5: real > nat > real] :
            ( ( member_real_nat_real2 @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_1003_Collect__restrict,axiom,
    ! [X2: set_real_real_b,P: ( real > real > b ) > $o] :
      ( ord_le227027178093677845real_b
      @ ( collect_real_real_b
        @ ^ [X5: real > real > b] :
            ( ( member_real_real_b @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_1004_Collect__restrict,axiom,
    ! [X2: set_real_real_nat,P: ( real > real > nat ) > $o] :
      ( ord_le491656556176229628al_nat
      @ ( collec3526718475268515771al_nat
        @ ^ [X5: real > real > nat] :
            ( ( member_real_real_nat @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_1005_Collect__restrict,axiom,
    ! [X2: set_real_set_nat,P: ( real > set_nat ) > $o] :
      ( ord_le7035988643939837671et_nat
      @ ( collect_real_set_nat
        @ ^ [X5: real > set_nat] :
            ( ( member_real_set_nat @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_1006_Collect__restrict,axiom,
    ! [X2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ X2 )
            & ( P @ X5 ) ) )
      @ X2 ) ).

% Collect_restrict
thf(fact_1007_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).

% Set.is_empty_def
thf(fact_1008_Set_Ois__empty__def,axiom,
    ( is_empty_real
    = ( ^ [A4: set_real] : ( A4 = bot_bot_set_real ) ) ) ).

% Set.is_empty_def
thf(fact_1009_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A4: set_o] : ( A4 = bot_bot_set_o ) ) ) ).

% Set.is_empty_def
thf(fact_1010_eucl__ivals_I6_J,axiom,
    ! [A2: real,B2: real] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X5: real] :
            ( ( topolo2105956845596822908s_real @ A2 @ X5 )
            & ( ord_less_eq_real @ X5 @ B2 ) ) )
      @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% eucl_ivals(6)
thf(fact_1011_eucl__ivals_I7_J,axiom,
    ! [A2: real,B2: real] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X5: real] :
            ( ( ord_less_eq_real @ A2 @ X5 )
            & ( topolo2105956845596822908s_real @ X5 @ B2 ) ) )
      @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% eucl_ivals(7)
thf(fact_1012_subset__singleton__iff__Uniq,axiom,
    ! [A: set_b] :
      ( ( ? [A3: b] : ( ord_less_eq_set_b @ A @ ( insert_b @ A3 @ bot_bot_set_b ) ) )
      = ( uniq_b
        @ ^ [X5: b] : ( member_b @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1013_subset__singleton__iff__Uniq,axiom,
    ! [A: set_nat] :
      ( ( ? [A3: nat] : ( ord_less_eq_set_nat @ A @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
      = ( uniq_nat
        @ ^ [X5: nat] : ( member_nat @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1014_subset__singleton__iff__Uniq,axiom,
    ! [A: set_real] :
      ( ( ? [A3: real] : ( ord_less_eq_set_real @ A @ ( insert_real @ A3 @ bot_bot_set_real ) ) )
      = ( uniq_real
        @ ^ [X5: real] : ( member_real @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1015_subset__singleton__iff__Uniq,axiom,
    ! [A: set_o] :
      ( ( ? [A3: $o] : ( ord_less_eq_set_o @ A @ ( insert_o @ A3 @ bot_bot_set_o ) ) )
      = ( uniq_o
        @ ^ [X5: $o] : ( member_o @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1016_subset__singleton__iff__Uniq,axiom,
    ! [A: set_set_nat] :
      ( ( ? [A3: set_nat] : ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ A3 @ bot_bot_set_set_nat ) ) )
      = ( uniq_set_nat
        @ ^ [X5: set_nat] : ( member_set_nat @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1017_subset__singleton__iff__Uniq,axiom,
    ! [A: set_real_nat] :
      ( ( ? [A3: real > nat] : ( ord_le6098800555920186673al_nat @ A @ ( insert_real_nat @ A3 @ bot_bot_set_real_nat ) ) )
      = ( uniq_real_nat
        @ ^ [X5: real > nat] : ( member_real_nat @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1018_subset__singleton__iff__Uniq,axiom,
    ! [A: set_real_b] :
      ( ( ? [A3: real > b] : ( ord_le5814440863667440394real_b @ A @ ( insert_real_b @ A3 @ bot_bot_set_real_b ) ) )
      = ( uniq_real_b
        @ ^ [X5: real > b] : ( member_real_b @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1019_subset__singleton__iff__Uniq,axiom,
    ! [A: set_nat_real] :
      ( ( ? [A3: nat > real] : ( ord_le2908806416726583473t_real @ A @ ( insert_nat_real @ A3 @ bot_bot_set_nat_real ) ) )
      = ( uniq_nat_real
        @ ^ [X5: nat > real] : ( member_nat_real @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1020_subset__singleton__iff__Uniq,axiom,
    ! [A: set_real_set_nat] :
      ( ( ? [A3: real > set_nat] : ( ord_le7035988643939837671et_nat @ A @ ( insert_real_set_nat @ A3 @ bot_bo6814059168456595739et_nat ) ) )
      = ( uniq_real_set_nat
        @ ^ [X5: real > set_nat] : ( member_real_set_nat @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1021_subset__singleton__iff__Uniq,axiom,
    ! [A: set_real_nat_real] :
      ( ( ? [A3: real > nat > real] : ( ord_le6912594875210535036t_real @ A @ ( insert_real_nat_real @ A3 @ bot_bo6533810469807102640t_real ) ) )
      = ( uniq_real_nat_real
        @ ^ [X5: real > nat > real] : ( member_real_nat_real2 @ X5 @ A ) ) ) ).

% subset_singleton_iff_Uniq
thf(fact_1022_eucl__ivals_I2_J,axiom,
    ! [A2: real] : ( member_set_real @ ( collect_real @ ( topolo2105956845596822908s_real @ A2 ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% eucl_ivals(2)
thf(fact_1023_eucl__ivals_I1_J,axiom,
    ! [A2: real] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X5: real] : ( topolo2105956845596822908s_real @ X5 @ A2 ) )
      @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% eucl_ivals(1)
thf(fact_1024_prod__set__defs_I2_J,axiom,
    ( basic_3503571567493806864real_b
    = ( ^ [P5: produc4373655991844504306real_b] : ( insert_set_real_b @ ( produc1187263614985732624real_b @ P5 ) @ bot_bo5566352981089410870real_b ) ) ) ).

% prod_set_defs(2)
thf(fact_1025_snds_Ointros,axiom,
    ! [P4: produc4373655991844504306real_b] : ( member_set_real_b @ ( produc1187263614985732624real_b @ P4 ) @ ( basic_3503571567493806864real_b @ P4 ) ) ).

% snds.intros
thf(fact_1026_snds_Osimps,axiom,
    ! [A2: set_real_b,P4: produc4373655991844504306real_b] :
      ( ( member_set_real_b @ A2 @ ( basic_3503571567493806864real_b @ P4 ) )
      = ( A2
        = ( produc1187263614985732624real_b @ P4 ) ) ) ).

% snds.simps
thf(fact_1027_snds_Ocases,axiom,
    ! [A2: set_real_b,P4: produc4373655991844504306real_b] :
      ( ( member_set_real_b @ A2 @ ( basic_3503571567493806864real_b @ P4 ) )
     => ( A2
        = ( produc1187263614985732624real_b @ P4 ) ) ) ).

