%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Quasi_Borel_Spaces/0003_Binary_CoProduct_QuasiBorel/prob_00618_024331__15232240_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2095 ( 685 unt; 812 typ;   0 def)
%            Number of atoms       : 3280 (1582 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 9451 ( 330   ~;  26   |; 192   &;7835   @)
%                                         (   0 <=>;1068  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   5 avg)
%            Number of types       :  148 ( 147 usr)
%            Number of type conns  : 2911 (2911   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  668 ( 665 usr;  66 con; 0-6 aty)
%            Number of variables   : 3223 ( 191   ^;2933   !;  99   ?;3223   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:09:17.014
%------------------------------------------------------------------------------
% Could-be-implicit typings (147)
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__d_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
    set_a_c: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
    set_a_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
    set_a_a: $tType ).

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

thf(ty_n_t__Sum____Type__Osum_I_Eo_Mtf__d_J,type,
    sum_sum_o_d: $tType ).

thf(ty_n_t__Sum____Type__Osum_I_Eo_Mtf__b_J,type,
    sum_sum_o_b: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_Eo_Mtf__a_J_J,type,
    set_o_a: $tType ).

thf(ty_n_t__Sum____Type__Osum_I_Eo_M_Eo_J,type,
    sum_sum_o_o: $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__d_J,type,
    set_d: $tType ).

thf(ty_n_t__Set__Oset_Itf__c_J,type,
    set_c: $tType ).

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

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

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

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

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

thf(ty_n_tf__d,type,
    d: $tType ).

thf(ty_n_tf__c,type,
    c: $tType ).

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

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (665)
thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    basic_2831768515958928805_set_o: sum_sum_set_o_set_o > set_set_o ).

thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    basic_127669078395480321nnreal: sum_su4962168936726258711nnreal > set_set_o ).

thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
    basic_112841166889062809et_nat: sum_su7609443801303718063et_nat > set_set_o ).

thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Real__Oreal_J,type,
    basic_5135449112562407285t_real: sum_su2944506142428048139t_real > set_set_o ).

thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_I_Eo_J,type,
    basic_68364610332180515_set_o: sum_su145664975714009857_set_o > set_se4580700918925141924nnreal ).

thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_It__Nat__Onat_J,type,
    basic_981975278752359067et_nat: sum_su1570082439829250771et_nat > set_se4580700918925141924nnreal ).

thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
    basic_650432819501328779_set_o: sum_su180556917423924329_set_o > set_set_nat ).

thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    basic_842870997940628123nnreal: sum_su7445071152329184979nnreal > set_set_nat ).

thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    basic_3910105186785813299et_nat: sum_su6357119772783438699et_nat > set_set_nat ).

thf(sy_c_Basic__BNFs_Osetl_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Real__Oreal_J,type,
    basic_4495003616656931855t_real: sum_su6439591028793565127t_real > set_set_nat ).

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

thf(sy_c_Basic__BNFs_Osetl_001tf__b_001tf__d,type,
    basic_setl_b_d: sum_sum_b_d > set_b ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    basic_1118969371388924075_set_o: sum_sum_set_o_set_o > set_set_o ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    basic_5549938302119316347nnreal: sum_su4962168936726258711nnreal > set_se4580700918925141924nnreal ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
    basic_5472916752366830611et_nat: sum_su7609443801303718063et_nat > set_set_nat ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Real__Oreal_J,type,
    basic_3640605392351816943t_real: sum_su2944506142428048139t_real > set_set_real ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_I_Eo_J,type,
    basic_5490633834056016541_set_o: sum_su145664975714009857_set_o > set_set_o ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_It__Nat__Onat_J,type,
    basic_8899975567016546977et_nat: sum_su1570082439829250771et_nat > set_set_nat ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
    basic_6010508404979096581_set_o: sum_su180556917423924329_set_o > set_set_o ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    basic_8760871286204816033nnreal: sum_su7445071152329184979nnreal > set_se4580700918925141924nnreal ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    basic_7585541094054265657et_nat: sum_su6357119772783438699et_nat > set_set_nat ).

thf(sy_c_Basic__BNFs_Osetr_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Real__Oreal_J,type,
    basic_4563158366440306965t_real: sum_su6439591028793565127t_real > set_set_real ).

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

thf(sy_c_Basic__BNFs_Osetr_001tf__b_001tf__d,type,
    basic_setr_b_d: sum_sum_b_d > set_d ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001_Eo_001_Eo,type,
    binary6836164603859296013bs_o_o: quasi_borel_o > quasi_borel_o > quasi_9151289748890374549um_o_o ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001_Eo_001t__Nat__Onat,type,
    binary2765902541835032475_o_nat: quasi_borel_o > quasi_borel_nat > quasi_6247536490924941631_o_nat ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001t__Nat__Onat_001_Eo,type,
    binary5868573902943217785_nat_o: quasi_borel_nat > quasi_borel_o > quasi_7295139613103245285_nat_o ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001t__Nat__Onat_001t__Nat__Onat,type,
    binary5195869006974583087at_nat: quasi_borel_nat > quasi_borel_nat > quasi_2537881152722034415at_nat ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001t__Real__Oreal_001t__Real__Oreal,type,
    binary3369073574056317543l_real: quasi_borel_real > quasi_borel_real > quasi_3918057494074722855l_real ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001tf__a_001tf__a,type,
    binary8555328655094383373bs_a_a: quasi_borel_a > quasi_borel_a > quasi_4115443774028964501um_a_a ).

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

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001tf__b_001tf__d,type,
    binary5767873073121707343bs_b_d: quasi_borel_b > quasi_borel_d > quasi_3692896893563015767um_b_d ).

thf(sy_c_Binary__CoProduct__QuasiBorel_Ocopair__qbs_001tf__c_001tf__c,type,
    binary2980417491149031309bs_c_c: quasi_borel_c > quasi_borel_c > quasi_2986213852911683093um_c_c ).

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

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

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001_Eo,type,
    borel_5500255247093592246orel_o: sigma_measure_o ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Extended____Nonnegative____Real__Oennreal,type,
    borel_6524799422816628122nnreal: sigma_7234349610311085201nnreal ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Nat__Onat,type,
    borel_8449730974584783410el_nat: sigma_measure_nat ).

thf(sy_c_Borel__Space_Otopological__space__class_Oborel_001t__Real__Oreal,type,
    borel_5078946678739801102l_real: sigma_measure_real ).

thf(sy_c_Convolution_Oconvolution_001t__Real__Oreal,type,
    convolution_real: sigma_measure_real > sigma_measure_real > sigma_measure_real ).

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

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

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

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

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

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

thf(sy_c_Countable__Set_Ocountable_001t__Set__Oset_I_Eo_J,type,
    counta9002483607034949683_set_o: set_set_o > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    counta2425349316461633011nnreal: set_se4580700918925141924nnreal > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Set__Oset_It__Nat__Onat_J,type,
    counta3299167949292459659et_nat: set_set_nat > $o ).

thf(sy_c_Countable__Set_Ocountable_001t__Set__Oset_It__Real__Oreal_J,type,
    counta8054315614235329383t_real: set_set_real > $o ).

thf(sy_c_Distribution__Functions_Ofinite__borel__measure__axioms,type,
    distri7440681460546723631axioms: sigma_measure_real > $o ).

thf(sy_c_Distribution__Functions_Oreal__distribution__axioms,type,
    distri5068091390715981392axioms: sigma_measure_real > $o ).

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

thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001_Eo,type,
    comp_o_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Real__Oreal,type,
    comp_o_o_real: ( $o > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    comp_o7356229531094944743nnreal: ( $o > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > $o ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_o_nat_nat: ( $o > nat ) > ( nat > $o ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Real__Oreal_001_Eo,type,
    comp_o_real_o: ( $o > real ) > ( $o > $o ) > $o > real ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_o_real_nat: ( $o > real ) > ( nat > $o ) > nat > real ).

thf(sy_c_Fun_Ocomp_001_Eo_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_o_real_real: ( $o > real ) > ( real > $o ) > real > real ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001_Eo_001_Eo,type,
    comp_E3606505270377598363al_o_o: ( extend8495563244428889912nnreal > $o ) > ( $o > extend8495563244428889912nnreal ) > $o > $o ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    comp_E7860224481218928525nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_E1259497758557687997at_nat: ( extend8495563244428889912nnreal > nat ) > ( nat > extend8495563244428889912nnreal ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    comp_E6146708109475903745nnreal: ( extend8495563244428889912nnreal > real ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > real ).

thf(sy_c_Fun_Ocomp_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_E3822617923592311797l_real: ( extend8495563244428889912nnreal > real ) > ( real > extend8495563244428889912nnreal ) > real > real ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001_Eo_001_Eo,type,
    comp_nat_o_o: ( nat > $o ) > ( $o > nat ) > $o > $o ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    comp_n4458900468863859749nnreal: ( nat > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > nat ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
    comp_nat_nat_real: ( nat > nat ) > ( real > nat ) > real > nat ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001_Eo,type,
    comp_nat_real_o: ( nat > real ) > ( $o > nat ) > $o > real ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).

thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_nat_real_real: ( nat > real ) > ( real > nat ) > real > real ).

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

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_Eo_001t__Real__Oreal,type,
    comp_real_o_real: ( real > $o ) > ( real > real ) > real > $o ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001_Eo_001t__Sum____Type__Osum_I_Eo_M_Eo_J,type,
    comp_r962287379166727918um_o_o: ( real > $o ) > ( sum_sum_o_o > real ) > sum_sum_o_o > $o ).

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

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

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Real__Oreal,type,
    comp_real_nat_real: ( real > nat ) > ( real > real ) > real > nat ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_r6376950801160420448at_nat: ( real > nat ) > ( sum_sum_nat_nat > real ) > sum_sum_nat_nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    comp_real_real_o: ( real > real ) > ( $o > real ) > $o > real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat,type,
    comp_real_real_nat: ( real > real ) > ( nat > real ) > nat > real ).

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

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_r5151246396438109300l_real: ( real > real ) > ( sum_sum_real_real > real ) > sum_sum_real_real > real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
    comp_real_real_a: ( real > real ) > ( a > real ) > a > real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001t__Real__Oreal_001tf__c,type,
    comp_real_real_c: ( real > real ) > ( c > real ) > c > real ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001_Eo,type,
    comp_real_a_o: ( real > a ) > ( $o > real ) > $o > a ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__a_001t__Real__Oreal,type,
    comp_real_a_real: ( real > a ) > ( real > real ) > real > a ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__b_001t__Real__Oreal,type,
    comp_real_b_real: ( real > b ) > ( real > real ) > real > b ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__c_001t__Real__Oreal,type,
    comp_real_c_real: ( real > c ) > ( real > real ) > real > c ).