% snds.cases
thf(fact_1028_subset__CollectI,axiom,
    ! [B: set_set_nat,A: set_set_nat,Q: set_nat > $o,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B @ A )
     => ( ! [X3: set_nat] :
            ( ( member_set_nat @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le6893508408891458716et_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1029_subset__CollectI,axiom,
    ! [B: set_real_nat,A: set_real_nat,Q: ( real > nat ) > $o,P: ( real > nat ) > $o] :
      ( ( ord_le6098800555920186673al_nat @ B @ A )
     => ( ! [X3: real > nat] :
            ( ( member_real_nat @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le6098800555920186673al_nat
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1030_subset__CollectI,axiom,
    ! [B: set_real_b,A: set_real_b,Q: ( real > b ) > $o,P: ( real > b ) > $o] :
      ( ( ord_le5814440863667440394real_b @ B @ A )
     => ( ! [X3: real > b] :
            ( ( member_real_b @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le5814440863667440394real_b
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1031_subset__CollectI,axiom,
    ! [B: set_nat_real,A: set_nat_real,Q: ( nat > real ) > $o,P: ( nat > real ) > $o] :
      ( ( ord_le2908806416726583473t_real @ B @ A )
     => ( ! [X3: nat > real] :
            ( ( member_nat_real @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le2908806416726583473t_real
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1032_subset__CollectI,axiom,
    ! [B: set_b,A: set_b,Q: b > $o,P: b > $o] :
      ( ( ord_less_eq_set_b @ B @ A )
     => ( ! [X3: b] :
            ( ( member_b @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1033_subset__CollectI,axiom,
    ! [B: set_real_nat_real,A: set_real_nat_real,Q: ( real > nat > real ) > $o,P: ( real > nat > real ) > $o] :
      ( ( ord_le6912594875210535036t_real @ B @ A )
     => ( ! [X3: real > nat > real] :
            ( ( member_real_nat_real2 @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le6912594875210535036t_real
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1034_subset__CollectI,axiom,
    ! [B: set_real_real_b,A: set_real_real_b,Q: ( real > real > b ) > $o,P: ( real > real > b ) > $o] :
      ( ( ord_le227027178093677845real_b @ B @ A )
     => ( ! [X3: real > real > b] :
            ( ( member_real_real_b @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le227027178093677845real_b
          @ ( collect_real_real_b
            @ ^ [X5: real > real > b] :
                ( ( member_real_real_b @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collect_real_real_b
            @ ^ [X5: real > real > b] :
                ( ( member_real_real_b @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1035_subset__CollectI,axiom,
    ! [B: set_real_real_nat,A: set_real_real_nat,Q: ( real > real > nat ) > $o,P: ( real > real > nat ) > $o] :
      ( ( ord_le491656556176229628al_nat @ B @ A )
     => ( ! [X3: real > real > nat] :
            ( ( member_real_real_nat @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le491656556176229628al_nat
          @ ( collec3526718475268515771al_nat
            @ ^ [X5: real > real > nat] :
                ( ( member_real_real_nat @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collec3526718475268515771al_nat
            @ ^ [X5: real > real > nat] :
                ( ( member_real_real_nat @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1036_subset__CollectI,axiom,
    ! [B: set_real_set_nat,A: set_real_set_nat,Q: ( real > set_nat ) > $o,P: ( real > set_nat ) > $o] :
      ( ( ord_le7035988643939837671et_nat @ B @ A )
     => ( ! [X3: real > set_nat] :
            ( ( member_real_set_nat @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_le7035988643939837671et_nat
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1037_subset__CollectI,axiom,
    ! [B: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B )
           => ( ( Q @ X3 )
             => ( P @ X3 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ B )
                & ( Q @ X5 ) ) )
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ A )
                & ( P @ X5 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_1038_subset__Collect__iff,axiom,
    ! [B: set_set_nat,A: set_set_nat,P: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ B @ A )
     => ( ( ord_le6893508408891458716et_nat @ B
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: set_nat] :
              ( ( member_set_nat @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1039_subset__Collect__iff,axiom,
    ! [B: set_real_nat,A: set_real_nat,P: ( real > nat ) > $o] :
      ( ( ord_le6098800555920186673al_nat @ B @ A )
     => ( ( ord_le6098800555920186673al_nat @ B
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: real > nat] :
              ( ( member_real_nat @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1040_subset__Collect__iff,axiom,
    ! [B: set_real_b,A: set_real_b,P: ( real > b ) > $o] :
      ( ( ord_le5814440863667440394real_b @ B @ A )
     => ( ( ord_le5814440863667440394real_b @ B
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: real > b] :
              ( ( member_real_b @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1041_subset__Collect__iff,axiom,
    ! [B: set_nat_real,A: set_nat_real,P: ( nat > real ) > $o] :
      ( ( ord_le2908806416726583473t_real @ B @ A )
     => ( ( ord_le2908806416726583473t_real @ B
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: nat > real] :
              ( ( member_nat_real @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1042_subset__Collect__iff,axiom,
    ! [B: set_b,A: set_b,P: b > $o] :
      ( ( ord_less_eq_set_b @ B @ A )
     => ( ( ord_less_eq_set_b @ B
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: b] :
              ( ( member_b @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1043_subset__Collect__iff,axiom,
    ! [B: set_real_nat_real,A: set_real_nat_real,P: ( real > nat > real ) > $o] :
      ( ( ord_le6912594875210535036t_real @ B @ A )
     => ( ( ord_le6912594875210535036t_real @ B
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: real > nat > real] :
              ( ( member_real_nat_real2 @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1044_subset__Collect__iff,axiom,
    ! [B: set_real_real_b,A: set_real_real_b,P: ( real > real > b ) > $o] :
      ( ( ord_le227027178093677845real_b @ B @ A )
     => ( ( ord_le227027178093677845real_b @ B
          @ ( collect_real_real_b
            @ ^ [X5: real > real > b] :
                ( ( member_real_real_b @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: real > real > b] :
              ( ( member_real_real_b @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1045_subset__Collect__iff,axiom,
    ! [B: set_real_real_nat,A: set_real_real_nat,P: ( real > real > nat ) > $o] :
      ( ( ord_le491656556176229628al_nat @ B @ A )
     => ( ( ord_le491656556176229628al_nat @ B
          @ ( collec3526718475268515771al_nat
            @ ^ [X5: real > real > nat] :
                ( ( member_real_real_nat @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: real > real > nat] :
              ( ( member_real_real_nat @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1046_subset__Collect__iff,axiom,
    ! [B: set_real_set_nat,A: set_real_set_nat,P: ( real > set_nat ) > $o] :
      ( ( ord_le7035988643939837671et_nat @ B @ A )
     => ( ( ord_le7035988643939837671et_nat @ B
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: real > set_nat] :
              ( ( member_real_set_nat @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1047_subset__Collect__iff,axiom,
    ! [B: set_nat,A: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ B
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ A )
                & ( P @ X5 ) ) ) )
        = ( ! [X5: nat] :
              ( ( member_nat @ X5 @ B )
             => ( P @ X5 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_1048_borel__measurableI__le,axiom,
    ! [M: sigma_measure_b,F: b > real] :
      ( ! [Y4: real] :
          ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_b @ M ) )
     => ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_le
thf(fact_1049_borel__measurableI__le,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ! [Y4: real] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_le
thf(fact_1050_borel__measurableI__le,axiom,
    ! [M: sigma_measure_o,F: $o > real] :
      ( ! [Y4: real] :
          ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_o @ M ) )
     => ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_le
thf(fact_1051_borel__measurableI__le,axiom,
    ! [M: sigma_measure_nat,F: nat > real] :
      ( ! [Y4: real] :
          ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_nat @ M ) )
     => ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_le
thf(fact_1052_borel__measurableI__le,axiom,
    ! [M: sigma_measure_b,F: b > nat] :
      ( ! [Y4: nat] :
          ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_b @ M ) )
     => ( member_b_nat @ F @ ( sigma_1308594411581951615_b_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_le
thf(fact_1053_borel__measurableI__le,axiom,
    ! [M: sigma_measure_nat,F: nat > nat] :
      ( ! [Y4: nat] :
          ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_nat @ M ) )
     => ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_le
thf(fact_1054_borel__measurableI__le,axiom,
    ! [M: sigma_measure_o,F: $o > nat] :
      ( ! [Y4: nat] :
          ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_o @ M ) )
     => ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_le
thf(fact_1055_borel__measurableI__le,axiom,
    ! [M: sigma_measure_real,F: real > nat] :
      ( ! [Y4: nat] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_le
thf(fact_1056_borel__measurableI__le,axiom,
    ! [M: sigma_3334325623652945375et_nat,F: set_nat > real] :
      ( ! [Y4: real] :
          ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_set_nat @ M ) )
     => ( member_set_nat_real @ F @ ( sigma_7357981997995286020t_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_le
thf(fact_1057_borel__measurableI__le,axiom,
    ! [M: sigma_3334325623652945375et_nat,F: set_nat > nat] :
      ( ! [Y4: nat] :
          ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ Y4 ) ) )
          @ ( sigma_sets_set_nat @ M ) )
     => ( member_set_nat_nat @ F @ ( sigma_6407790436254459688at_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_le
thf(fact_1058_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_b,F: b > real] :
      ( ! [Y4: real] :
          ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( ord_less_eq_real @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_b @ M ) )
     => ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_ge
thf(fact_1059_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_real,F: real > real] :
      ( ! [Y4: real] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_real @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_ge
thf(fact_1060_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_o,F: $o > real] :
      ( ! [Y4: real] :
          ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( ord_less_eq_real @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_o @ M ) )
     => ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_ge
thf(fact_1061_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_nat,F: nat > real] :
      ( ! [Y4: real] :
          ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( ord_less_eq_real @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_nat @ M ) )
     => ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_ge
thf(fact_1062_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_b,F: b > nat] :
      ( ! [Y4: nat] :
          ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( ord_less_eq_nat @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_b @ M ) )
     => ( member_b_nat @ F @ ( sigma_1308594411581951615_b_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_ge
thf(fact_1063_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_nat,F: nat > nat] :
      ( ! [Y4: nat] :
          ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( ord_less_eq_nat @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_nat @ M ) )
     => ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_ge
thf(fact_1064_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_o,F: $o > nat] :
      ( ! [Y4: nat] :
          ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( ord_less_eq_nat @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_o @ M ) )
     => ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_ge
thf(fact_1065_borel__measurableI__ge,axiom,
    ! [M: sigma_measure_real,F: real > nat] :
      ( ! [Y4: nat] :
          ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_nat @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_real @ M ) )
     => ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_ge
thf(fact_1066_borel__measurableI__ge,axiom,
    ! [M: sigma_3334325623652945375et_nat,F: set_nat > real] :
      ( ! [Y4: real] :
          ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( ord_less_eq_real @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_set_nat @ M ) )
     => ( member_set_nat_real @ F @ ( sigma_7357981997995286020t_real @ M @ borel_5078946678739801102l_real ) ) ) ).

% borel_measurableI_ge
thf(fact_1067_borel__measurableI__ge,axiom,
    ! [M: sigma_3334325623652945375et_nat,F: set_nat > nat] :
      ( ! [Y4: nat] :
          ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( ord_less_eq_nat @ Y4 @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_set_nat @ M ) )
     => ( member_set_nat_nat @ F @ ( sigma_6407790436254459688at_nat @ M @ borel_8449730974584783410el_nat ) ) ) ).

% borel_measurableI_ge
thf(fact_1068_borel__measurable__le,axiom,
    ! [F: b > real,M: sigma_measure_b,G: b > real] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_b_real @ G @ ( sigma_5489272243865111003b_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [W2: b] :
                ( ( member_b @ W2 @ ( sigma_space_b @ M ) )
                & ( ord_less_eq_real @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1069_borel__measurable__le,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [W2: real] :
                ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_real @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1070_borel__measurable__le,axiom,
    ! [F: $o > real,M: sigma_measure_o,G: $o > real] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_o_real @ G @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [W2: $o] :
                ( ( member_o @ W2 @ ( sigma_space_o @ M ) )
                & ( ord_less_eq_real @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1071_borel__measurable__le,axiom,
    ! [F: nat > real,M: sigma_measure_nat,G: nat > real] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [W2: nat] :
                ( ( member_nat @ W2 @ ( sigma_space_nat @ M ) )
                & ( ord_less_eq_real @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1072_borel__measurable__le,axiom,
    ! [F: b > nat,M: sigma_measure_b,G: b > nat] :
      ( ( member_b_nat @ F @ ( sigma_1308594411581951615_b_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_b_nat @ G @ ( sigma_1308594411581951615_b_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [W2: b] :
                ( ( member_b @ W2 @ ( sigma_space_b @ M ) )
                & ( ord_less_eq_nat @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1073_borel__measurable__le,axiom,
    ! [F: nat > nat,M: sigma_measure_nat,G: nat > nat] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_nat_nat @ G @ ( sigma_4350458207664084850at_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [W2: nat] :
                ( ( member_nat @ W2 @ ( sigma_space_nat @ M ) )
                & ( ord_less_eq_nat @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1074_borel__measurable__le,axiom,
    ! [F: $o > nat,M: sigma_measure_o,G: $o > nat] :
      ( ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_o_nat @ G @ ( sigma_1999164137574644376_o_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [W2: $o] :
                ( ( member_o @ W2 @ ( sigma_space_o @ M ) )
                & ( ord_less_eq_nat @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1075_borel__measurable__le,axiom,
    ! [F: real > nat,M: sigma_measure_real,G: real > nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [W2: real] :
                ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1076_borel__measurable__le,axiom,
    ! [F: set_nat > real,M: sigma_3334325623652945375et_nat,G: set_nat > real] :
      ( ( member_set_nat_real @ F @ ( sigma_7357981997995286020t_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_set_nat_real @ G @ ( sigma_7357981997995286020t_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [W2: set_nat] :
                ( ( member_set_nat @ W2 @ ( sigma_space_set_nat @ M ) )
                & ( ord_less_eq_real @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_set_nat @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1077_borel__measurable__le,axiom,
    ! [F: set_nat > nat,M: sigma_3334325623652945375et_nat,G: set_nat > nat] :
      ( ( member_set_nat_nat @ F @ ( sigma_6407790436254459688at_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_set_nat_nat @ G @ ( sigma_6407790436254459688at_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [W2: set_nat] :
                ( ( member_set_nat @ W2 @ ( sigma_space_set_nat @ M ) )
                & ( ord_less_eq_nat @ ( F @ W2 ) @ ( G @ W2 ) ) ) )
          @ ( sigma_sets_set_nat @ M ) ) ) ) ).

% borel_measurable_le
thf(fact_1078_measurable__inequality__set_I1_J,axiom,
    ! [F: b > real,M: sigma_measure_b,G: b > real] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_b_real @ G @ ( sigma_5489272243865111003b_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1079_measurable__inequality__set_I1_J,axiom,
    ! [F: real > real,M: sigma_measure_real,G: real > real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_real_real @ G @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1080_measurable__inequality__set_I1_J,axiom,
    ! [F: $o > real,M: sigma_measure_o,G: $o > real] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_o_real @ G @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1081_measurable__inequality__set_I1_J,axiom,
    ! [F: nat > real,M: sigma_measure_nat,G: nat > real] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1082_measurable__inequality__set_I1_J,axiom,
    ! [F: b > nat,M: sigma_measure_b,G: b > nat] :
      ( ( member_b_nat @ F @ ( sigma_1308594411581951615_b_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_b_nat @ G @ ( sigma_1308594411581951615_b_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1083_measurable__inequality__set_I1_J,axiom,
    ! [F: nat > nat,M: sigma_measure_nat,G: nat > nat] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_nat_nat @ G @ ( sigma_4350458207664084850at_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1084_measurable__inequality__set_I1_J,axiom,
    ! [F: $o > nat,M: sigma_measure_o,G: $o > nat] :
      ( ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_o_nat @ G @ ( sigma_1999164137574644376_o_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1085_measurable__inequality__set_I1_J,axiom,
    ! [F: real > nat,M: sigma_measure_real,G: real > nat] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1086_measurable__inequality__set_I1_J,axiom,
    ! [F: set_nat > real,M: sigma_3334325623652945375et_nat,G: set_nat > real] :
      ( ( member_set_nat_real @ F @ ( sigma_7357981997995286020t_real @ M @ borel_5078946678739801102l_real ) )
     => ( ( member_set_nat_real @ G @ ( sigma_7357981997995286020t_real @ M @ borel_5078946678739801102l_real ) )
       => ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_set_nat @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1087_measurable__inequality__set_I1_J,axiom,
    ! [F: set_nat > nat,M: sigma_3334325623652945375et_nat,G: set_nat > nat] :
      ( ( member_set_nat_nat @ F @ ( sigma_6407790436254459688at_nat @ M @ borel_8449730974584783410el_nat ) )
     => ( ( member_set_nat_nat @ G @ ( sigma_6407790436254459688at_nat @ M @ borel_8449730974584783410el_nat ) )
       => ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) ) )
          @ ( sigma_sets_set_nat @ M ) ) ) ) ).

% measurable_inequality_set(1)
thf(fact_1088_sets_Otop,axiom,
    ! [M: sigma_measure_nat] : ( member_set_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) ).

% sets.top
thf(fact_1089_sets_Otop,axiom,
    ! [M: sigma_measure_real] : ( member_set_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).

% sets.top
thf(fact_1090_sets_Otop,axiom,
    ! [M: sigma_measure_o] : ( member_set_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) ).

% sets.top
thf(fact_1091_space__borel,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% space_borel
thf(fact_1092_space__borel,axiom,
    ( ( sigma_space_nat @ borel_8449730974584783410el_nat )
    = top_top_set_nat ) ).

% space_borel
thf(fact_1093_space__borel,axiom,
    ( ( sigma_space_o @ borel_5500255247093592246orel_o )
    = top_top_set_o ) ).

% space_borel
thf(fact_1094_space__bot,axiom,
    ( ( sigma_space_nat @ bot_bo6718502177978453909re_nat )
    = bot_bot_set_nat ) ).

% space_bot
thf(fact_1095_space__bot,axiom,
    ( ( sigma_space_real @ bot_bo5982154664989874033e_real )
    = bot_bot_set_real ) ).

% space_bot
thf(fact_1096_space__bot,axiom,
    ( ( sigma_space_o @ bot_bo5758314138661044393sure_o )
    = bot_bot_set_o ) ).

% space_bot
thf(fact_1097_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_1098_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_1099_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_1100_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_measure_b,N2: sigma_measure_b,F: real > b,G: real > b] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ M2 ) )
            = ( member_real_b @ G @ ( sigma_523072396149930113real_b @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1101_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_3396294578489551860t_real,N2: sigma_3396294578489551860t_real,F: real > nat > real,G: real > nat > real] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_nat_real2 @ F @ ( sigma_783869947231497753t_real @ M @ M2 ) )
            = ( member_real_nat_real2 @ G @ ( sigma_783869947231497753t_real @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1102_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_measure_real_b,N2: sigma_measure_real_b,F: real > real > b,G: real > real > b] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_real_b @ F @ ( sigma_5735160446100821900real_b @ M @ M2 ) )
            = ( member_real_real_b @ G @ ( sigma_5735160446100821900real_b @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1103_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_6586288717683155060al_nat,N2: sigma_6586288717683155060al_nat,F: real > real > nat,G: real > real > nat] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_real_nat @ F @ ( sigma_6032194292609393305al_nat @ M @ M2 ) )
            = ( member_real_real_nat @ G @ ( sigma_6032194292609393305al_nat @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1104_measurable__cong__simp,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real,M2: sigma_3334325623652945375et_nat,N2: sigma_3334325623652945375et_nat,F: real > set_nat,G: real > set_nat] :
      ( ( M = N )
     => ( ( M2 = N2 )
       => ( ! [W: real] :
              ( ( member_real @ W @ ( sigma_space_real @ M ) )
             => ( ( F @ W )
                = ( G @ W ) ) )
         => ( ( member_real_set_nat @ F @ ( sigma_5099707562218272132et_nat @ M @ M2 ) )
            = ( member_real_set_nat @ G @ ( sigma_5099707562218272132et_nat @ N @ N2 ) ) ) ) ) ) ).

% measurable_cong_simp
thf(fact_1105_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_1106_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_1107_measurable__space,axiom,
    ! [F: b > b,M: sigma_measure_b,A: sigma_measure_b,X: b] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ A ) )
     => ( ( member_b @ X @ ( sigma_space_b @ M ) )
       => ( member_b @ ( F @ X ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_1108_measurable__space,axiom,
    ! [F: b > nat,M: sigma_measure_b,A: sigma_measure_nat,X: b] :
      ( ( member_b_nat @ F @ ( sigma_1308594411581951615_b_nat @ M @ A ) )
     => ( ( member_b @ X @ ( sigma_space_b @ M ) )
       => ( member_nat @ ( F @ X ) @ ( sigma_space_nat @ A ) ) ) ) ).

% measurable_space
thf(fact_1109_measurable__space,axiom,
    ! [F: b > real,M: sigma_measure_b,A: sigma_measure_real,X: b] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M @ A ) )
     => ( ( member_b @ X @ ( sigma_space_b @ M ) )
       => ( member_real @ ( F @ X ) @ ( sigma_space_real @ A ) ) ) ) ).

% measurable_space
thf(fact_1110_measurable__space,axiom,
    ! [F: b > $o,M: sigma_measure_b,A: sigma_measure_o,X: b] :
      ( ( member_b_o @ F @ ( sigma_measurable_b_o @ M @ A ) )
     => ( ( member_b @ X @ ( sigma_space_b @ M ) )
       => ( member_o @ ( F @ X ) @ ( sigma_space_o @ A ) ) ) ) ).

% measurable_space
thf(fact_1111_measurable__space,axiom,
    ! [F: nat > b,M: sigma_measure_nat,A: sigma_measure_b,X: nat] :
      ( ( member_nat_b @ F @ ( sigma_4105081583803843549_nat_b @ M @ A ) )
     => ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
       => ( member_b @ ( F @ X ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_1112_measurable__space,axiom,
    ! [F: nat > nat,M: sigma_measure_nat,A: sigma_measure_nat,X: nat] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ A ) )
     => ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
       => ( member_nat @ ( F @ X ) @ ( sigma_space_nat @ A ) ) ) ) ).