thf(sy_c_Fun_Ocomp_001t__Real__Oreal_001tf__d_001t__Real__Oreal,type,
    comp_real_d_real: ( real > d ) > ( real > real ) > real > d ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_M_Eo_J_001t__Real__Oreal_001t__Sum____Type__Osum_It__Extended____Nonnegative____Real__Oennreal_M_Eo_J,type,
    comp_S8575781285974020201real_o: ( sum_sum_real_o > real ) > ( sum_su1359252823600007511real_o > sum_sum_real_o ) > sum_su1359252823600007511real_o > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Real__Oreal_001t__Sum____Type__Osum_I_Eo_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    comp_S3118471914838181037nnreal: ( sum_su3240232783174752707nnreal > real ) > ( sum_su7753341093581952129nnreal > sum_su3240232783174752707nnreal ) > sum_su7753341093581952129nnreal > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Real__Oreal_001t__Sum____Type__Osum_It__Extended____Nonnegative____Real__Oennreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    comp_S8072141257669888163nnreal: ( sum_su3240232783174752707nnreal > real ) > ( sum_su4415445757542774223nnreal > sum_su3240232783174752707nnreal ) > sum_su4415445757542774223nnreal > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J_001t__Real__Oreal_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    comp_S1133027055745533243nnreal: ( sum_su3240232783174752707nnreal > real ) > ( sum_su3730406437774119527nnreal > sum_su3240232783174752707nnreal ) > sum_su3730406437774119527nnreal > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Nat__Onat_J_001t__Real__Oreal_001t__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J,type,
    comp_S1246679031322203821_o_nat: ( sum_sum_real_nat > real ) > ( sum_sum_o_nat > sum_sum_real_nat ) > sum_sum_o_nat > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Nat__Onat_J_001t__Real__Oreal_001t__Sum____Type__Osum_It__Extended____Nonnegative____Real__Oennreal_Mt__Nat__Onat_J,type,
    comp_S3004035650380784083al_nat: ( sum_sum_real_nat > real ) > ( sum_su1883948583941721703al_nat > sum_sum_real_nat ) > sum_su1883948583941721703al_nat > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Nat__Onat_J_001t__Real__Oreal_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_S3112721647402360427at_nat: ( sum_sum_real_nat > real ) > ( sum_sum_nat_nat > sum_sum_real_nat ) > sum_sum_nat_nat > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001_Eo_001t__Sum____Type__Osum_I_Eo_M_Eo_J,type,
    comp_S7834891836943127203um_o_o: ( sum_sum_real_real > $o ) > ( sum_sum_o_o > sum_sum_real_real ) > sum_sum_o_o > $o ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    comp_S193255956842892907at_nat: ( sum_sum_real_real > nat ) > ( sum_sum_nat_nat > sum_sum_real_real ) > sum_sum_nat_nat > nat ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Sum____Type__Osum_I_Eo_Mt__Real__Oreal_J,type,
    comp_S1736399213409601069o_real: ( sum_sum_real_real > real ) > ( sum_sum_o_real > sum_sum_real_real ) > sum_sum_o_real > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Sum____Type__Osum_It__Extended____Nonnegative____Real__Oennreal_Mt__Real__Oreal_J,type,
    comp_S3462165939084609547l_real: ( sum_sum_real_real > real ) > ( sum_su3194684483830730051l_real > sum_sum_real_real ) > sum_su3194684483830730051l_real > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Real__Oreal_J,type,
    comp_S7774070043488426147t_real: ( sum_sum_real_real > real ) > ( sum_sum_nat_real > sum_sum_real_real ) > sum_sum_nat_real > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Real__Oreal_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_S1658032853064855039l_real: ( sum_sum_real_real > real ) > ( sum_sum_real_real > sum_sum_real_real ) > sum_sum_real_real > real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Sum____Type__Osum_I_Eo_M_Eo_J_001t__Sum____Type__Osum_I_Eo_M_Eo_J,type,
    comp_S7546904374943229414um_o_o: ( sum_sum_real_real > sum_sum_o_o ) > ( sum_sum_o_o > sum_sum_real_real ) > sum_sum_o_o > sum_sum_o_o ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J_001t__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J,type,
    comp_S4704009564200261978_o_nat: ( sum_sum_real_real > sum_sum_o_nat ) > ( sum_sum_o_nat > sum_sum_real_real ) > sum_sum_o_nat > sum_sum_o_nat ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Sum____Type__Osum_I_Eo_Mt__Real__Oreal_J_001t__Sum____Type__Osum_I_Eo_Mt__Real__Oreal_J,type,
    comp_S4972280585966916698o_real: ( sum_sum_real_real > sum_sum_o_real ) > ( sum_sum_o_real > sum_sum_real_real ) > sum_sum_o_real > sum_sum_o_real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Sum____Type__Osum_It__Real__Oreal_M_Eo_J_001t__Sum____Type__Osum_It__Real__Oreal_M_Eo_J,type,
    comp_S5772965672812085466real_o: ( sum_sum_real_real > sum_sum_real_o ) > ( sum_sum_real_o > sum_sum_real_real ) > sum_sum_real_o > sum_sum_real_o ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Nat__Onat_J_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Nat__Onat_J,type,
    comp_S1631106420198528594al_nat: ( sum_sum_real_real > sum_sum_real_nat ) > ( sum_sum_real_nat > sum_sum_real_real ) > sum_sum_real_nat > sum_sum_real_nat ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_S4948336051826200842l_real: ( sum_sum_real_real > sum_sum_real_real ) > ( sum_sum_real_real > sum_sum_real_real ) > sum_sum_real_real > sum_sum_real_real ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mtf__a_J_001t__Sum____Type__Osum_I_Eo_Mtf__b_J_001t__Sum____Type__Osum_I_Eo_Mt__Real__Oreal_J,type,
    comp_S1583458553065613633o_real: ( sum_sum_real_a > sum_sum_o_b ) > ( sum_sum_o_real > sum_sum_real_a ) > sum_sum_o_real > sum_sum_o_b ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mtf__a_J_001t__Sum____Type__Osum_It__Real__Oreal_Mtf__b_J_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_S5849561226221799003l_real: ( sum_sum_real_a > sum_sum_real_b ) > ( sum_sum_real_real > sum_sum_real_a ) > sum_sum_real_real > sum_sum_real_b ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mtf__c_J_001t__Sum____Type__Osum_I_Eo_Mtf__d_J_001t__Sum____Type__Osum_I_Eo_Mt__Real__Oreal_J,type,
    comp_S8369131124857326909o_real: ( sum_sum_real_c > sum_sum_o_d ) > ( sum_sum_o_real > sum_sum_real_c ) > sum_sum_o_real > sum_sum_o_d ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_It__Real__Oreal_Mtf__c_J_001t__Sum____Type__Osum_It__Real__Oreal_Mtf__d_J_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_S8379285214102871771l_real: ( sum_sum_real_c > sum_sum_real_d ) > ( sum_sum_real_real > sum_sum_real_c ) > sum_sum_real_real > sum_sum_real_d ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__a_Mtf__a_J_001tf__b_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_S8357179028215158318l_real: ( sum_sum_a_a > b ) > ( sum_sum_real_real > sum_sum_a_a ) > sum_sum_real_real > b ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001tf__a,type,
    comp_S7118176607350915467_a_c_a: ( sum_sum_a_c > sum_sum_a_c ) > ( a > sum_sum_a_c ) > a > sum_sum_a_c ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001tf__c,type,
    comp_S7118176607350915469_a_c_c: ( sum_sum_a_c > sum_sum_a_c ) > ( c > sum_sum_a_c ) > c > sum_sum_a_c ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001t__Real__Oreal,type,
    comp_S4176151853342328607d_real: ( sum_sum_a_c > sum_sum_b_d ) > ( real > sum_sum_a_c ) > real > sum_sum_b_d ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001tf__a,type,
    comp_S2855374185079283659_b_d_a: ( sum_sum_a_c > sum_sum_b_d ) > ( a > sum_sum_a_c ) > a > sum_sum_b_d ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001tf__c,type,
    comp_S2855374185079283661_b_d_c: ( sum_sum_a_c > sum_sum_b_d ) > ( c > sum_sum_a_c ) > c > sum_sum_b_d ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001tf__b,type,
    comp_S1474653431308885964_a_c_b: ( sum_sum_b_d > sum_sum_a_c ) > ( b > sum_sum_b_d ) > b > sum_sum_a_c ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001tf__d,type,
    comp_S1474653431308885966_a_c_d: ( sum_sum_b_d > sum_sum_a_c ) > ( d > sum_sum_b_d ) > d > sum_sum_a_c ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001tf__b,type,
    comp_S6435223045892029964_b_d_b: ( sum_sum_b_d > sum_sum_b_d ) > ( b > sum_sum_b_d ) > b > sum_sum_b_d ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001tf__d,type,
    comp_S6435223045892029966_b_d_d: ( sum_sum_b_d > sum_sum_b_d ) > ( d > sum_sum_b_d ) > d > sum_sum_b_d ).

thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__c_Mtf__c_J_001tf__d_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_S5966200288902075824l_real: ( sum_sum_c_c > d ) > ( sum_sum_real_real > sum_sum_c_c ) > sum_sum_real_real > d ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_a_real_real: ( a > real ) > ( real > a ) > real > real ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001tf__a,type,
    comp_a_Sum_sum_a_c_a: ( a > sum_sum_a_c ) > ( a > a ) > a > sum_sum_a_c ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001tf__b,type,
    comp_a_Sum_sum_a_c_b: ( a > sum_sum_a_c ) > ( b > a ) > b > sum_sum_a_c ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Real__Oreal,type,
    comp_a_a_real: ( a > a ) > ( real > a ) > real > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001t__Real__Oreal,type,
    comp_a_b_real: ( a > b ) > ( real > a ) > real > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_a5455185540716242459l_real: ( a > b ) > ( sum_sum_real_real > a ) > sum_sum_real_real > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__a,type,
    comp_a_b_a: ( a > b ) > ( a > a ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__c,type,
    comp_a_b_c: ( a > b ) > ( c > a ) > c > b ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001t__Real__Oreal,type,
    comp_a_c_real: ( a > c ) > ( real > a ) > real > c ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__d_001t__Real__Oreal,type,
    comp_a_d_real: ( a > d ) > ( real > a ) > real > d ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_b_real_real: ( b > real ) > ( real > b ) > real > real ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Real__Oreal_001tf__a,type,
    comp_b_real_a: ( b > real ) > ( a > b ) > a > real ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001tf__a,type,
    comp_b_Sum_sum_b_d_a: ( b > sum_sum_b_d ) > ( a > b ) > a > sum_sum_b_d ).

thf(sy_c_Fun_Ocomp_001tf__b_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001tf__b,type,
    comp_b_Sum_sum_b_d_b: ( b > sum_sum_b_d ) > ( b > b ) > b > sum_sum_b_d ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001t__Real__Oreal,type,
    comp_b_b_real: ( b > b ) > ( real > b ) > real > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__a,type,
    comp_b_b_a: ( b > b ) > ( a > b ) > a > b ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__d_001t__Real__Oreal,type,
    comp_b_d_real: ( b > d ) > ( real > b ) > real > d ).

thf(sy_c_Fun_Ocomp_001tf__b_001tf__d_001tf__a,type,
    comp_b_d_a: ( b > d ) > ( a > b ) > a > d ).

thf(sy_c_Fun_Ocomp_001tf__c_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_c_real_real: ( c > real ) > ( real > c ) > real > real ).

thf(sy_c_Fun_Ocomp_001tf__c_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001tf__c,type,
    comp_c_Sum_sum_a_c_c: ( c > sum_sum_a_c ) > ( c > c ) > c > sum_sum_a_c ).

thf(sy_c_Fun_Ocomp_001tf__c_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_001tf__d,type,
    comp_c_Sum_sum_a_c_d: ( c > sum_sum_a_c ) > ( d > c ) > d > sum_sum_a_c ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001t__Real__Oreal,type,
    comp_c_a_real: ( c > a ) > ( real > c ) > real > a ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001t__Real__Oreal,type,
    comp_c_b_real: ( c > b ) > ( real > c ) > real > b ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001t__Real__Oreal,type,
    comp_c_c_real: ( c > c ) > ( real > c ) > real > c ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__d_001t__Real__Oreal,type,
    comp_c_d_real: ( c > d ) > ( real > c ) > real > d ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__d_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J,type,
    comp_c8591738523173874331l_real: ( c > d ) > ( sum_sum_real_real > c ) > sum_sum_real_real > d ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__d_001tf__a,type,
    comp_c_d_a: ( c > d ) > ( a > c ) > a > d ).

thf(sy_c_Fun_Ocomp_001tf__c_001tf__d_001tf__c,type,
    comp_c_d_c: ( c > d ) > ( c > c ) > c > d ).

thf(sy_c_Fun_Ocomp_001tf__d_001t__Real__Oreal_001t__Real__Oreal,type,
    comp_d_real_real: ( d > real ) > ( real > d ) > real > real ).