% measurable_space
thf(fact_1113_measurable__space,axiom,
    ! [F: nat > $o,M: sigma_measure_nat,A: sigma_measure_o,X: nat] :
      ( ( member_nat_o @ F @ ( sigma_5101835498682829686_nat_o @ M @ A ) )
     => ( ( member_nat @ X @ ( sigma_space_nat @ M ) )
       => ( member_o @ ( F @ X ) @ ( sigma_space_o @ A ) ) ) ) ).

% measurable_space
thf(fact_1114_measurable__space,axiom,
    ! [F: real > b,M: sigma_measure_real,A: sigma_measure_b,X: real] :
      ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ A ) )
     => ( ( member_real @ X @ ( sigma_space_real @ M ) )
       => ( member_b @ ( F @ X ) @ ( sigma_space_b @ A ) ) ) ) ).

% measurable_space
thf(fact_1115_measurable__space,axiom,
    ! [F: real > real,M: sigma_measure_real,A: sigma_measure_real,X: real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ A ) )
     => ( ( member_real @ X @ ( sigma_space_real @ M ) )
       => ( member_real @ ( F @ X ) @ ( sigma_space_real @ A ) ) ) ) ).

% measurable_space
thf(fact_1116_measurable__space,axiom,
    ! [F: real > $o,M: sigma_measure_real,A: sigma_measure_o,X: real] :
      ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ A ) )
     => ( ( member_real @ X @ ( sigma_space_real @ M ) )
       => ( member_o @ ( F @ X ) @ ( sigma_space_o @ A ) ) ) ) ).

% measurable_space
thf(fact_1117_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > b,G: real > b,M2: sigma_measure_b] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ M2 ) )
        = ( member_real_b @ G @ ( sigma_523072396149930113real_b @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1118_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > nat > real,G: real > nat > real,M2: sigma_3396294578489551860t_real] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_nat_real2 @ F @ ( sigma_783869947231497753t_real @ M @ M2 ) )
        = ( member_real_nat_real2 @ G @ ( sigma_783869947231497753t_real @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1119_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > real > b,G: real > real > b,M2: sigma_measure_real_b] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_real_b @ F @ ( sigma_5735160446100821900real_b @ M @ M2 ) )
        = ( member_real_real_b @ G @ ( sigma_5735160446100821900real_b @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1120_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > real > nat,G: real > real > nat,M2: sigma_6586288717683155060al_nat] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_real_nat @ F @ ( sigma_6032194292609393305al_nat @ M @ M2 ) )
        = ( member_real_real_nat @ G @ ( sigma_6032194292609393305al_nat @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1121_measurable__cong,axiom,
    ! [M: sigma_measure_real,F: real > set_nat,G: real > set_nat,M2: sigma_3334325623652945375et_nat] :
      ( ! [W: real] :
          ( ( member_real @ W @ ( sigma_space_real @ M ) )
         => ( ( F @ W )
            = ( G @ W ) ) )
     => ( ( member_real_set_nat @ F @ ( sigma_5099707562218272132et_nat @ M @ M2 ) )
        = ( member_real_set_nat @ G @ ( sigma_5099707562218272132et_nat @ M @ M2 ) ) ) ) ).

% measurable_cong
thf(fact_1122_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_1123_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_1124_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_b,Pb: $o] :
      ( member_set_b
      @ ( collect_b
        @ ^ [X5: b] :
            ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
            & Pb ) )
      @ ( sigma_sets_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1125_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_nat,Pb: $o] :
      ( member_set_nat
      @ ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
            & Pb ) )
      @ ( sigma_sets_nat @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1126_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_real,Pb: $o] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X5: real] :
            ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
            & Pb ) )
      @ ( sigma_sets_real @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1127_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_o,Pb: $o] :
      ( member_set_o
      @ ( collect_o
        @ ^ [X5: $o] :
            ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
            & Pb ) )
      @ ( sigma_sets_o @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1128_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_3334325623652945375et_nat,Pb: $o] :
      ( member_set_set_nat
      @ ( collect_set_nat
        @ ^ [X5: set_nat] :
            ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
            & Pb ) )
      @ ( sigma_sets_set_nat @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1129_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_6586288717683155060al_nat,Pb: $o] :
      ( member_set_real_nat
      @ ( collect_real_nat
        @ ^ [X5: real > nat] :
            ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
            & Pb ) )
      @ ( sigma_sets_real_nat @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1130_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_measure_real_b,Pb: $o] :
      ( member_set_real_b
      @ ( collect_real_b
        @ ^ [X5: real > b] :
            ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
            & Pb ) )
      @ ( sigma_sets_real_b @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1131_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_3396294578489551860t_real,Pb: $o] :
      ( member_set_nat_real2
      @ ( collect_nat_real
        @ ^ [X5: nat > real] :
            ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
            & Pb ) )
      @ ( sigma_sets_nat_real @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1132_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_7120471488171859498et_nat,Pb: $o] :
      ( member8337256853318788382et_nat
      @ ( collect_real_set_nat
        @ ^ [X5: real > set_nat] :
            ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
            & Pb ) )
      @ ( sigma_1129187412287173170et_nat @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1133_sets_Osets__Collect_I5_J,axiom,
    ! [M: sigma_4069289485848494527t_real,Pb: $o] :
      ( member890523794726346931t_real
      @ ( collec336724336074912571t_real
        @ ^ [X5: real > nat > real] :
            ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
            & Pb ) )
      @ ( sigma_787286210617551431t_real @ M ) ) ).

% sets.sets_Collect(5)
thf(fact_1134_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X5: b] :
              ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_b @ M ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1135_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
      ( ( member_set_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1136_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X5: real] :
              ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_real @ M ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1137_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
      ( ( member_set_o
        @ ( collect_o
          @ ^ [X5: $o] :
              ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_o @ M ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1138_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_3334325623652945375et_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( member_set_set_nat
        @ ( collect_set_nat
          @ ^ [X5: set_nat] :
              ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_set_nat @ M ) )
     => ( ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_set_nat @ M ) )
       => ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_sets_set_nat @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1139_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_6586288717683155060al_nat,P: ( real > nat ) > $o,Q: ( real > nat ) > $o] :
      ( ( member_set_real_nat
        @ ( collect_real_nat
          @ ^ [X5: real > nat] :
              ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real_nat @ M ) )
     => ( ( member_set_real_nat
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_real_nat @ M ) )
       => ( member_set_real_nat
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_sets_real_nat @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1140_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_measure_real_b,P: ( real > b ) > $o,Q: ( real > b ) > $o] :
      ( ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X5: real > b] :
              ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real_b @ M ) )
     => ( ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_real_b @ M ) )
       => ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_sets_real_b @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1141_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_3396294578489551860t_real,P: ( nat > real ) > $o,Q: ( nat > real ) > $o] :
      ( ( member_set_nat_real2
        @ ( collect_nat_real
          @ ^ [X5: nat > real] :
              ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_nat_real @ M ) )
     => ( ( member_set_nat_real2
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_nat_real @ M ) )
       => ( member_set_nat_real2
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_sets_nat_real @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1142_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_7120471488171859498et_nat,P: ( real > set_nat ) > $o,Q: ( real > set_nat ) > $o] :
      ( ( member8337256853318788382et_nat
        @ ( collect_real_set_nat
          @ ^ [X5: real > set_nat] :
              ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_1129187412287173170et_nat @ M ) )
     => ( ( member8337256853318788382et_nat
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_1129187412287173170et_nat @ M ) )
       => ( member8337256853318788382et_nat
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_1129187412287173170et_nat @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1143_sets_Osets__Collect__imp,axiom,
    ! [M: sigma_4069289485848494527t_real,P: ( real > nat > real ) > $o,Q: ( real > nat > real ) > $o] :
      ( ( member890523794726346931t_real
        @ ( collec336724336074912571t_real
          @ ^ [X5: real > nat > real] :
              ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_787286210617551431t_real @ M ) )
     => ( ( member890523794726346931t_real
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_787286210617551431t_real @ M ) )
       => ( member890523794726346931t_real
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
                & ( ( Q @ X5 )
                 => ( P @ X5 ) ) ) )
          @ ( sigma_787286210617551431t_real @ M ) ) ) ) ).

% sets.sets_Collect_imp
thf(fact_1144_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_b,P: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X5: b] :
              ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( member_set_b
        @ ( collect_b
          @ ^ [X5: b] :
              ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_sets_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1145_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_nat,P: nat > $o] :
      ( ( member_set_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_nat @ M ) )
     => ( member_set_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_sets_nat @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1146_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_real,P: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X5: real] :
              ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( member_set_real
        @ ( collect_real
          @ ^ [X5: real] :
              ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_sets_real @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1147_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_o,P: $o > $o] :
      ( ( member_set_o
        @ ( collect_o
          @ ^ [X5: $o] :
              ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_o @ M ) )
     => ( member_set_o
        @ ( collect_o
          @ ^ [X5: $o] :
              ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_sets_o @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1148_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_3334325623652945375et_nat,P: set_nat > $o] :
      ( ( member_set_set_nat
        @ ( collect_set_nat
          @ ^ [X5: set_nat] :
              ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_set_nat @ M ) )
     => ( member_set_set_nat
        @ ( collect_set_nat
          @ ^ [X5: set_nat] :
              ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_sets_set_nat @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1149_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_6586288717683155060al_nat,P: ( real > nat ) > $o] :
      ( ( member_set_real_nat
        @ ( collect_real_nat
          @ ^ [X5: real > nat] :
              ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real_nat @ M ) )
     => ( member_set_real_nat
        @ ( collect_real_nat
          @ ^ [X5: real > nat] :
              ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_sets_real_nat @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1150_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_measure_real_b,P: ( real > b ) > $o] :
      ( ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X5: real > b] :
              ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real_b @ M ) )
     => ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X5: real > b] :
              ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_sets_real_b @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1151_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_3396294578489551860t_real,P: ( nat > real ) > $o] :
      ( ( member_set_nat_real2
        @ ( collect_nat_real
          @ ^ [X5: nat > real] :
              ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_nat_real @ M ) )
     => ( member_set_nat_real2
        @ ( collect_nat_real
          @ ^ [X5: nat > real] :
              ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_sets_nat_real @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1152_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_7120471488171859498et_nat,P: ( real > set_nat ) > $o] :
      ( ( member8337256853318788382et_nat
        @ ( collect_real_set_nat
          @ ^ [X5: real > set_nat] :
              ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_1129187412287173170et_nat @ M ) )
     => ( member8337256853318788382et_nat
        @ ( collect_real_set_nat
          @ ^ [X5: real > set_nat] :
              ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_1129187412287173170et_nat @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1153_sets_Osets__Collect__neg,axiom,
    ! [M: sigma_4069289485848494527t_real,P: ( real > nat > real ) > $o] :
      ( ( member890523794726346931t_real
        @ ( collec336724336074912571t_real
          @ ^ [X5: real > nat > real] :
              ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_787286210617551431t_real @ M ) )
     => ( member890523794726346931t_real
        @ ( collec336724336074912571t_real
          @ ^ [X5: real > nat > real] :
              ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
              & ~ ( P @ X5 ) ) )
        @ ( sigma_787286210617551431t_real @ M ) ) ) ).