thf(sy_c_Fun_Ocomp_001tf__d_001t__Real__Oreal_001tf__c,type,
    comp_d_real_c: ( d > real ) > ( c > d ) > c > real ).

thf(sy_c_Fun_Ocomp_001tf__d_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001tf__c,type,
    comp_d_Sum_sum_b_d_c: ( d > sum_sum_b_d ) > ( c > d ) > c > sum_sum_b_d ).

thf(sy_c_Fun_Ocomp_001tf__d_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_001tf__d,type,
    comp_d_Sum_sum_b_d_d: ( d > sum_sum_b_d ) > ( d > d ) > d > sum_sum_b_d ).

thf(sy_c_Fun_Ocomp_001tf__d_001tf__b_001t__Real__Oreal,type,
    comp_d_b_real: ( d > b ) > ( real > d ) > real > b ).

thf(sy_c_Fun_Ocomp_001tf__d_001tf__b_001tf__c,type,
    comp_d_b_c: ( d > b ) > ( c > d ) > c > b ).

thf(sy_c_Fun_Ocomp_001tf__d_001tf__d_001t__Real__Oreal,type,
    comp_d_d_real: ( d > d ) > ( real > d ) > real > d ).

thf(sy_c_Fun_Ocomp_001tf__d_001tf__d_001tf__c,type,
    comp_d_d_c: ( d > d ) > ( c > d ) > c > d ).

thf(sy_c_Fun_Oid_001_062_I_Eo_M_Eo_J,type,
    id_o_o: ( $o > $o ) > $o > $o ).

thf(sy_c_Fun_Oid_001_062_I_Eo_Mt__Real__Oreal_J,type,
    id_o_real: ( $o > real ) > $o > real ).

thf(sy_c_Fun_Oid_001_062_It__Extended____Nonnegative____Real__Oennreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    id_Ext6301196394018042846nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Oid_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    id_nat_nat: ( nat > nat ) > nat > nat ).

thf(sy_c_Fun_Oid_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
    id_real_real: ( real > real ) > real > real ).

thf(sy_c_Fun_Oid_001_062_It__Real__Oreal_Mtf__a_J,type,
    id_real_a: ( real > a ) > real > a ).

thf(sy_c_Fun_Oid_001_Eo,type,
    id_o: $o > $o ).

thf(sy_c_Fun_Oid_001t__Extended____Nonnegative____Real__Oennreal,type,
    id_Ext8331313139072774535nnreal: extend8495563244428889912nnreal > extend8495563244428889912nnreal ).

thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
    id_nat: nat > nat ).

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

thf(sy_c_Fun_Oid_001t__Set__Oset_I_Eo_J,type,
    id_set_o: set_o > set_o ).

thf(sy_c_Fun_Oid_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(sy_c_Fun_Oid_001tf__d,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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thf(sy_c_Lebesgue__Measure_Olborel_001t__Real__Oreal,type,
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thf(sy_c_Measure__Space_OSup__measure_H_001_Eo,type,
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thf(sy_c_Measure__Space_OSup__measure_H_001t__Real__Oreal,type,
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thf(sy_c_Measure__Space_Osup__measure_H_001_Eo,type,
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thf(sy_c_Measure__Space_Osup__measure_H_001t__Extended____Nonnegative____Real__Oennreal,type,
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thf(sy_c_Measure__Space_Osup__measure_H_001t__Nat__Onat,type,
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thf(sy_c_Measure__Space_Osup__measure_H_001t__Real__Oreal,type,
    measur2147279183506585690e_real: sigma_measure_real > sigma_measure_real > sigma_measure_real ).

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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Real__Oreal_J_M_Eo_J,type,
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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__Real__Oreal_M_Eo_J_J,type,
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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,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
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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_062_It__Real__Oreal_Mtf__c_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__d_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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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_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__Extended____Nonnegative____Real__Oennreal_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sigma____Algebra__Omeasure_I_Eo_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sigma____Algebra__Omeasure_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sigma____Algebra__Omeasure_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sigma____Algebra__Omeasure_It__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
    bot_bot_set_a: set_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__b_J,type,
    bot_bot_set_b: set_b ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__c_J,type,
    bot_bot_set_c: set_c ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__d_J,type,
    bot_bot_set_d: set_d ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_I_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nonnegative____Real__Oennreal,type,
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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__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
    ord_le5743406823621094409real_a: set_real_a > set_real_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
    ord_less_eq_set_a_b: set_a_b > set_a_b > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__c_Mtf__d_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    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,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_I_Eo_Mt__Real__Oreal_J_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_I_062_It__Real__Oreal_Mtf__a_J_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Extended____Nonnegative____Real__Oennreal_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_I_Eo_J_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Set__Oset_It__Real__Oreal_J_M_Eo_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    top_to7994903218803871134nnreal: set_Ex3793607809372303086nnreal ).

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_I_Eo_J_J,type,
    top_top_set_set_o: set_set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
    top_to3356475028079756884nnreal: set_se4580700918925141924nnreal ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    top_top_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Real__Oreal_J_J,type,
    top_top_set_set_real: set_set_real ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001_Eo,type,
    empty_quasi_borel_o: quasi_borel_o ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001t__Extended____Nonnegative____Real__Oennreal,type,
    empty_1788085430566700506nnreal: quasi_9015997321629101608nnreal ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001t__Nat__Onat,type,
    empty_8278123436611590770el_nat: quasi_borel_nat ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001t__Real__Oreal,type,
    empty_1876425439295802446l_real: quasi_borel_real ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001tf__a,type,
    empty_quasi_borel_a: quasi_borel_a ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001tf__b,type,
    empty_quasi_borel_b: quasi_borel_b ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001tf__c,type,
    empty_quasi_borel_c: quasi_borel_c ).

thf(sy_c_QuasiBorel_Oempty__quasi__borel_001tf__d,type,
    empty_quasi_borel_d: quasi_borel_d ).

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

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

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

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

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

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

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Set__Oset_I_Eo_J,type,
    qbs_Mx_set_o: quasi_borel_set_o > set_real_set_o ).

thf(sy_c_QuasiBorel_Oqbs__Mx_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    qbs_Mx3319822389276443575nnreal: quasi_953260806197706462nnreal > set_re634636480907793903nnreal ).

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_001t__Set__Oset_It__Real__Oreal_J,type,
    qbs_Mx_set_real: quasi_borel_set_real > set_real_set_real ).

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

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

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

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

thf(sy_c_QuasiBorel_Oqbs__Mx_001tf__d,type,
    qbs_Mx_d: quasi_borel_d > set_real_d ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_Eo_001_Eo,type,
    qbs_morphism_o_o: quasi_borel_o > quasi_borel_o > set_o_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001_Eo_001t__Nat__Onat,type,
    qbs_morphism_o_nat: quasi_borel_o > quasi_borel_nat > set_o_nat ).

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

thf(sy_c_QuasiBorel_Oqbs__morphism_001_Eo_001tf__a,type,
    qbs_morphism_o_a: quasi_borel_o > quasi_borel_a > set_o_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    qbs_mo660571752308592106nnreal: quasi_9015997321629101608nnreal > quasi_9015997321629101608nnreal > set_Ex7514979451064110021nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Nat__Onat_001_Eo,type,
    qbs_morphism_nat_o: quasi_borel_nat > quasi_borel_o > set_nat_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Nat__Onat_001t__Nat__Onat,type,
    qbs_morphism_nat_nat: quasi_borel_nat > quasi_borel_nat > set_nat_nat ).

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

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

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

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

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
    qbs_mo4711702798760812422et_nat: quasi_borel_set_o > quasi_borel_set_nat > set_set_o_set_nat ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J,type,
    qbs_mo5249294451373078392_set_o: quasi_borel_set_nat > quasi_borel_set_o > set_set_nat_set_o ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    qbs_mo3729789103443073518nnreal: quasi_borel_set_nat > quasi_953260806197706462nnreal > set_se5513212049808698057nnreal ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    qbs_mo4089447833561950854et_nat: quasi_borel_set_nat > quasi_borel_set_nat > set_set_nat_set_nat ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Real__Oreal_J,type,
    qbs_mo34581953175210466t_real: quasi_borel_set_nat > quasi_borel_set_real > set_set_nat_set_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Sum____Type__Osum_I_Eo_M_Eo_J_001t__Real__Oreal,type,
    qbs_mo4523485202552518603o_real: quasi_9151289748890374549um_o_o > quasi_borel_real > set_Sum_sum_o_o_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J_001t__Real__Oreal,type,
    qbs_mo7285195558795298381t_real: quasi_6247536490924941631_o_nat > quasi_borel_real > set_Su4870880633484326550t_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Sum____Type__Osum_It__Nat__Onat_M_Eo_J_001t__Real__Oreal,type,
    qbs_mo5015923643019482355o_real: quasi_7295139613103245285_nat_o > quasi_borel_real > set_Su8975513444386755132o_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Real__Oreal,type,
    qbs_mo3635929294422082341t_real: quasi_2537881152722034415at_nat > quasi_borel_real > set_Su1304851494220063104t_real ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_001tf__a,type,
    qbs_mo4449801170876743949real_a: quasi_3918057494074722855l_real > quasi_borel_a > set_Su256150577512953374real_a ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Sum____Type__Osum_Itf__a_Mtf__a_J_001tf__b,type,
    qbs_mo1966825068709392160_a_a_b: quasi_4115443774028964501um_a_a > quasi_borel_b > set_Sum_sum_a_a_b ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001t__Sum____Type__Osum_Itf__c_Mtf__c_J_001tf__d,type,
    qbs_mo2664592261020904354_c_c_d: quasi_2986213852911683093um_c_c > quasi_borel_d > set_Sum_sum_c_c_d ).

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

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001tf__a,type,
    qbs_morphism_a_a: quasi_borel_a > quasi_borel_a > set_a_a ).

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

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001tf__c,type,
    qbs_morphism_a_c: quasi_borel_a > quasi_borel_c > set_a_c ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__a_001tf__d,type,
    qbs_morphism_a_d: quasi_borel_a > quasi_borel_d > set_a_d ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__b_001t__Sum____Type__Osum_Itf__b_Mtf__d_J,type,
    qbs_mo3380411100751004654um_b_d: quasi_borel_b > quasi_3692896893563015767um_b_d > set_b_Sum_sum_b_d ).

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

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__c_001tf__a,type,
    qbs_morphism_c_a: quasi_borel_c > quasi_borel_a > set_c_a ).

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

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__c_001tf__c,type,
    qbs_morphism_c_c: quasi_borel_c > quasi_borel_c > set_c_c ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__c_001tf__d,type,
    qbs_morphism_c_d: quasi_borel_c > quasi_borel_d > set_c_d ).

thf(sy_c_QuasiBorel_Oqbs__morphism_001tf__d_001t__Sum____Type__Osum_Itf__b_Mtf__d_J,type,
    qbs_mo1214661810789969904um_b_d: quasi_borel_d > quasi_3692896893563015767um_b_d > set_d_Sum_sum_b_d ).

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

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

thf(sy_c_QuasiBorel_Oqbs__space_001_Eo,type,
    qbs_space_o: quasi_borel_o > set_o ).

thf(sy_c_QuasiBorel_Oqbs__space_001t__Extended____Nonnegative____Real__Oennreal,type,
    qbs_sp175953267596557954nnreal: quasi_9015997321629101608nnreal > set_Ex3793607809372303086nnreal ).

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

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

thf(sy_c_QuasiBorel_Oqbs__space_001t__Set__Oset_I_Eo_J,type,
    qbs_space_set_o: quasi_borel_set_o > set_set_o ).

thf(sy_c_QuasiBorel_Oqbs__space_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    qbs_sp6328763913151411768nnreal: quasi_953260806197706462nnreal > set_se4580700918925141924nnreal ).

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_001t__Set__Oset_It__Real__Oreal_J,type,
    qbs_space_set_real: quasi_borel_set_real > set_set_real ).

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

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

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

thf(sy_c_QuasiBorel_Oqbs__space_001tf__d,type,
    qbs_space_d: quasi_borel_d > set_d ).

thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist_001_Eo_001_Eo,type,
    iso_tu5524162117909064517st_o_o: ( ( $o > $o ) > $o > $o ) > ( $o > $o ) > $o > ( $o > $o ) > $o > $o > $o ).

thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    iso_tu1500430003656523831nnreal: ( ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > extend8495563244428889912nnreal > extend8495563244428889912nnreal > $o ).

thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist_001t__Nat__Onat_001t__Nat__Onat,type,
    iso_tu3079242893353865575at_nat: ( ( nat > nat ) > nat > nat ) > ( nat > nat ) > nat > ( nat > nat ) > nat > nat > $o ).

thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist_001t__Real__Oreal_001t__Real__Oreal,type,
    iso_tu1764886537532566175l_real: ( ( real > real ) > real > real ) > ( real > real ) > real > ( real > real ) > real > real > $o ).

thf(sy_c_Set_OCollect_001_062_I_Eo_Mt__Real__Oreal_J,type,
    collect_o_real: ( ( $o > real ) > $o ) > set_o_real ).

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_Mtf__a_J,type,
    collect_real_a: ( ( real > a ) > $o ) > set_real_a ).

thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__b_J,type,
    collect_a_b: ( ( a > b ) > $o ) > set_a_b ).

thf(sy_c_Set_OCollect_001_062_Itf__c_Mtf__d_J,type,
    collect_c_d: ( ( c > d ) > $o ) > set_c_d ).

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

thf(sy_c_Set_OCollect_001t__Extended____Nonnegative____Real__Oennreal,type,
    collec6648975593938027277nnreal: ( extend8495563244428889912nnreal > $o ) > set_Ex3793607809372303086nnreal ).

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

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

thf(sy_c_Set_OCollect_001t__Set__Oset_I_Eo_J,type,
    collect_set_o: ( set_o > $o ) > set_set_o ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    collec4858231573021281987nnreal: ( set_Ex3793607809372303086nnreal > $o ) > set_se4580700918925141924nnreal ).

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

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Real__Oreal_J,type,
    collect_set_real: ( set_real > $o ) > set_set_real ).

thf(sy_c_Set_OPow_001_062_I_Eo_Mt__Real__Oreal_J,type,
    pow_o_real: set_o_real > set_set_o_real ).

thf(sy_c_Set_OPow_001_062_It__Real__Oreal_Mtf__a_J,type,
    pow_real_a: set_real_a > set_set_real_a ).

thf(sy_c_Set_OPow_001_Eo,type,
    pow_o: set_o > set_set_o ).

thf(sy_c_Set_OPow_001t__Extended____Nonnegative____Real__Oennreal,type,
    pow_Ex5372160365422184283nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal ).

thf(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

thf(sy_c_Set_OPow_001t__Real__Oreal,type,
    pow_real: set_real > set_set_real ).

thf(sy_c_Set_OPow_001t__Set__Oset_I_Eo_J,type,
    pow_set_o: set_set_o > set_set_set_o ).

thf(sy_c_Set_OPow_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    pow_se7371645914972035857nnreal: set_se4580700918925141924nnreal > set_se8256708918794385754nnreal ).

thf(sy_c_Set_OPow_001t__Set__Oset_It__Nat__Onat_J,type,
    pow_set_nat: set_set_nat > set_set_set_nat ).

thf(sy_c_Set_OPow_001t__Set__Oset_It__Real__Oreal_J,type,
    pow_set_real: set_set_real > set_set_set_real ).

thf(sy_c_Set_Oinsert_001_062_I_Eo_Mt__Real__Oreal_J,type,
    insert_o_real: ( $o > real ) > set_o_real > set_o_real ).

thf(sy_c_Set_Oinsert_001_062_It__Real__Oreal_Mtf__a_J,type,
    insert_real_a: ( real > a ) > set_real_a > set_real_a ).

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

thf(sy_c_Set_Oinsert_001t__Extended____Nonnegative____Real__Oennreal,type,
    insert7407984058720857448nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
    insert_set_o: set_o > set_set_o > set_set_o ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    insert1343806209672318238nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > set_se4580700918925141924nnreal ).

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__a,type,
    insert_a: a > set_a > set_a ).

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

thf(sy_c_Set_Oinsert_001tf__c,type,
    insert_c: c > set_c > set_c ).

thf(sy_c_Set_Oinsert_001tf__d,type,
    insert_d: d > set_d > set_d ).

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

thf(sy_c_Set_Ois__empty_001t__Extended____Nonnegative____Real__Oennreal,type,
    is_emp182806100662350310nnreal: set_Ex3793607809372303086nnreal > $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_I_Eo_Mt__Real__Oreal_J,type,
    is_singleton_o_real: set_o_real > $o ).

thf(sy_c_Set_Ois__singleton_001_062_It__Real__Oreal_Mtf__a_J,type,
    is_singleton_real_a: set_real_a > $o ).

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

thf(sy_c_Set_Ois__singleton_001t__Extended____Nonnegative____Real__Oennreal,type,
    is_sin3654761921782142788nnreal: set_Ex3793607809372303086nnreal > $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_I_Eo_J,type,
    is_singleton_set_o: set_set_o > $o ).

thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    is_sin9058363718368806650nnreal: set_se4580700918925141924nnreal > $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_001t__Set__Oset_It__Real__Oreal_J,type,
    is_sin3548895728136638702t_real: set_set_real > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Sigma__Algebra_Omeasurable_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal,type,
    sigma_7049758200512112822l_real: sigma_7234349610311085201nnreal > sigma_measure_real > set_Ex5658717452565810105l_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__Real__Oreal_001_Eo,type,
    sigma_3939073009482781210real_o: sigma_measure_real > sigma_measure_o > set_real_o ).

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

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

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

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

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

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

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

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

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

thf(sy_c_Sigma__Algebra_Osets_001_Eo,type,
    sigma_sets_o: sigma_measure_o > set_set_o ).

thf(sy_c_Sigma__Algebra_Osets_001t__Extended____Nonnegative____Real__Oennreal,type,
    sigma_5465916536984168985nnreal: sigma_7234349610311085201nnreal > set_se4580700918925141924nnreal ).

thf(sy_c_Sigma__Algebra_Osets_001t__Nat__Onat,type,
    sigma_sets_nat: sigma_measure_nat > set_set_nat ).

thf(sy_c_Sigma__Algebra_Osets_001t__Real__Oreal,type,
    sigma_sets_real: sigma_measure_real > set_set_real ).

thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_I_Eo_J,type,
    sigma_sets_set_o: sigma_measure_set_o > set_set_set_o ).

thf(sy_c_Sigma__Algebra_Osets_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    sigma_5308793447563920847nnreal: sigma_523634232904505671nnreal > set_se8256708918794385754nnreal ).

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_001t__Set__Oset_It__Real__Oreal_J,type,
    sigma_sets_set_real: sigma_3733394171116455995t_real > set_set_set_real ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_I_Eo_J,type,
    sigma_space_set_o: sigma_measure_set_o > set_set_o ).

thf(sy_c_Sigma__Algebra_Ospace_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    sigma_2539764534872131430nnreal: sigma_523634232904505671nnreal > set_se4580700918925141924nnreal ).

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_001t__Set__Oset_It__Real__Oreal_J,type,
    sigma_space_set_real: sigma_3733394171116455995t_real > set_set_real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Sum__Type_OInl_001tf__b_001tf__d,type,
    sum_Inl_b_d: b > sum_sum_b_d ).

thf(sy_c_Sum__Type_OInr_001tf__c_001tf__a,type,
    sum_Inr_c_a: c > sum_sum_a_c ).

thf(sy_c_Sum__Type_OInr_001tf__d_001tf__b,type,
    sum_Inr_d_b: d > sum_sum_b_d ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001_Eo_001_Eo_001_Eo,type,
    sum_map_sum_o_o_o_o: ( $o > $o ) > ( $o > $o ) > sum_sum_o_o > sum_sum_o_o ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001_Eo_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ma344318232541114282at_nat: ( $o > $o ) > ( nat > nat ) > sum_sum_o_nat > sum_sum_o_nat ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001_Eo_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ma4692674304805643746l_real: ( $o > $o ) > ( real > real ) > sum_sum_o_real > sum_sum_o_real ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001_Eo_001t__Real__Oreal_001tf__b,type,
    sum_ma2721287238701537865real_b: ( $o > $o ) > ( real > b ) > sum_sum_o_real > sum_sum_o_b ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001_Eo_001t__Real__Oreal_001tf__d,type,
    sum_ma2721287238701537867real_d: ( $o > $o ) > ( real > d ) > sum_sum_o_real > sum_sum_o_d ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001t__Real__Oreal_001_Eo_001t__Real__Oreal,type,
    sum_ma2390433805885911944o_real: ( $o > real ) > ( $o > real ) > sum_sum_o_o > sum_sum_real_real ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ma3146212978127159086nnreal: ( $o > real ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > sum_su7753341093581952129nnreal > sum_su3240232783174752707nnreal ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ma3983538065750851678at_nat: ( $o > real ) > ( nat > nat ) > sum_sum_o_nat > sum_sum_real_nat ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001t__Real__Oreal_001t__Nat__Onat_001t__Real__Oreal,type,
    sum_ma9203121963192613946t_real: ( $o > real ) > ( nat > real ) > sum_sum_o_nat > sum_sum_real_real ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ma7581390967160423062l_real: ( $o > real ) > ( real > real ) > sum_sum_o_real > sum_sum_real_real ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
    sum_ma6566039659836893332real_a: ( $o > real ) > ( real > a ) > sum_sum_o_real > sum_sum_real_a ).

thf(sy_c_Sum__Type_Omap__sum_001_Eo_001t__Real__Oreal_001t__Real__Oreal_001tf__c,type,
    sum_ma6566039659836893334real_c: ( $o > real ) > ( real > c ) > sum_sum_o_real > sum_sum_real_c ).

thf(sy_c_Sum__Type_Omap__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001_Eo_001_Eo,type,
    sum_ma5762510830048976762al_o_o: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( $o > $o ) > sum_su1359252823600007511real_o > sum_su1359252823600007511real_o ).

thf(sy_c_Sum__Type_Omap__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ma2548858965165278700nnreal: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > sum_su4415445757542774223nnreal > sum_su4415445757542774223nnreal ).

thf(sy_c_Sum__Type_Omap__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ma840650751729215260at_nat: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( nat > nat ) > sum_su1883948583941721703al_nat > sum_su1883948583941721703al_nat ).

thf(sy_c_Sum__Type_Omap__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ma5613765533332990804l_real: ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > ( real > real ) > sum_su3194684483830730051l_real > sum_su3194684483830730051l_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001_Eo_001_Eo,type,
    sum_ma4122737713070071790al_o_o: ( extend8495563244428889912nnreal > real ) > ( $o > $o ) > sum_su1359252823600007511real_o > sum_sum_real_o ).

thf(sy_c_Sum__Type_Omap__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ma7572881292572212320nnreal: ( extend8495563244428889912nnreal > real ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > sum_su4415445757542774223nnreal > sum_su3240232783174752707nnreal ).

thf(sy_c_Sum__Type_Omap__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ma2425729402563468688at_nat: ( extend8495563244428889912nnreal > real ) > ( nat > nat ) > sum_su1883948583941721703al_nat > sum_sum_real_nat ).

thf(sy_c_Sum__Type_Omap__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ma436539099328933320l_real: ( extend8495563244428889912nnreal > real ) > ( real > real ) > sum_su3194684483830730051l_real > sum_sum_real_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ma5444172813184304924nnreal: ( nat > nat ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > sum_su3730406437774119527nnreal > sum_su3730406437774119527nnreal ).