% sets.sets_Collect_neg
thf(fact_1154_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X5: b] :
              ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_b @ M ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1155_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
      ( ( member_set_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1156_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X5: real] :
              ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_real @ M ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1157_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
      ( ( member_set_o
        @ ( collect_o
          @ ^ [X5: $o] :
              ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_o @ M ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1158_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_3334325623652945375et_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( member_set_set_nat
        @ ( collect_set_nat
          @ ^ [X5: set_nat] :
              ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_set_nat @ M ) )
     => ( ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_set_nat @ M ) )
       => ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_set_nat @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1159_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_6586288717683155060al_nat,P: ( real > nat ) > $o,Q: ( real > nat ) > $o] :
      ( ( member_set_real_nat
        @ ( collect_real_nat
          @ ^ [X5: real > nat] :
              ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real_nat @ M ) )
     => ( ( member_set_real_nat
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_real_nat @ M ) )
       => ( member_set_real_nat
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_real_nat @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1160_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_measure_real_b,P: ( real > b ) > $o,Q: ( real > b ) > $o] :
      ( ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X5: real > b] :
              ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real_b @ M ) )
     => ( ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_real_b @ M ) )
       => ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_real_b @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1161_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_3396294578489551860t_real,P: ( nat > real ) > $o,Q: ( nat > real ) > $o] :
      ( ( member_set_nat_real2
        @ ( collect_nat_real
          @ ^ [X5: nat > real] :
              ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_nat_real @ M ) )
     => ( ( member_set_nat_real2
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_nat_real @ M ) )
       => ( member_set_nat_real2
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_nat_real @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1162_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_7120471488171859498et_nat,P: ( real > set_nat ) > $o,Q: ( real > set_nat ) > $o] :
      ( ( member8337256853318788382et_nat
        @ ( collect_real_set_nat
          @ ^ [X5: real > set_nat] :
              ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_1129187412287173170et_nat @ M ) )
     => ( ( member8337256853318788382et_nat
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_1129187412287173170et_nat @ M ) )
       => ( member8337256853318788382et_nat
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_1129187412287173170et_nat @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1163_sets_Osets__Collect__conj,axiom,
    ! [M: sigma_4069289485848494527t_real,P: ( real > nat > real ) > $o,Q: ( real > nat > real ) > $o] :
      ( ( member890523794726346931t_real
        @ ( collec336724336074912571t_real
          @ ^ [X5: real > nat > real] :
              ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_787286210617551431t_real @ M ) )
     => ( ( member890523794726346931t_real
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_787286210617551431t_real @ M ) )
       => ( member890523794726346931t_real
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
                & ( Q @ X5 )
                & ( P @ X5 ) ) )
          @ ( sigma_787286210617551431t_real @ M ) ) ) ) ).

% sets.sets_Collect_conj
thf(fact_1164_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_b,P: b > $o,Q: b > $o] :
      ( ( member_set_b
        @ ( collect_b
          @ ^ [X5: b] :
              ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_b @ M ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_b @ M ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1165_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_nat,P: nat > $o,Q: nat > $o] :
      ( ( member_set_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1166_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_real,P: real > $o,Q: real > $o] :
      ( ( member_set_real
        @ ( collect_real
          @ ^ [X5: real] :
              ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_real @ M ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1167_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_o,P: $o > $o,Q: $o > $o] :
      ( ( member_set_o
        @ ( collect_o
          @ ^ [X5: $o] :
              ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_o @ M ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1168_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_3334325623652945375et_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( member_set_set_nat
        @ ( collect_set_nat
          @ ^ [X5: set_nat] :
              ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_set_nat @ M ) )
     => ( ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_set_nat @ M ) )
       => ( member_set_set_nat
          @ ( collect_set_nat
            @ ^ [X5: set_nat] :
                ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_sets_set_nat @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1169_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_6586288717683155060al_nat,P: ( real > nat ) > $o,Q: ( real > nat ) > $o] :
      ( ( member_set_real_nat
        @ ( collect_real_nat
          @ ^ [X5: real > nat] :
              ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real_nat @ M ) )
     => ( ( member_set_real_nat
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_real_nat @ M ) )
       => ( member_set_real_nat
          @ ( collect_real_nat
            @ ^ [X5: real > nat] :
                ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_sets_real_nat @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1170_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_measure_real_b,P: ( real > b ) > $o,Q: ( real > b ) > $o] :
      ( ( member_set_real_b
        @ ( collect_real_b
          @ ^ [X5: real > b] :
              ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_real_b @ M ) )
     => ( ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_real_b @ M ) )
       => ( member_set_real_b
          @ ( collect_real_b
            @ ^ [X5: real > b] :
                ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_sets_real_b @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1171_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_3396294578489551860t_real,P: ( nat > real ) > $o,Q: ( nat > real ) > $o] :
      ( ( member_set_nat_real2
        @ ( collect_nat_real
          @ ^ [X5: nat > real] :
              ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_sets_nat_real @ M ) )
     => ( ( member_set_nat_real2
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_sets_nat_real @ M ) )
       => ( member_set_nat_real2
          @ ( collect_nat_real
            @ ^ [X5: nat > real] :
                ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_sets_nat_real @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1172_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_7120471488171859498et_nat,P: ( real > set_nat ) > $o,Q: ( real > set_nat ) > $o] :
      ( ( member8337256853318788382et_nat
        @ ( collect_real_set_nat
          @ ^ [X5: real > set_nat] :
              ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_1129187412287173170et_nat @ M ) )
     => ( ( member8337256853318788382et_nat
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_1129187412287173170et_nat @ M ) )
       => ( member8337256853318788382et_nat
          @ ( collect_real_set_nat
            @ ^ [X5: real > set_nat] :
                ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_1129187412287173170et_nat @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1173_sets_Osets__Collect__disj,axiom,
    ! [M: sigma_4069289485848494527t_real,P: ( real > nat > real ) > $o,Q: ( real > nat > real ) > $o] :
      ( ( member890523794726346931t_real
        @ ( collec336724336074912571t_real
          @ ^ [X5: real > nat > real] :
              ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
              & ( P @ X5 ) ) )
        @ ( sigma_787286210617551431t_real @ M ) )
     => ( ( member890523794726346931t_real
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
                & ( Q @ X5 ) ) )
          @ ( sigma_787286210617551431t_real @ M ) )
       => ( member890523794726346931t_real
          @ ( collec336724336074912571t_real
            @ ^ [X5: real > nat > real] :
                ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
                & ( ( Q @ X5 )
                  | ( P @ X5 ) ) ) )
          @ ( sigma_787286210617551431t_real @ M ) ) ) ) ).

% sets.sets_Collect_disj
thf(fact_1174_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_b,P: $o] :
      ( member_set_b
      @ ( collect_b
        @ ^ [X5: b] :
            ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
            & P ) )
      @ ( sigma_sets_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1175_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_nat,P: $o] :
      ( member_set_nat
      @ ( collect_nat
        @ ^ [X5: nat] :
            ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
            & P ) )
      @ ( sigma_sets_nat @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1176_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_real,P: $o] :
      ( member_set_real
      @ ( collect_real
        @ ^ [X5: real] :
            ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
            & P ) )
      @ ( sigma_sets_real @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1177_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_o,P: $o] :
      ( member_set_o
      @ ( collect_o
        @ ^ [X5: $o] :
            ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
            & P ) )
      @ ( sigma_sets_o @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1178_sets_Osets__Collect__const,axiom,
    ! [M: sigma_3334325623652945375et_nat,P: $o] :
      ( member_set_set_nat
      @ ( collect_set_nat
        @ ^ [X5: set_nat] :
            ( ( member_set_nat @ X5 @ ( sigma_space_set_nat @ M ) )
            & P ) )
      @ ( sigma_sets_set_nat @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1179_sets_Osets__Collect__const,axiom,
    ! [M: sigma_6586288717683155060al_nat,P: $o] :
      ( member_set_real_nat
      @ ( collect_real_nat
        @ ^ [X5: real > nat] :
            ( ( member_real_nat @ X5 @ ( sigma_space_real_nat @ M ) )
            & P ) )
      @ ( sigma_sets_real_nat @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1180_sets_Osets__Collect__const,axiom,
    ! [M: sigma_measure_real_b,P: $o] :
      ( member_set_real_b
      @ ( collect_real_b
        @ ^ [X5: real > b] :
            ( ( member_real_b @ X5 @ ( sigma_space_real_b @ M ) )
            & P ) )
      @ ( sigma_sets_real_b @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1181_sets_Osets__Collect__const,axiom,
    ! [M: sigma_3396294578489551860t_real,P: $o] :
      ( member_set_nat_real2
      @ ( collect_nat_real
        @ ^ [X5: nat > real] :
            ( ( member_nat_real @ X5 @ ( sigma_space_nat_real @ M ) )
            & P ) )
      @ ( sigma_sets_nat_real @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1182_sets_Osets__Collect__const,axiom,
    ! [M: sigma_7120471488171859498et_nat,P: $o] :
      ( member8337256853318788382et_nat
      @ ( collect_real_set_nat
        @ ^ [X5: real > set_nat] :
            ( ( member_real_set_nat @ X5 @ ( sigma_8533228468570120649et_nat @ M ) )
            & P ) )
      @ ( sigma_1129187412287173170et_nat @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1183_sets_Osets__Collect__const,axiom,
    ! [M: sigma_4069289485848494527t_real,P: $o] :
      ( member890523794726346931t_real
      @ ( collec336724336074912571t_real
        @ ^ [X5: real > nat > real] :
            ( ( member_real_nat_real2 @ X5 @ ( sigma_7241629334780537822t_real @ M ) )
            & P ) )
      @ ( sigma_787286210617551431t_real @ M ) ) ).