thf(sy_c_Sum__Type_Omap__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ma7952235013896551556l_real: ( nat > nat ) > ( real > real ) > sum_sum_nat_real > sum_sum_nat_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Nat__Onat_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ma2610832426335247608nnreal: ( nat > real ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > sum_su3730406437774119527nnreal > sum_su3240232783174752707nnreal ).

thf(sy_c_Sum__Type_Omap__sum_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ma2704325357117275688at_nat: ( nat > real ) > ( nat > nat ) > sum_sum_nat_nat > sum_sum_real_nat ).

thf(sy_c_Sum__Type_Omap__sum_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat_001t__Real__Oreal,type,
    sum_ma3776711750342522116t_real: ( nat > real ) > ( nat > real ) > sum_sum_nat_nat > sum_sum_real_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ma1321509161104881248l_real: ( nat > real ) > ( real > real ) > sum_sum_nat_real > sum_sum_real_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001_Eo_001t__Real__Oreal_001_Eo,type,
    sum_ma8868230608733716744real_o: ( real > $o ) > ( real > $o ) > sum_sum_real_real > sum_sum_o_o ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001_Eo_001t__Real__Oreal_001t__Nat__Onat,type,
    sum_ma3912033132148697696al_nat: ( real > $o ) > ( real > nat ) > sum_sum_real_real > sum_sum_o_nat ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001_Eo_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ma5402501128500593980l_real: ( real > $o ) > ( real > real ) > sum_sum_real_real > sum_sum_o_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001_Eo_001tf__a_001tf__b,type,
    sum_ma5244931126318244963_o_a_b: ( real > $o ) > ( a > b ) > sum_sum_real_a > sum_sum_o_b ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001_Eo_001tf__c_001tf__d,type,
    sum_ma8893391999227668707_o_c_d: ( real > $o ) > ( c > d ) > sum_sum_real_c > sum_sum_o_d ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001_Eo_001_Eo,type,
    sum_ma5398380231479159522al_o_o: ( real > real ) > ( $o > $o ) > sum_sum_real_o > sum_sum_real_o ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001_Eo_001t__Real__Oreal,type,
    sum_ma3100260629580862178o_real: ( real > real ) > ( $o > real ) > sum_sum_real_o > sum_sum_real_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ma4707617466468859220nnreal: ( real > real ) > ( extend8495563244428889912nnreal > extend8495563244428889912nnreal ) > sum_su3240232783174752707nnreal > sum_su3240232783174752707nnreal ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ma5138984332203957892at_nat: ( real > real ) > ( nat > nat ) > sum_sum_real_nat > sum_sum_real_nat ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat_001t__Real__Oreal,type,
    sum_ma4051207219129444192t_real: ( real > real ) > ( nat > real ) > sum_sum_real_nat > sum_sum_real_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    sum_ma4609325004622031752real_o: ( real > real ) > ( real > $o ) > sum_sum_real_real > sum_sum_real_o ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat,type,
    sum_ma8618515792258342880al_nat: ( real > real ) > ( real > nat ) > sum_sum_real_real > sum_sum_real_nat ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ma9028575376852974268l_real: ( real > real ) > ( real > real ) > sum_sum_real_real > sum_sum_real_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal_001tf__a,type,
    sum_ma1303030344635120366real_a: ( real > real ) > ( real > a ) > sum_sum_real_real > sum_sum_real_a ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal_001tf__b,type,
    sum_ma1303030344635120367real_b: ( real > real ) > ( real > b ) > sum_sum_real_real > sum_sum_real_b ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal_001tf__c,type,
    sum_ma1303030344635120368real_c: ( real > real ) > ( real > c ) > sum_sum_real_real > sum_sum_real_c ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal_001tf__d,type,
    sum_ma1303030344635120369real_d: ( real > real ) > ( real > d ) > sum_sum_real_real > sum_sum_real_d ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001tf__a_001tf__b,type,
    sum_ma1807311858354067939al_a_b: ( real > real ) > ( a > b ) > sum_sum_real_a > sum_sum_real_b ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001t__Real__Oreal_001tf__c_001tf__d,type,
    sum_ma5455772731263491683al_c_d: ( real > real ) > ( c > d ) > sum_sum_real_c > sum_sum_real_d ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001tf__a_001t__Real__Oreal_001tf__a,type,
    sum_ma6640897443653631624real_a: ( real > a ) > ( real > a ) > sum_sum_real_real > sum_sum_a_a ).

thf(sy_c_Sum__Type_Omap__sum_001t__Real__Oreal_001tf__c_001t__Real__Oreal_001tf__c,type,
    sum_ma393449481388452872real_c: ( real > c ) > ( real > c ) > sum_sum_real_real > sum_sum_c_c ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    sum_ma5536731787355361032_set_o: ( set_o > set_o ) > ( set_o > set_o ) > sum_sum_set_o_set_o > sum_sum_set_o_set_o ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    sum_ma1877970594666251750nnreal: ( set_o > set_o ) > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ) > sum_su4962168936726258711nnreal > sum_su4962168936726258711nnreal ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    sum_ma5195006407206924054et_nat: ( set_o > set_o ) > ( set_nat > set_nat ) > sum_su7609443801303718063et_nat > sum_su7609443801303718063et_nat ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
    sum_ma1644795144998597454t_real: ( set_o > set_o ) > ( set_real > set_real ) > sum_su2944506142428048139t_real > sum_su2944506142428048139t_real ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    sum_ma2058262178995162342_set_o: ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ) > ( set_o > set_o ) > sum_su145664975714009857_set_o > sum_su145664975714009857_set_o ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    sum_ma204727134861501940et_nat: ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ) > ( set_nat > set_nat ) > sum_su1570082439829250771et_nat > sum_su1570082439829250771et_nat ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
    sum_ma5689907174169415190_set_o: ( set_nat > set_nat ) > ( set_o > set_o ) > sum_su180556917423924329_set_o > sum_su180556917423924329_set_o ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_001t__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J,type,
    sum_ma4567893330926093812nnreal: ( set_nat > set_nat ) > ( set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal ) > sum_su7445071152329184979nnreal > sum_su7445071152329184979nnreal ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    sum_ma5866040581394992932et_nat: ( set_nat > set_nat ) > ( set_nat > set_nat ) > sum_su6357119772783438699et_nat > sum_su6357119772783438699et_nat ).

thf(sy_c_Sum__Type_Omap__sum_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Real__Oreal_J_001t__Set__Oset_It__Real__Oreal_J,type,
    sum_ma2514991182478280028t_real: ( set_nat > set_nat ) > ( set_real > set_real ) > sum_su6439591028793565127t_real > sum_su6439591028793565127t_real ).

thf(sy_c_Sum__Type_Omap__sum_001tf__a_001tf__a_001tf__c_001tf__c,type,
    sum_map_sum_a_a_c_c: ( a > a ) > ( c > c ) > sum_sum_a_c > sum_sum_a_c ).

thf(sy_c_Sum__Type_Omap__sum_001tf__a_001tf__b_001tf__c_001tf__d,type,
    sum_map_sum_a_b_c_d: ( a > b ) > ( c > d ) > sum_sum_a_c > sum_sum_b_d ).

thf(sy_c_Sum__Type_Omap__sum_001tf__b_001tf__a_001tf__d_001tf__c,type,
    sum_map_sum_b_a_d_c: ( b > a ) > ( d > c ) > sum_sum_b_d > sum_sum_a_c ).

thf(sy_c_Sum__Type_Omap__sum_001tf__b_001tf__b_001tf__d_001tf__d,type,
    sum_map_sum_b_b_d_d: ( b > b ) > ( d > d ) > sum_sum_b_d > sum_sum_b_d ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001_Eo_001_Eo_001_Eo,type,
    sum_case_sum_o_o_o: ( $o > $o ) > ( $o > $o ) > sum_sum_o_o > $o ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001_Eo_001t__Real__Oreal_001_Eo,type,
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thf(sy_c_Sum__Type_Osum_Ocase__sum_001_Eo_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ca412452145296005092nnreal: ( $o > real ) > ( extend8495563244428889912nnreal > real ) > sum_su7753341093581952129nnreal > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001_Eo_001t__Real__Oreal_001t__Nat__Onat,type,
    sum_ca4335601045291449468al_nat: ( $o > real ) > ( nat > real ) > sum_sum_o_nat > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001_Eo_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ca576396664188569944l_real: ( $o > real ) > ( real > real ) > sum_sum_o_real > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001_Eo,type,
    sum_ca5495973872699983128real_o: ( extend8495563244428889912nnreal > real ) > ( $o > real ) > sum_su1359252823600007511real_o > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ca1130728446324819384nnreal: ( extend8495563244428889912nnreal > real ) > ( extend8495563244428889912nnreal > real ) > sum_su4415445757542774223nnreal > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Nat__Onat,type,
    sum_ca112682941461744208al_nat: ( extend8495563244428889912nnreal > real ) > ( nat > real ) > sum_su1883948583941721703al_nat > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Extended____Nonnegative____Real__Oennreal_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ca6763686470577984908at_nat: ( nat > nat ) > ( nat > nat ) > sum_sum_nat_nat > nat ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Real__Oreal_001_Eo,type,
    sum_ca7527334803461514880real_o: ( nat > real ) > ( $o > real ) > sum_sum_nat_o > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ca6708266607595310672nnreal: ( nat > real ) > ( extend8495563244428889912nnreal > real ) > sum_su3730406437774119527nnreal > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
    sum_ca8334624595930125032al_nat: ( nat > real ) > ( nat > real ) > sum_sum_nat_nat > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ca6225210458162260420l_real: ( nat > real ) > ( real > real ) > sum_sum_nat_real > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001_Eo_001t__Real__Oreal,type,
    sum_ca7497528202123613950o_real: ( real > $o ) > ( real > $o ) > sum_sum_real_real > $o ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001t__Nat__Onat_001t__Real__Oreal,type,
    sum_ca2049687194608230980t_real: ( real > nat ) > ( real > nat ) > sum_sum_real_real > nat ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
    sum_ca9006592577164783524real_o: ( real > real ) > ( $o > real ) > sum_sum_real_o > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Extended____Nonnegative____Real__Oennreal,type,
    sum_ca5039698333674796588nnreal: ( real > real ) > ( extend8495563244428889912nnreal > real ) > sum_su3240232783174752707nnreal > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Nat__Onat,type,
    sum_ca6616995767737129668al_nat: ( real > real ) > ( nat > real ) > sum_sum_real_nat > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001t__Real__Oreal_001t__Real__Oreal,type,
    sum_ca8732840427581260704l_real: ( real > real ) > ( real > real ) > sum_sum_real_real > real ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001tf__a_001t__Real__Oreal,type,
    sum_ca3691009268231894756a_real: ( real > a ) > ( real > a ) > sum_sum_real_real > a ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001tf__b_001t__Real__Oreal,type,
    sum_ca63855846565249637b_real: ( real > b ) > ( real > b ) > sum_sum_real_real > b ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001tf__c_001t__Real__Oreal,type,
    sum_ca5660074461753380326c_real: ( real > c ) > ( real > c ) > sum_sum_real_real > c ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Real__Oreal_001tf__d_001t__Real__Oreal,type,
    sum_ca2032921040086735207d_real: ( real > d ) > ( real > d ) > sum_sum_real_real > d ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001tf__a_001tf__b_001tf__a,type,
    sum_case_sum_a_b_a: ( a > b ) > ( a > b ) > sum_sum_a_a > b ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001tf__c_001tf__d_001tf__c,type,
    sum_case_sum_c_d_c: ( c > d ) > ( c > d ) > sum_sum_c_c > d ).

thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Real__Oreal,type,
    topolo4860482606490270245n_real: set_real > $o ).

thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded_001t__Real__Oreal,type,
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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,
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thf(sy_c_member_001_062_I_Eo_Mtf__a_J,type,
    member_o_a: ( $o > a ) > set_o_a > $o ).

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    member_nat_o: ( nat > $o ) > set_nat_o > $o ).

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    member_nat_nat: ( nat > nat ) > set_nat_nat > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Real__Oreal_J,type,
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thf(sy_c_member_001_062_It__Real__Oreal_M_062_I_Eo_Mt__Real__Oreal_J_J,type,
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    member_real_o: ( real > $o ) > set_real_o > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Extended____Nonnegative____Real__Oennreal_J,type,
    member2919562650594848410nnreal: ( real > extend8495563244428889912nnreal ) > set_re5328672808648366137nnreal > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Nat__Onat_J,type,
    member_real_nat: ( real > nat ) > set_real_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_I_Eo_J_J,type,
    member_real_set_o: ( real > set_o ) > set_real_set_o > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
    member8689841359643572048nnreal: ( real > set_Ex3793607809372303086nnreal ) > set_re634636480907793903nnreal > $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_Mt__Set__Oset_It__Real__Oreal_J_J,type,
    member_real_set_real: ( real > set_real ) > set_real_set_real > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mt__Sum____Type__Osum_Itf__a_Mtf__c_J_J,type,
    member2264291325230826761um_a_c: ( real > sum_sum_a_c ) > set_real_Sum_sum_a_c > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mtf__a_J,type,
    member_real_a: ( real > a ) > set_real_a > $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__Real__Oreal_Mtf__c_J,type,
    member_real_c: ( real > c ) > set_real_c > $o ).

thf(sy_c_member_001_062_It__Real__Oreal_Mtf__d_J,type,
    member_real_d: ( real > d ) > set_real_d > $o ).

thf(sy_c_member_001_062_It__Set__Oset_I_Eo_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member_set_o_set_nat: ( set_o > set_nat ) > set_set_o_set_nat > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_I_Eo_J_J,type,
    member_set_nat_set_o: ( set_nat > set_o ) > set_set_nat_set_o > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
    member2136063858382286634nnreal: ( set_nat > set_Ex3793607809372303086nnreal ) > set_se5513212049808698057nnreal > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member1686471427249568706et_nat: ( set_nat > set_nat ) > set_set_nat_set_nat > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Real__Oreal_J_J,type,
    member6118920896213660830t_real: ( set_nat > set_real ) > set_set_nat_set_real > $o ).

thf(sy_c_member_001_062_It__Sum____Type__Osum_I_Eo_M_Eo_J_Mt__Real__Oreal_J,type,
    member3421099978642967687o_real: ( sum_sum_o_o > real ) > set_Sum_sum_o_o_real > $o ).

thf(sy_c_member_001_062_It__Sum____Type__Osum_I_Eo_Mt__Nat__Onat_J_Mt__Real__Oreal_J,type,
    member8111550167385560159t_real: ( sum_sum_o_nat > real ) > set_Su4870880633484326550t_real > $o ).

thf(sy_c_member_001_062_It__Sum____Type__Osum_It__Nat__Onat_M_Eo_J_Mt__Real__Oreal_J,type,
    member5842278251609744133o_real: ( sum_sum_nat_o > real ) > set_Su8975513444386755132o_real > $o ).

thf(sy_c_member_001_062_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Real__Oreal_J,type,
    member1647107379014654305t_real: ( sum_sum_nat_nat > real ) > set_Su1304851494220063104t_real > $o ).

thf(sy_c_member_001_062_It__Sum____Type__Osum_It__Real__Oreal_Mt__Real__Oreal_J_Mtf__a_J,type,
    member6630180758533022695real_a: ( sum_sum_real_real > a ) > set_Su256150577512953374real_a > $o ).

thf(sy_c_member_001_062_It__Sum____Type__Osum_Itf__a_Mtf__a_J_Mtf__b_J,type,
    member_Sum_sum_a_a_b: ( sum_sum_a_a > b ) > set_Sum_sum_a_a_b > $o ).

thf(sy_c_member_001_062_It__Sum____Type__Osum_Itf__c_Mtf__c_J_Mtf__d_J,type,
    member_Sum_sum_c_c_d: ( sum_sum_c_c > d ) > set_Sum_sum_c_c_d > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__Sum____Type__Osum_Itf__a_Mtf__c_J_J,type,
    member_a_Sum_sum_a_c: ( a > sum_sum_a_c ) > set_a_Sum_sum_a_c > $o ).

thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
    member_a_a: ( a > a ) > set_a_a > $o ).

thf(sy_c_member_001_062_Itf__a_Mtf__b_J,type,
    member_a_b: ( a > b ) > set_a_b > $o ).

thf(sy_c_member_001_062_Itf__a_Mtf__c_J,type,
    member_a_c: ( a > c ) > set_a_c > $o ).

thf(sy_c_member_001_062_Itf__a_Mtf__d_J,type,
    member_a_d: ( a > d ) > set_a_d > $o ).

thf(sy_c_member_001_062_Itf__b_Mt__Sum____Type__Osum_Itf__b_Mtf__d_J_J,type,
    member_b_Sum_sum_b_d: ( b > sum_sum_b_d ) > set_b_Sum_sum_b_d > $o ).

thf(sy_c_member_001_062_Itf__c_Mt__Sum____Type__Osum_Itf__a_Mtf__c_J_J,type,
    member_c_Sum_sum_a_c: ( c > sum_sum_a_c ) > set_c_Sum_sum_a_c > $o ).

thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
    member_c_a: ( c > a ) > set_c_a > $o ).

thf(sy_c_member_001_062_Itf__c_Mtf__b_J,type,
    member_c_b: ( c > b ) > set_c_b > $o ).

thf(sy_c_member_001_062_Itf__c_Mtf__c_J,type,
    member_c_c: ( c > c ) > set_c_c > $o ).

thf(sy_c_member_001_062_Itf__c_Mtf__d_J,type,
    member_c_d: ( c > d ) > set_c_d > $o ).

thf(sy_c_member_001_062_Itf__d_Mt__Sum____Type__Osum_Itf__b_Mtf__d_J_J,type,
    member_d_Sum_sum_b_d: ( d > sum_sum_b_d ) > set_d_Sum_sum_b_d > $o ).

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

thf(sy_c_member_001t__Extended____Nonnegative____Real__Oennreal,type,
    member7908768830364227535nnreal: extend8495563244428889912nnreal > set_Ex3793607809372303086nnreal > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_I_062_I_Eo_Mt__Real__Oreal_J_J,type,
    member_set_o_real: set_o_real > set_set_o_real > $o ).

thf(sy_c_member_001t__Set__Oset_I_062_It__Real__Oreal_Mtf__a_J_J,type,
    member_set_real_a: set_real_a > set_set_real_a > $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__Extended____Nonnegative____Real__Oennreal_J,type,
    member603777416030116741nnreal: set_Ex3793607809372303086nnreal > set_se4580700918925141924nnreal > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
    member_set_set_o: set_set_o > set_set_set_o > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Extended____Nonnegative____Real__Oennreal_J_J,type,
    member6568240578637133883nnreal: set_se4580700918925141924nnreal > set_se8256708918794385754nnreal > $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_It__Set__Oset_It__Real__Oreal_J_J,type,
    member_set_set_real: set_set_real > set_set_set_real > $o ).

thf(sy_c_member_001t__Sigma____Algebra__Omeasure_I_Eo_J,type,
    member1844656263901471916sure_o: sigma_measure_o > set_Sigma_measure_o > $o ).

thf(sy_c_member_001t__Sigma____Algebra__Omeasure_It__Extended____Nonnegative____Real__Oennreal_J,type,
    member6261374078160781754nnreal: sigma_7234349610311085201nnreal > set_Si97717610131227249nnreal > $o ).

thf(sy_c_member_001t__Sigma____Algebra__Omeasure_It__Nat__Onat_J,type,
    member4416920341759242834re_nat: sigma_measure_nat > set_Si3048223896905877257re_nat > $o ).

thf(sy_c_member_001t__Sigma____Algebra__Omeasure_It__Real__Oreal_J,type,
    member4553183543495551918e_real: sigma_measure_real > set_Si6059263944882162789e_real > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

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

thf(sy_c_member_001tf__c,type,
    member_c: c > set_c > $o ).

thf(sy_c_member_001tf__d,type,
    member_d: d > set_d > $o ).

thf(sy_v_S____,type,
    s: set_real ).

thf(sy_v_X,type,
    x: quasi_borel_a ).

thf(sy_v_X_H,type,
    x2: quasi_borel_c ).

thf(sy_v_Y,type,
    y: quasi_borel_b ).

thf(sy_v_Y_H,type,
    y2: quasi_borel_d ).

thf(sy_v__092_060alpha_0621____,type,
    alpha_1: real > a ).

thf(sy_v__092_060alpha_0622____,type,
    alpha_2: real > c ).

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

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

thf(sy_v_f_H____,type,
    f2: real > b ).

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

thf(sy_v_g_H____,type,
    g2: real > d ).

% Relevant facts (1277)
thf(fact_0__092_060open_062S_A_092_060noteq_062_A_123_125_A_092_060and_062_AS_A_092_060noteq_062_AUNIV_092_060close_062,axiom,
    ( ( s != bot_bot_set_real )
    & ( s != top_top_set_real ) ) ).

% \<open>S \<noteq> {} \<and> S \<noteq> UNIV\<close>
thf(fact_1__092_060open_062_092_060And_062thesis_O_A_092_060lbrakk_062S_A_061_A_123_125_A_092_060Longrightarrow_062_Athesis_059_AS_A_061_AUNIV_A_092_060Longrightarrow_062_Athesis_059_AS_A_092_060noteq_062_A_123_125_A_092_060and_062_AS_A_092_060noteq_062_AUNIV_A_092_060Longrightarrow_062_Athesis_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ( ( s != bot_bot_set_real )
   => ( ( s != top_top_set_real )
     => ( ( s != bot_bot_set_real )
        & ( s != top_top_set_real ) ) ) ) ).

% \<open>\<And>thesis. \<lbrakk>S = {} \<Longrightarrow> thesis; S = UNIV \<Longrightarrow> thesis; S \<noteq> {} \<and> S \<noteq> UNIV \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis\<close>
thf(fact_2__092_060open_062_092_060alpha_062_A_092_060in_062_Acopair__qbs__Mx_AX_AX_H_092_060close_062,axiom,
    member2264291325230826761um_a_c @ alpha @ ( binary8286901584692334522Mx_a_c @ x @ x2 ) ).

% \<open>\<alpha> \<in> copair_qbs_Mx X X'\<close>
thf(fact_3_hs,axiom,
    ( ( member_set_real @ s @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
    & ( ( s = bot_bot_set_real )
     => ? [X: real > a] :
          ( ( member_real_a @ X @ ( qbs_Mx_a @ x ) )
          & ( alpha
            = ( ^ [R: real] : ( sum_Inl_a_c @ ( X @ R ) ) ) ) ) )
    & ( ( s = top_top_set_real )
     => ? [X: real > c] :
          ( ( member_real_c @ X @ ( qbs_Mx_c @ x2 ) )
          & ( alpha
            = ( ^ [R: real] : ( sum_Inr_c_a @ ( X @ R ) ) ) ) ) )
    & ( ( ( s != bot_bot_set_real )
        & ( s != top_top_set_real ) )
     => ? [X: real > a] :
          ( ( member_real_a @ X @ ( qbs_Mx_a @ x ) )
          & ? [Xa: real > c] :
              ( ( member_real_c @ Xa @ ( qbs_Mx_c @ x2 ) )
              & ( alpha
                = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ s ) @ ( sum_Inl_a_c @ ( X @ R ) ) @ ( sum_Inr_c_a @ ( Xa @ R ) ) ) ) ) ) ) ) ) ).

% hs
thf(fact_4_space__in__borel,axiom,
    member_set_real @ top_top_set_real @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).

% space_in_borel
thf(fact_5_space__in__borel,axiom,
    member603777416030116741nnreal @ top_to7994903218803871134nnreal @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ).

% space_in_borel
thf(fact_6_space__in__borel,axiom,
    member_set_o @ top_top_set_o @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ).