% sets.sets_Collect_const
thf(fact_1184_measurable__const,axiom,
    ! [C: set_nat,M2: sigma_3334325623652945375et_nat,M: sigma_measure_real] :
      ( ( member_set_nat @ C @ ( sigma_space_set_nat @ M2 ) )
     => ( member_real_set_nat
        @ ^ [X5: real] : C
        @ ( sigma_5099707562218272132et_nat @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_1185_measurable__const,axiom,
    ! [C: real > nat,M2: sigma_6586288717683155060al_nat,M: sigma_measure_real] :
      ( ( member_real_nat @ C @ ( sigma_space_real_nat @ M2 ) )
     => ( member_real_real_nat
        @ ^ [X5: real] : C
        @ ( sigma_6032194292609393305al_nat @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_1186_measurable__const,axiom,
    ! [C: real > b,M2: sigma_measure_real_b,M: sigma_measure_real] :
      ( ( member_real_b @ C @ ( sigma_space_real_b @ M2 ) )
     => ( member_real_real_b
        @ ^ [X5: real] : C
        @ ( sigma_5735160446100821900real_b @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_1187_measurable__const,axiom,
    ! [C: nat > real,M2: sigma_3396294578489551860t_real,M: sigma_measure_real] :
      ( ( member_nat_real @ C @ ( sigma_space_nat_real @ M2 ) )
     => ( member_real_nat_real2
        @ ^ [X5: real] : C
        @ ( sigma_783869947231497753t_real @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_1188_measurable__const,axiom,
    ! [C: b,M2: sigma_measure_b,M: sigma_measure_real] :
      ( ( member_b @ C @ ( sigma_space_b @ M2 ) )
     => ( member_real_b
        @ ^ [X5: real] : C
        @ ( sigma_523072396149930113real_b @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_1189_measurable__const,axiom,
    ! [C: nat,M2: sigma_measure_nat,M: sigma_measure_real] :
      ( ( member_nat @ C @ ( sigma_space_nat @ M2 ) )
     => ( member_real_nat
        @ ^ [X5: real] : C
        @ ( sigma_6315060578831106510al_nat @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_1190_measurable__const,axiom,
    ! [C: real,M2: sigma_measure_real,M: sigma_measure_nat] :
      ( ( member_real @ C @ ( sigma_space_real @ M2 ) )
     => ( member_nat_real
        @ ^ [X5: nat] : C
        @ ( sigma_1747752005702207822t_real @ M @ M2 ) ) ) ).

% measurable_const
thf(fact_1191_sets_Osets__into__space,axiom,
    ! [X: set_nat,M: sigma_measure_nat] :
      ( ( member_set_nat @ X @ ( sigma_sets_nat @ M ) )
     => ( ord_less_eq_set_nat @ X @ ( sigma_space_nat @ M ) ) ) ).

% sets.sets_into_space
thf(fact_1192_sets_Osets__into__space,axiom,
    ! [X: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ X @ ( sigma_sets_real @ M ) )
     => ( ord_less_eq_set_real @ X @ ( sigma_space_real @ M ) ) ) ).

% sets.sets_into_space
thf(fact_1193_sets_Osets__into__space,axiom,
    ! [X: set_o,M: sigma_measure_o] :
      ( ( member_set_o @ X @ ( sigma_sets_o @ M ) )
     => ( ord_less_eq_set_o @ X @ ( sigma_space_o @ M ) ) ) ).

% sets.sets_into_space
thf(fact_1194_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_nat,B: sigma_measure_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ A ) @ ( sigma_sets_nat @ B ) )
     => ( ord_less_eq_set_nat @ ( sigma_space_nat @ A ) @ ( sigma_space_nat @ B ) ) ) ).

% sets_le_imp_space_le
thf(fact_1195_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_real,B: sigma_measure_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B ) )
     => ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B ) ) ) ).

% sets_le_imp_space_le
thf(fact_1196_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_o,B: sigma_measure_o] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ A ) @ ( sigma_sets_o @ B ) )
     => ( ord_less_eq_set_o @ ( sigma_space_o @ A ) @ ( sigma_space_o @ B ) ) ) ).

% sets_le_imp_space_le
thf(fact_1197_measurable__empty__iff,axiom,
    ! [N: sigma_measure_b,F: real > b,M: sigma_measure_real] :
      ( ( ( sigma_space_b @ N )
        = bot_bot_set_b )
     => ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ N ) )
        = ( ( sigma_space_real @ M )
          = bot_bot_set_real ) ) ) ).

% measurable_empty_iff
thf(fact_1198_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_1199_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_1200_measurable__empty__iff,axiom,
    ! [N: sigma_measure_real,F: real > real,M: sigma_measure_real] :
      ( ( ( sigma_space_real @ N )
        = bot_bot_set_real )
     => ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
        = ( ( sigma_space_real @ M )
          = bot_bot_set_real ) ) ) ).

% measurable_empty_iff
thf(fact_1201_measurable__empty__iff,axiom,
    ! [N: sigma_measure_real,F: $o > real,M: sigma_measure_o] :
      ( ( ( sigma_space_real @ N )
        = bot_bot_set_real )
     => ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ N ) )
        = ( ( sigma_space_o @ M )
          = bot_bot_set_o ) ) ) ).

% measurable_empty_iff
thf(fact_1202_measurable__empty__iff,axiom,
    ! [N: sigma_measure_o,F: nat > $o,M: sigma_measure_nat] :
      ( ( ( sigma_space_o @ N )
        = bot_bot_set_o )
     => ( ( member_nat_o @ F @ ( sigma_5101835498682829686_nat_o @ M @ N ) )
        = ( ( sigma_space_nat @ M )
          = bot_bot_set_nat ) ) ) ).

% measurable_empty_iff
thf(fact_1203_measurable__empty__iff,axiom,
    ! [N: sigma_measure_o,F: real > $o,M: sigma_measure_real] :
      ( ( ( sigma_space_o @ N )
        = bot_bot_set_o )
     => ( ( member_real_o @ F @ ( sigma_3939073009482781210real_o @ M @ N ) )
        = ( ( sigma_space_real @ M )
          = bot_bot_set_real ) ) ) ).

% measurable_empty_iff
thf(fact_1204_measurable__empty__iff,axiom,
    ! [N: sigma_measure_o,F: $o > $o,M: sigma_measure_o] :
      ( ( ( sigma_space_o @ N )
        = bot_bot_set_o )
     => ( ( member_o_o @ F @ ( sigma_measurable_o_o @ M @ N ) )
        = ( ( sigma_space_o @ M )
          = bot_bot_set_o ) ) ) ).

% measurable_empty_iff
thf(fact_1205_measurable__empty__iff,axiom,
    ! [N: sigma_measure_nat,F: real > nat,M: sigma_measure_real] :
      ( ( ( sigma_space_nat @ N )
        = bot_bot_set_nat )
     => ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ N ) )
        = ( ( sigma_space_real @ M )
          = bot_bot_set_real ) ) ) ).

% measurable_empty_iff
thf(fact_1206_measurable__empty__iff,axiom,
    ! [N: sigma_measure_real,F: nat > real,M: sigma_measure_nat] :
      ( ( ( sigma_space_real @ N )
        = bot_bot_set_real )
     => ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ N ) )
        = ( ( sigma_space_nat @ M )
          = bot_bot_set_nat ) ) ) ).

% measurable_empty_iff
thf(fact_1207_le__measureD1,axiom,
    ! [A: sigma_measure_nat,B: sigma_measure_nat] :
      ( ( ord_le2862109966718184649re_nat @ A @ B )
     => ( ord_less_eq_set_nat @ ( sigma_space_nat @ A ) @ ( sigma_space_nat @ B ) ) ) ).

% le_measureD1
thf(fact_1208_le__measureD1,axiom,
    ! [A: sigma_measure_real,B: sigma_measure_real] :
      ( ( ord_le487379304121309861e_real @ A @ B )
     => ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B ) ) ) ).

% le_measureD1
thf(fact_1209_le__measureD1,axiom,
    ! [A: sigma_measure_o,B: sigma_measure_o] :
      ( ( ord_le478349814012620405sure_o @ A @ B )
     => ( ord_less_eq_set_o @ ( sigma_space_o @ A ) @ ( sigma_space_o @ B ) ) ) ).

% le_measureD1
thf(fact_1210_le__measureD2,axiom,
    ! [A: sigma_measure_nat,B: sigma_measure_nat] :
      ( ( ord_le2862109966718184649re_nat @ A @ B )
     => ( ( ( sigma_space_nat @ A )
          = ( sigma_space_nat @ B ) )
       => ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ A ) @ ( sigma_sets_nat @ B ) ) ) ) ).

% le_measureD2
thf(fact_1211_le__measureD2,axiom,
    ! [A: sigma_measure_real,B: sigma_measure_real] :
      ( ( ord_le487379304121309861e_real @ A @ B )
     => ( ( ( sigma_space_real @ A )
          = ( sigma_space_real @ B ) )
       => ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B ) ) ) ) ).

% le_measureD2
thf(fact_1212_le__measureD2,axiom,
    ! [A: sigma_measure_o,B: sigma_measure_o] :
      ( ( ord_le478349814012620405sure_o @ A @ B )
     => ( ( ( sigma_space_o @ A )
          = ( sigma_space_o @ B ) )
       => ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ A ) @ ( sigma_sets_o @ B ) ) ) ) ).