% space_in_borel
thf(fact_7_space__in__borel,axiom,
    member_set_nat @ top_top_set_nat @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ).

% space_in_borel
thf(fact_8_sets__Ball,axiom,
    ! [I: set_real_a,A: ( real > a ) > set_real,M: ( real > a ) > sigma_measure_real,I2: real > a] :
      ( ! [X: real > a] :
          ( ( member_real_a @ X @ I )
         => ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_real_a @ I2 @ I )
       => ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_9_sets__Ball,axiom,
    ! [I: set_o_real,A: ( $o > real ) > set_real,M: ( $o > real ) > sigma_measure_real,I2: $o > real] :
      ( ! [X: $o > real] :
          ( ( member_o_real @ X @ I )
         => ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_o_real @ I2 @ I )
       => ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_10_sets__Ball,axiom,
    ! [I: set_nat_real,A: ( nat > real ) > set_real,M: ( nat > real ) > sigma_measure_real,I2: nat > real] :
      ( ! [X: nat > real] :
          ( ( member_nat_real @ X @ I )
         => ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_nat_real @ I2 @ I )
       => ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_11_sets__Ball,axiom,
    ! [I: set_c_d,A: ( c > d ) > set_real,M: ( c > d ) > sigma_measure_real,I2: c > d] :
      ( ! [X: c > d] :
          ( ( member_c_d @ X @ I )
         => ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_c_d @ I2 @ I )
       => ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_12_sets__Ball,axiom,
    ! [I: set_a_b,A: ( a > b ) > set_real,M: ( a > b ) > sigma_measure_real,I2: a > b] :
      ( ! [X: a > b] :
          ( ( member_a_b @ X @ I )
         => ( member_set_real @ ( A @ X ) @ ( sigma_sets_real @ ( M @ X ) ) ) )
     => ( ( member_a_b @ I2 @ I )
       => ( member_set_real @ ( A @ I2 ) @ ( sigma_sets_real @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_13_sets__Ball,axiom,
    ! [I: set_real_a,A: ( real > a ) > set_nat,M: ( real > a ) > sigma_measure_nat,I2: real > a] :
      ( ! [X: real > a] :
          ( ( member_real_a @ X @ I )
         => ( member_set_nat @ ( A @ X ) @ ( sigma_sets_nat @ ( M @ X ) ) ) )
     => ( ( member_real_a @ I2 @ I )
       => ( member_set_nat @ ( A @ I2 ) @ ( sigma_sets_nat @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_14_sets__Ball,axiom,
    ! [I: set_o_real,A: ( $o > real ) > set_nat,M: ( $o > real ) > sigma_measure_nat,I2: $o > real] :
      ( ! [X: $o > real] :
          ( ( member_o_real @ X @ I )
         => ( member_set_nat @ ( A @ X ) @ ( sigma_sets_nat @ ( M @ X ) ) ) )
     => ( ( member_o_real @ I2 @ I )
       => ( member_set_nat @ ( A @ I2 ) @ ( sigma_sets_nat @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_15_sets__Ball,axiom,
    ! [I: set_nat_real,A: ( nat > real ) > set_nat,M: ( nat > real ) > sigma_measure_nat,I2: nat > real] :
      ( ! [X: nat > real] :
          ( ( member_nat_real @ X @ I )
         => ( member_set_nat @ ( A @ X ) @ ( sigma_sets_nat @ ( M @ X ) ) ) )
     => ( ( member_nat_real @ I2 @ I )
       => ( member_set_nat @ ( A @ I2 ) @ ( sigma_sets_nat @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_16_sets__Ball,axiom,
    ! [I: set_c_d,A: ( c > d ) > set_nat,M: ( c > d ) > sigma_measure_nat,I2: c > d] :
      ( ! [X: c > d] :
          ( ( member_c_d @ X @ I )
         => ( member_set_nat @ ( A @ X ) @ ( sigma_sets_nat @ ( M @ X ) ) ) )
     => ( ( member_c_d @ I2 @ I )
       => ( member_set_nat @ ( A @ I2 ) @ ( sigma_sets_nat @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_17_sets__Ball,axiom,
    ! [I: set_a_b,A: ( a > b ) > set_nat,M: ( a > b ) > sigma_measure_nat,I2: a > b] :
      ( ! [X: a > b] :
          ( ( member_a_b @ X @ I )
         => ( member_set_nat @ ( A @ X ) @ ( sigma_sets_nat @ ( M @ X ) ) ) )
     => ( ( member_a_b @ I2 @ I )
       => ( member_set_nat @ ( A @ I2 ) @ ( sigma_sets_nat @ ( M @ I2 ) ) ) ) ) ).

% sets_Ball
thf(fact_18_real__distribution__axioms_Ointro,axiom,
    ! [M: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( distri5068091390715981392axioms @ M ) ) ).

% real_distribution_axioms.intro
thf(fact_19_finite__borel__measure__axioms_Ointro,axiom,
    ! [M: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( distri7440681460546723631axioms @ M ) ) ).

% finite_borel_measure_axioms.intro
thf(fact_20_real__distribution__axioms__def,axiom,
    ( distri5068091390715981392axioms
    = ( ^ [M2: sigma_measure_real] :
          ( ( sigma_sets_real @ M2 )
          = ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).

% real_distribution_axioms_def
thf(fact_21_finite__borel__measure__axioms__def,axiom,
    ( distri7440681460546723631axioms
    = ( ^ [M2: sigma_measure_real] :
          ( ( sigma_sets_real @ M2 )
          = ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ) ).

% finite_borel_measure_axioms_def
thf(fact_22_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_real] : ( member_set_real @ bot_bot_set_real @ ( sigma_sets_real @ M ) ) ).

% sets.empty_sets
thf(fact_23_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_nat] : ( member_set_nat @ bot_bot_set_nat @ ( sigma_sets_nat @ M ) ) ).

% sets.empty_sets
thf(fact_24_sets_Oempty__sets,axiom,
    ! [M: sigma_measure_o] : ( member_set_o @ bot_bot_set_o @ ( sigma_sets_o @ M ) ) ).

% sets.empty_sets
thf(fact_25_sets_Oempty__sets,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.empty_sets
thf(fact_26_sets__interval__measure,axiom,
    ! [F: real > real] :
      ( ( sigma_sets_real @ ( lebesg8227263024992965735easure @ F ) )
      = ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% sets_interval_measure
thf(fact_27_sets__convolution,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real] :
      ( ( sigma_sets_real @ ( convolution_real @ M @ N ) )
      = ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ).

% sets_convolution
thf(fact_28__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060alpha_0621_A_092_060alpha_0622_O_A_092_060alpha_0621_A_092_060in_062_Aqbs__Mx_AX_A_092_060and_062_A_092_060alpha_0622_A_092_060in_062_Aqbs__Mx_AX_H_A_092_060and_062_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_Aif_Ar_A_092_060in_062_AS_Athen_AInl_A_I_092_060alpha_0621_Ar_J_Aelse_AInr_A_I_092_060alpha_0622_Ar_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Alpha_1: real > a,Alpha_2: real > c] :
        ~ ( ( member_real_a @ Alpha_1 @ ( qbs_Mx_a @ x ) )
          & ( member_real_c @ Alpha_2 @ ( qbs_Mx_c @ x2 ) )
          & ( alpha
            = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ s ) @ ( sum_Inl_a_c @ ( Alpha_1 @ R ) ) @ ( sum_Inr_c_a @ ( Alpha_2 @ R ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>\<alpha>1 \<alpha>2. \<alpha>1 \<in> qbs_Mx X \<and> \<alpha>2 \<in> qbs_Mx X' \<and> \<alpha> = (\<lambda>r. if r \<in> S then Inl (\<alpha>1 r) else Inr (\<alpha>2 r)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_29__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062S_O_AS_A_092_060in_062_Asets_Areal__borel_A_092_060and_062_A_IS_A_061_A_123_125_A_092_060longrightarrow_062_A_I_092_060exists_062_092_060alpha_0621_092_060in_062qbs__Mx_AX_O_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_AInl_A_I_092_060alpha_0621_Ar_J_J_J_J_A_092_060and_062_A_IS_A_061_AUNIV_A_092_060longrightarrow_062_A_I_092_060exists_062_092_060alpha_0622_092_060in_062qbs__Mx_AX_H_O_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_AInr_A_I_092_060alpha_0622_Ar_J_J_J_J_A_092_060and_062_A_IS_A_092_060noteq_062_A_123_125_A_092_060and_062_AS_A_092_060noteq_062_AUNIV_A_092_060longrightarrow_062_A_I_092_060exists_062_092_060alpha_0621_092_060in_062qbs__Mx_AX_O_A_092_060exists_062_092_060alpha_0622_092_060in_062qbs__Mx_AX_H_O_A_092_060alpha_062_A_061_A_I_092_060lambda_062r_O_Aif_Ar_A_092_060in_062_AS_Athen_AInl_A_I_092_060alpha_0621_Ar_J_Aelse_AInr_A_I_092_060alpha_0622_Ar_J_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [S: set_real] :
        ~ ( ( member_set_real @ S @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
          & ( ( S = bot_bot_set_real )
           => ? [X: real > a] :
                ( ( member_real_a @ X @ ( qbs_Mx_a @ x ) )
                & ( alpha
                  = ( ^ [R: real] : ( sum_Inl_a_c @ ( X @ R ) ) ) ) ) )
          & ( ( S = top_top_set_real )
           => ? [X: real > c] :
                ( ( member_real_c @ X @ ( qbs_Mx_c @ x2 ) )
                & ( alpha
                  = ( ^ [R: real] : ( sum_Inr_c_a @ ( X @ R ) ) ) ) ) )
          & ( ( ( S != bot_bot_set_real )
              & ( S != top_top_set_real ) )
           => ? [X: real > a] :
                ( ( member_real_a @ X @ ( qbs_Mx_a @ x ) )
                & ? [Xa: real > c] :
                    ( ( member_real_c @ Xa @ ( qbs_Mx_c @ x2 ) )
                    & ( alpha
                      = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ S ) @ ( sum_Inl_a_c @ ( X @ R ) ) @ ( sum_Inr_c_a @ ( Xa @ R ) ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>S. S \<in> sets real_borel \<and> (S = {} \<longrightarrow> (\<exists>\<alpha>1\<in>qbs_Mx X. \<alpha> = (\<lambda>r. Inl (\<alpha>1 r)))) \<and> (S = UNIV \<longrightarrow> (\<exists>\<alpha>2\<in>qbs_Mx X'. \<alpha> = (\<lambda>r. Inr (\<alpha>2 r)))) \<and> (S \<noteq> {} \<and> S \<noteq> UNIV \<longrightarrow> (\<exists>\<alpha>1\<in>qbs_Mx X. \<exists>\<alpha>2\<in>qbs_Mx X'. \<alpha> = (\<lambda>r. if r \<in> S then Inl (\<alpha>1 r) else Inr (\<alpha>2 r)))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_30_h,axiom,
    ( ( member_real_a @ alpha_1 @ ( qbs_Mx_a @ x ) )
    & ( member_real_c @ alpha_2 @ ( qbs_Mx_c @ x2 ) )
    & ( alpha
      = ( ^ [R: real] : ( if_Sum_sum_a_c @ ( member_real @ R @ s ) @ ( sum_Inl_a_c @ ( alpha_1 @ R ) ) @ ( sum_Inr_c_a @ ( alpha_2 @ R ) ) ) ) ) ) ).

% h
thf(fact_31_Inl__Inr__False,axiom,
    ! [X2: a,Y: c] :
      ( ( sum_Inl_a_c @ X2 )
     != ( sum_Inr_c_a @ Y ) ) ).

% Inl_Inr_False
thf(fact_32_Inl__Inr__False,axiom,
    ! [X2: b,Y: d] :
      ( ( sum_Inl_b_d @ X2 )
     != ( sum_Inr_d_b @ Y ) ) ).