% le_measureD2
thf(fact_1213_space__empty__eq__bot,axiom,
    ! [A2: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ A2 )
        = bot_bot_set_nat )
      = ( A2 = bot_bo6718502177978453909re_nat ) ) ).

% space_empty_eq_bot
thf(fact_1214_space__empty__eq__bot,axiom,
    ! [A2: sigma_measure_real] :
      ( ( ( sigma_space_real @ A2 )
        = bot_bot_set_real )
      = ( A2 = bot_bo5982154664989874033e_real ) ) ).

% space_empty_eq_bot
thf(fact_1215_space__empty__eq__bot,axiom,
    ! [A2: sigma_measure_o] :
      ( ( ( sigma_space_o @ A2 )
        = bot_bot_set_o )
      = ( A2 = bot_bo5758314138661044393sure_o ) ) ).

% space_empty_eq_bot
thf(fact_1216_measurable__sets__Collect,axiom,
    ! [F: b > b,M: sigma_measure_b,N: sigma_measure_b,P: b > $o] :
      ( ( member_b_b @ F @ ( sigma_measurable_b_b @ M @ N ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_b @ N ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1217_measurable__sets__Collect,axiom,
    ! [F: nat > b,M: sigma_measure_nat,N: sigma_measure_b,P: b > $o] :
      ( ( member_nat_b @ F @ ( sigma_4105081583803843549_nat_b @ M @ N ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_b @ N ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1218_measurable__sets__Collect,axiom,
    ! [F: real > b,M: sigma_measure_real,N: sigma_measure_b,P: b > $o] :
      ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ N ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_b @ N ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1219_measurable__sets__Collect,axiom,
    ! [F: $o > b,M: sigma_measure_o,N: sigma_measure_b,P: b > $o] :
      ( ( member_o_b @ F @ ( sigma_measurable_o_b @ M @ N ) )
     => ( ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_b @ N ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1220_measurable__sets__Collect,axiom,
    ! [F: b > nat,M: sigma_measure_b,N: sigma_measure_nat,P: nat > $o] :
      ( ( member_b_nat @ F @ ( sigma_1308594411581951615_b_nat @ M @ N ) )
     => ( ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_nat @ N ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1221_measurable__sets__Collect,axiom,
    ! [F: nat > nat,M: sigma_measure_nat,N: sigma_measure_nat,P: nat > $o] :
      ( ( member_nat_nat @ F @ ( sigma_4350458207664084850at_nat @ M @ N ) )
     => ( ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_nat @ N ) )
       => ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_nat @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1222_measurable__sets__Collect,axiom,
    ! [F: $o > nat,M: sigma_measure_o,N: sigma_measure_nat,P: nat > $o] :
      ( ( member_o_nat @ F @ ( sigma_1999164137574644376_o_nat @ M @ N ) )
     => ( ( member_set_nat
          @ ( collect_nat
            @ ^ [X5: nat] :
                ( ( member_nat @ X5 @ ( sigma_space_nat @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_nat @ N ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1223_measurable__sets__Collect,axiom,
    ! [F: b > real,M: sigma_measure_b,N: sigma_measure_real,P: real > $o] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M @ N ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_real @ N ) )
       => ( member_set_b
          @ ( collect_b
            @ ^ [X5: b] :
                ( ( member_b @ X5 @ ( sigma_space_b @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_b @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1224_measurable__sets__Collect,axiom,
    ! [F: real > real,M: sigma_measure_real,N: sigma_measure_real,P: real > $o] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ N ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_real @ N ) )
       => ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_real @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1225_measurable__sets__Collect,axiom,
    ! [F: $o > real,M: sigma_measure_o,N: sigma_measure_real,P: real > $o] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ N ) )
     => ( ( member_set_real
          @ ( collect_real
            @ ^ [X5: real] :
                ( ( member_real @ X5 @ ( sigma_space_real @ N ) )
                & ( P @ X5 ) ) )
          @ ( sigma_sets_real @ N ) )
       => ( member_set_o
          @ ( collect_o
            @ ^ [X5: $o] :
                ( ( member_o @ X5 @ ( sigma_space_o @ M ) )
                & ( P @ ( F @ X5 ) ) ) )
          @ ( sigma_sets_o @ M ) ) ) ) ).

% measurable_sets_Collect
thf(fact_1226_measurable__If,axiom,
    ! [F: real > b,M: sigma_measure_real,M2: sigma_measure_b,G: real > b,P: real > $o] :
      ( ( member_real_b @ F @ ( sigma_523072396149930113real_b @ M @ M2 ) )
     => ( ( member_real_b @ G @ ( sigma_523072396149930113real_b @ M @ M2 ) )
       => ( ( member_set_real
            @ ( collect_real
              @ ^ [X5: real] :
                  ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                  & ( P @ X5 ) ) )
            @ ( sigma_sets_real @ M ) )
         => ( member_real_b
            @ ^ [X5: real] : ( if_b @ ( P @ X5 ) @ ( F @ X5 ) @ ( G @ X5 ) )
            @ ( sigma_523072396149930113real_b @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_1227_measurable__If,axiom,
    ! [F: real > nat > real,M: sigma_measure_real,M2: sigma_3396294578489551860t_real,G: real > nat > real,P: real > $o] :
      ( ( member_real_nat_real2 @ F @ ( sigma_783869947231497753t_real @ M @ M2 ) )
     => ( ( member_real_nat_real2 @ G @ ( sigma_783869947231497753t_real @ M @ M2 ) )
       => ( ( member_set_real
            @ ( collect_real
              @ ^ [X5: real] :
                  ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                  & ( P @ X5 ) ) )
            @ ( sigma_sets_real @ M ) )
         => ( member_real_nat_real2
            @ ^ [X5: real] : ( if_nat_real @ ( P @ X5 ) @ ( F @ X5 ) @ ( G @ X5 ) )
            @ ( sigma_783869947231497753t_real @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_1228_measurable__If,axiom,
    ! [F: real > real > b,M: sigma_measure_real,M2: sigma_measure_real_b,G: real > real > b,P: real > $o] :
      ( ( member_real_real_b @ F @ ( sigma_5735160446100821900real_b @ M @ M2 ) )
     => ( ( member_real_real_b @ G @ ( sigma_5735160446100821900real_b @ M @ M2 ) )
       => ( ( member_set_real
            @ ( collect_real
              @ ^ [X5: real] :
                  ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                  & ( P @ X5 ) ) )
            @ ( sigma_sets_real @ M ) )
         => ( member_real_real_b
            @ ^ [X5: real] : ( if_real_b @ ( P @ X5 ) @ ( F @ X5 ) @ ( G @ X5 ) )
            @ ( sigma_5735160446100821900real_b @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_1229_measurable__If,axiom,
    ! [F: real > real > nat,M: sigma_measure_real,M2: sigma_6586288717683155060al_nat,G: real > real > nat,P: real > $o] :
      ( ( member_real_real_nat @ F @ ( sigma_6032194292609393305al_nat @ M @ M2 ) )
     => ( ( member_real_real_nat @ G @ ( sigma_6032194292609393305al_nat @ M @ M2 ) )
       => ( ( member_set_real
            @ ( collect_real
              @ ^ [X5: real] :
                  ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                  & ( P @ X5 ) ) )
            @ ( sigma_sets_real @ M ) )
         => ( member_real_real_nat
            @ ^ [X5: real] : ( if_real_nat @ ( P @ X5 ) @ ( F @ X5 ) @ ( G @ X5 ) )
            @ ( sigma_6032194292609393305al_nat @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_1230_measurable__If,axiom,
    ! [F: real > set_nat,M: sigma_measure_real,M2: sigma_3334325623652945375et_nat,G: real > set_nat,P: real > $o] :
      ( ( member_real_set_nat @ F @ ( sigma_5099707562218272132et_nat @ M @ M2 ) )
     => ( ( member_real_set_nat @ G @ ( sigma_5099707562218272132et_nat @ M @ M2 ) )
       => ( ( member_set_real
            @ ( collect_real
              @ ^ [X5: real] :
                  ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                  & ( P @ X5 ) ) )
            @ ( sigma_sets_real @ M ) )
         => ( member_real_set_nat
            @ ^ [X5: real] : ( if_set_nat @ ( P @ X5 ) @ ( F @ X5 ) @ ( G @ X5 ) )
            @ ( sigma_5099707562218272132et_nat @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_1231_measurable__If,axiom,
    ! [F: real > nat,M: sigma_measure_real,M2: sigma_measure_nat,G: real > nat,P: real > $o] :
      ( ( member_real_nat @ F @ ( sigma_6315060578831106510al_nat @ M @ M2 ) )
     => ( ( member_real_nat @ G @ ( sigma_6315060578831106510al_nat @ M @ M2 ) )
       => ( ( member_set_real
            @ ( collect_real
              @ ^ [X5: real] :
                  ( ( member_real @ X5 @ ( sigma_space_real @ M ) )
                  & ( P @ X5 ) ) )
            @ ( sigma_sets_real @ M ) )
         => ( member_real_nat
            @ ^ [X5: real] : ( if_nat @ ( P @ X5 ) @ ( F @ X5 ) @ ( G @ X5 ) )
            @ ( sigma_6315060578831106510al_nat @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_1232_measurable__If,axiom,
    ! [F: nat > real,M: sigma_measure_nat,M2: sigma_measure_real,G: nat > real,P: nat > $o] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ M2 ) )
     => ( ( member_nat_real @ G @ ( sigma_1747752005702207822t_real @ M @ M2 ) )
       => ( ( member_set_nat
            @ ( collect_nat
              @ ^ [X5: nat] :
                  ( ( member_nat @ X5 @ ( sigma_space_nat @ M ) )
                  & ( P @ X5 ) ) )
            @ ( sigma_sets_nat @ M ) )
         => ( member_nat_real
            @ ^ [X5: nat] : ( if_real @ ( P @ X5 ) @ ( F @ X5 ) @ ( G @ X5 ) )
            @ ( sigma_1747752005702207822t_real @ M @ M2 ) ) ) ) ) ).

% measurable_If
thf(fact_1233_borel__measurable__iff__ge,axiom,
    ! [F: b > real,M: sigma_measure_b] :
      ( ( member_b_real @ F @ ( sigma_5489272243865111003b_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member_set_b
            @ ( collect_b
              @ ^ [W2: b] :
                  ( ( member_b @ W2 @ ( sigma_space_b @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_b @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1234_borel__measurable__iff__ge,axiom,
    ! [F: real > real,M: sigma_measure_real] :
      ( ( member_real_real @ F @ ( sigma_5267869275261027754l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member_set_real
            @ ( collect_real
              @ ^ [W2: real] :
                  ( ( member_real @ W2 @ ( sigma_space_real @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_real @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1235_borel__measurable__iff__ge,axiom,
    ! [F: $o > real,M: sigma_measure_o] :
      ( ( member_o_real @ F @ ( sigma_2430008634441611636o_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member_set_o
            @ ( collect_o
              @ ^ [W2: $o] :
                  ( ( member_o @ W2 @ ( sigma_space_o @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_o @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1236_borel__measurable__iff__ge,axiom,
    ! [F: nat > real,M: sigma_measure_nat] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member_set_nat
            @ ( collect_nat
              @ ^ [W2: nat] :
                  ( ( member_nat @ W2 @ ( sigma_space_nat @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_nat @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1237_borel__measurable__iff__ge,axiom,
    ! [F: set_nat > real,M: sigma_3334325623652945375et_nat] :
      ( ( member_set_nat_real @ F @ ( sigma_7357981997995286020t_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member_set_set_nat
            @ ( collect_set_nat
              @ ^ [W2: set_nat] :
                  ( ( member_set_nat @ W2 @ ( sigma_space_set_nat @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_set_nat @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1238_borel__measurable__iff__ge,axiom,
    ! [F: ( real > nat ) > real,M: sigma_6586288717683155060al_nat] :
      ( ( member_real_nat_real @ F @ ( sigma_2883472102289741465t_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member_set_real_nat
            @ ( collect_real_nat
              @ ^ [W2: real > nat] :
                  ( ( member_real_nat @ W2 @ ( sigma_space_real_nat @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_real_nat @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1239_borel__measurable__iff__ge,axiom,
    ! [F: ( real > b ) > real,M: sigma_measure_real_b] :
      ( ( member_real_b_real @ F @ ( sigma_4595654474610825574b_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member_set_real_b
            @ ( collect_real_b
              @ ^ [W2: real > b] :
                  ( ( member_real_b @ W2 @ ( sigma_space_real_b @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_real_b @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1240_borel__measurable__iff__ge,axiom,
    ! [F: ( nat > real ) > real,M: sigma_3396294578489551860t_real] :
      ( ( member_nat_real_real @ F @ ( sigma_8475865534613037593l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member_set_nat_real2
            @ ( collect_nat_real
              @ ^ [W2: nat > real] :
                  ( ( member_nat_real @ W2 @ ( sigma_space_nat_real @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_sets_nat_real @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1241_borel__measurable__iff__ge,axiom,
    ! [F: ( real > set_nat ) > real,M: sigma_7120471488171859498et_nat] :
      ( ( member5086141250106945459t_real @ F @ ( sigma_2288790926702686287t_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member8337256853318788382et_nat
            @ ( collect_real_set_nat
              @ ^ [W2: real > set_nat] :
                  ( ( member_real_set_nat @ W2 @ ( sigma_8533228468570120649et_nat @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_1129187412287173170et_nat @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1242_borel__measurable__iff__ge,axiom,
    ! [F: ( real > nat > real ) > real,M: sigma_4069289485848494527t_real] :
      ( ( member6568573216281280328l_real @ F @ ( sigma_7540513786767735268l_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member890523794726346931t_real
            @ ( collec336724336074912571t_real
              @ ^ [W2: real > nat > real] :
                  ( ( member_real_nat_real2 @ W2 @ ( sigma_7241629334780537822t_real @ M ) )
                  & ( ord_less_eq_real @ A3 @ ( F @ W2 ) ) ) )
            @ ( sigma_787286210617551431t_real @ M ) ) ) ) ).

% borel_measurable_iff_ge
thf(fact_1243_borel__measurable__iff__le,axiom,
    ! [F: nat > real,M: sigma_measure_nat] :
      ( ( member_nat_real @ F @ ( sigma_1747752005702207822t_real @ M @ borel_5078946678739801102l_real ) )
      = ( ! [A3: real] :
            ( member_set_nat
            @ ( collect_nat
              @ ^ [W2: nat] :
                  ( ( member_nat @ W2 @ ( sigma_space_nat @ M ) )
                  & ( ord_less_eq_real @ ( F @ W2 ) @ A3 ) ) )
            @ ( sigma_sets_nat @ M ) ) ) ) ).

% borel_measurable_iff_le
thf(fact_1244_nat_Osingleton__sets,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
     => ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% nat.singleton_sets
thf(fact_1245_real_Osingleton__sets,axiom,
    ! [X: real] :
      ( ( member_real @ X @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% real.singleton_sets
thf(fact_1246_real_Ospace__UNIV,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% real.space_UNIV
thf(fact_1247_nat_Ospace__UNIV,axiom,
    ( ( sigma_space_nat @ borel_8449730974584783410el_nat )
    = top_top_set_nat ) ).

% nat.space_UNIV
thf(fact_1248_bool_Ospace__UNIV,axiom,
    ( ( sigma_space_o @ borel_5500255247093592246orel_o )
    = top_top_set_o ) ).

% bool.space_UNIV
thf(fact_1249_bool_Osingleton__sets,axiom,
    ! [X: $o] :
      ( ( member_o @ X @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
     => ( member_set_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% bool.singleton_sets
thf(fact_1250_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_1251_nat_Ogf__comp__id_I2_J,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
     => ( ( standard_g_nat @ borel_8449730974584783410el_nat @ ( standard_f_nat @ borel_8449730974584783410el_nat @ X ) )
        = X ) ) ).

% nat.gf_comp_id(2)
thf(fact_1252_nat_Ogf__comp__id_H_I2_J,axiom,
    ! [X: nat] :
      ( ( standard_g_nat @ borel_8449730974584783410el_nat @ ( standard_f_nat @ borel_8449730974584783410el_nat @ X ) )
      = X ) ).

% nat.gf_comp_id'(2)
thf(fact_1253_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_1254_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_1255_nat_Ocountable__space__discrete,axiom,
    ( ( counta1168086296615599829le_nat @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
   => ( ( sigma_sets_nat @ borel_8449730974584783410el_nat )
      = ( sigma_sets_nat @ ( sigma_7685844798829912695ce_nat @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) ) ) ) ) ).

% nat.countable_space_discrete
thf(fact_1256_bool_Ocountable__space__discrete,axiom,
    ( ( counta5976203206615340371able_o @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
   => ( ( sigma_sets_o @ borel_5500255247093592246orel_o )
      = ( sigma_sets_o @ ( sigma_count_space_o @ ( sigma_space_o @ borel_5500255247093592246orel_o ) ) ) ) ) ).

% bool.countable_space_discrete
thf(fact_1257_finite__Collect__le__nat,axiom,
    ! [K: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).

% finite_Collect_le_nat
thf(fact_1258_finite__less__ub,axiom,
    ! [F: nat > nat,U2: nat] :
      ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U2 ) ) ) ) ).

% finite_less_ub
thf(fact_1259_bounded__Max__nat,axiom,
    ! [P: nat > $o,X: nat,M: nat] :
      ( ( P @ X )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( ord_less_eq_nat @ X3 @ M ) )
       => ~ ! [M4: nat] :
              ( ( P @ M4 )
             => ~ ! [X7: nat] :
                    ( ( P @ X7 )
                   => ( ord_less_eq_nat @ X7 @ M4 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1260_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M5: nat] :
        ! [X5: nat] :
          ( ( member_nat @ X5 @ N5 )
         => ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1261_uncountable__UNIV__real,axiom,
    ~ ( counta7319604579010473777e_real @ top_top_set_real ) ).

% uncountable_UNIV_real
thf(fact_1262_real__non__denum,axiom,
    ~ ? [F3: nat > real] :
        ( ( image_nat_real @ F3 @ top_top_set_nat )
        = top_top_set_real ) ).

% real_non_denum
thf(fact_1263_nat_Og__surj_H,axiom,
    ( ( image_real_nat @ ( standard_g_nat @ borel_8449730974584783410el_nat ) @ top_top_set_real )
    = top_top_set_nat ) ).

% nat.g_surj'

% Helper facts (15)
thf(help_If_2_1_If_001tf__b_T,axiom,
    ! [X: b,Y: b] :
      ( ( if_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__b_T,axiom,
    ! [X: b,Y: b] :
      ( ( if_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Real__Oreal_Mtf__b_J_T,axiom,
    ! [X: real > b,Y: real > b] :
      ( ( if_real_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Real__Oreal_Mtf__b_J_T,axiom,
    ! [X: real > b,Y: real > b] :
      ( ( if_real_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Real__Oreal_J_T,axiom,
    ! [X: nat > real,Y: nat > real] :
      ( ( if_nat_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Real__Oreal_J_T,axiom,
    ! [X: nat > real,Y: nat > real] :
      ( ( if_nat_real @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001_062_It__Real__Oreal_Mt__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001_062_It__Real__Oreal_Mt__Nat__Onat_J_T,axiom,
    ! [X: real > nat,Y: real > nat] :
      ( ( if_real_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Real__Oreal_Mt__Nat__Onat_J_T,axiom,
    ! [X: real > nat,Y: real > nat] :
      ( ( if_real_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( member_real_b
    @ ^ [R: real] : ( fi @ ( p @ R ) @ R )
    @ ( qbs_Mx_b @ x ) ) ).

%------------------------------------------------------------------------------