% Inl_Inr_False
thf(fact_33_Inr__Inl__False,axiom,
    ! [X2: c,Y: a] :
      ( ( sum_Inr_c_a @ X2 )
     != ( sum_Inl_a_c @ Y ) ) ).

% Inr_Inl_False
thf(fact_34_Inr__Inl__False,axiom,
    ! [X2: d,Y: b] :
      ( ( sum_Inr_d_b @ X2 )
     != ( sum_Inl_b_d @ Y ) ) ).

% Inr_Inl_False
thf(fact_35_sum_Oinject_I1_J,axiom,
    ! [X1: a,Y1: a] :
      ( ( ( sum_Inl_a_c @ X1 )
        = ( sum_Inl_a_c @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% sum.inject(1)
thf(fact_36_sum_Oinject_I1_J,axiom,
    ! [X1: b,Y1: b] :
      ( ( ( sum_Inl_b_d @ X1 )
        = ( sum_Inl_b_d @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% sum.inject(1)
thf(fact_37_old_Osum_Oinject_I1_J,axiom,
    ! [A2: a,A3: a] :
      ( ( ( sum_Inl_a_c @ A2 )
        = ( sum_Inl_a_c @ A3 ) )
      = ( A2 = A3 ) ) ).

% old.sum.inject(1)
thf(fact_38_old_Osum_Oinject_I1_J,axiom,
    ! [A2: b,A3: b] :
      ( ( ( sum_Inl_b_d @ A2 )
        = ( sum_Inl_b_d @ A3 ) )
      = ( A2 = A3 ) ) ).

% old.sum.inject(1)
thf(fact_39_sum_Oinject_I2_J,axiom,
    ! [X22: c,Y2: c] :
      ( ( ( sum_Inr_c_a @ X22 )
        = ( sum_Inr_c_a @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% sum.inject(2)
thf(fact_40_sum_Oinject_I2_J,axiom,
    ! [X22: d,Y2: d] :
      ( ( ( sum_Inr_d_b @ X22 )
        = ( sum_Inr_d_b @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% sum.inject(2)
thf(fact_41_old_Osum_Oinject_I2_J,axiom,
    ! [B: c,B2: c] :
      ( ( ( sum_Inr_c_a @ B )
        = ( sum_Inr_c_a @ B2 ) )
      = ( B = B2 ) ) ).

% old.sum.inject(2)
thf(fact_42_old_Osum_Oinject_I2_J,axiom,
    ! [B: d,B2: d] :
      ( ( ( sum_Inr_d_b @ B )
        = ( sum_Inr_d_b @ B2 ) )
      = ( B = B2 ) ) ).

% old.sum.inject(2)
thf(fact_43_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_44_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_45_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_46_empty__iff,axiom,
    ! [C: extend8495563244428889912nnreal] :
      ~ ( member7908768830364227535nnreal @ C @ bot_bo4854962954004695426nnreal ) ).

% empty_iff
thf(fact_47_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_48_empty__iff,axiom,
    ! [C: set_o] :
      ~ ( member_set_o @ C @ bot_bot_set_set_o ) ).

% empty_iff
thf(fact_49_empty__iff,axiom,
    ! [C: set_Ex3793607809372303086nnreal] :
      ~ ( member603777416030116741nnreal @ C @ bot_bo2988155216863113784nnreal ) ).

% empty_iff
thf(fact_50_empty__iff,axiom,
    ! [C: set_real] :
      ~ ( member_set_real @ C @ bot_bot_set_set_real ) ).

% empty_iff
thf(fact_51_empty__iff,axiom,
    ! [C: real > a] :
      ~ ( member_real_a @ C @ bot_bot_set_real_a ) ).

% empty_iff
thf(fact_52_empty__iff,axiom,
    ! [C: $o > real] :
      ~ ( member_o_real @ C @ bot_bot_set_o_real ) ).

% empty_iff
thf(fact_53_all__not__in__conv,axiom,
    ! [A: set_real] :
      ( ( ! [X3: real] :
            ~ ( member_real @ X3 @ A ) )
      = ( A = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_54_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X3: nat] :
            ~ ( member_nat @ X3 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_55_all__not__in__conv,axiom,
    ! [A: set_o] :
      ( ( ! [X3: $o] :
            ~ ( member_o @ X3 @ A ) )
      = ( A = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_56_all__not__in__conv,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( ! [X3: extend8495563244428889912nnreal] :
            ~ ( member7908768830364227535nnreal @ X3 @ A ) )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% all_not_in_conv
thf(fact_57_all__not__in__conv,axiom,
    ! [A: set_set_nat] :
      ( ( ! [X3: set_nat] :
            ~ ( member_set_nat @ X3 @ A ) )
      = ( A = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_58_all__not__in__conv,axiom,
    ! [A: set_set_o] :
      ( ( ! [X3: set_o] :
            ~ ( member_set_o @ X3 @ A ) )
      = ( A = bot_bot_set_set_o ) ) ).

% all_not_in_conv
thf(fact_59_all__not__in__conv,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ( ! [X3: set_Ex3793607809372303086nnreal] :
            ~ ( member603777416030116741nnreal @ X3 @ A ) )
      = ( A = bot_bo2988155216863113784nnreal ) ) ).

% all_not_in_conv
thf(fact_60_all__not__in__conv,axiom,
    ! [A: set_set_real] :
      ( ( ! [X3: set_real] :
            ~ ( member_set_real @ X3 @ A ) )
      = ( A = bot_bot_set_set_real ) ) ).

% all_not_in_conv
thf(fact_61_all__not__in__conv,axiom,
    ! [A: set_real_a] :
      ( ( ! [X3: real > a] :
            ~ ( member_real_a @ X3 @ A ) )
      = ( A = bot_bot_set_real_a ) ) ).

% all_not_in_conv
thf(fact_62_all__not__in__conv,axiom,
    ! [A: set_o_real] :
      ( ( ! [X3: $o > real] :
            ~ ( member_o_real @ X3 @ A ) )
      = ( A = bot_bot_set_o_real ) ) ).

% all_not_in_conv
thf(fact_63_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X3: real] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_64_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_65_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X3: $o] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_66_Collect__empty__eq,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ( ( ( collec6648975593938027277nnreal @ P )
        = bot_bo4854962954004695426nnreal )
      = ( ! [X3: extend8495563244428889912nnreal] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_67_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X3: real] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_68_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_69_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X3: $o] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_70_empty__Collect__eq,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ( ( bot_bo4854962954004695426nnreal
        = ( collec6648975593938027277nnreal @ P ) )
      = ( ! [X3: extend8495563244428889912nnreal] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_71_UNIV__I,axiom,
    ! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).

% UNIV_I
thf(fact_72_UNIV__I,axiom,
    ! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).

% UNIV_I
thf(fact_73_UNIV__I,axiom,
    ! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_74_UNIV__I,axiom,
    ! [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ top_to7994903218803871134nnreal ) ).

% UNIV_I
thf(fact_75_UNIV__I,axiom,
    ! [X2: set_nat] : ( member_set_nat @ X2 @ top_top_set_set_nat ) ).

% UNIV_I
thf(fact_76_UNIV__I,axiom,
    ! [X2: set_o] : ( member_set_o @ X2 @ top_top_set_set_o ) ).

% UNIV_I
thf(fact_77_UNIV__I,axiom,
    ! [X2: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X2 @ top_to3356475028079756884nnreal ) ).

% UNIV_I
thf(fact_78_UNIV__I,axiom,
    ! [X2: set_real] : ( member_set_real @ X2 @ top_top_set_set_real ) ).

% UNIV_I
thf(fact_79_UNIV__I,axiom,
    ! [X2: real > a] : ( member_real_a @ X2 @ top_top_set_real_a ) ).

% UNIV_I
thf(fact_80_UNIV__I,axiom,
    ! [X2: $o > real] : ( member_o_real @ X2 @ top_top_set_o_real ) ).

% UNIV_I
thf(fact_81_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_82_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

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

% bot_set_def
thf(fact_84_bot__set__def,axiom,
    ( bot_bo4854962954004695426nnreal
    = ( collec6648975593938027277nnreal @ bot_bo412624608084785539real_o ) ) ).

% bot_set_def
thf(fact_85_UNIV__witness,axiom,
    ? [X: real] : ( member_real @ X @ top_top_set_real ) ).

% UNIV_witness
thf(fact_86_UNIV__witness,axiom,
    ? [X: $o] : ( member_o @ X @ top_top_set_o ) ).

% UNIV_witness
thf(fact_87_UNIV__witness,axiom,
    ? [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_88_UNIV__witness,axiom,
    ? [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ top_to7994903218803871134nnreal ) ).

% UNIV_witness
thf(fact_89_UNIV__witness,axiom,
    ? [X: set_nat] : ( member_set_nat @ X @ top_top_set_set_nat ) ).

% UNIV_witness
thf(fact_90_UNIV__witness,axiom,
    ? [X: set_o] : ( member_set_o @ X @ top_top_set_set_o ) ).

% UNIV_witness
thf(fact_91_UNIV__witness,axiom,
    ? [X: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X @ top_to3356475028079756884nnreal ) ).

% UNIV_witness
thf(fact_92_UNIV__witness,axiom,
    ? [X: set_real] : ( member_set_real @ X @ top_top_set_set_real ) ).

% UNIV_witness
thf(fact_93_UNIV__witness,axiom,
    ? [X: real > a] : ( member_real_a @ X @ top_top_set_real_a ) ).

% UNIV_witness
thf(fact_94_UNIV__witness,axiom,
    ? [X: $o > real] : ( member_o_real @ X @ top_top_set_o_real ) ).

% UNIV_witness
thf(fact_95_UNIV__eq__I,axiom,
    ! [A: set_real] :
      ( ! [X: real] : ( member_real @ X @ A )
     => ( top_top_set_real = A ) ) ).

% UNIV_eq_I
thf(fact_96_UNIV__eq__I,axiom,
    ! [A: set_o] :
      ( ! [X: $o] : ( member_o @ X @ A )
     => ( top_top_set_o = A ) ) ).

% UNIV_eq_I
thf(fact_97_UNIV__eq__I,axiom,
    ! [A: set_nat] :
      ( ! [X: nat] : ( member_nat @ X @ A )
     => ( top_top_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_98_UNIV__eq__I,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ! [X: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X @ A )
     => ( top_to7994903218803871134nnreal = A ) ) ).

% UNIV_eq_I
thf(fact_99_UNIV__eq__I,axiom,
    ! [A: set_set_nat] :
      ( ! [X: set_nat] : ( member_set_nat @ X @ A )
     => ( top_top_set_set_nat = A ) ) ).

% UNIV_eq_I
thf(fact_100_UNIV__eq__I,axiom,
    ! [A: set_set_o] :
      ( ! [X: set_o] : ( member_set_o @ X @ A )
     => ( top_top_set_set_o = A ) ) ).

% UNIV_eq_I
thf(fact_101_UNIV__eq__I,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ! [X: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X @ A )
     => ( top_to3356475028079756884nnreal = A ) ) ).

% UNIV_eq_I
thf(fact_102_UNIV__eq__I,axiom,
    ! [A: set_set_real] :
      ( ! [X: set_real] : ( member_set_real @ X @ A )
     => ( top_top_set_set_real = A ) ) ).

% UNIV_eq_I
thf(fact_103_UNIV__eq__I,axiom,
    ! [A: set_real_a] :
      ( ! [X: real > a] : ( member_real_a @ X @ A )
     => ( top_top_set_real_a = A ) ) ).

% UNIV_eq_I
thf(fact_104_UNIV__eq__I,axiom,
    ! [A: set_o_real] :
      ( ! [X: $o > real] : ( member_o_real @ X @ A )
     => ( top_top_set_o_real = A ) ) ).

% UNIV_eq_I
thf(fact_105_ex__in__conv,axiom,
    ! [A: set_real] :
      ( ( ? [X3: real] : ( member_real @ X3 @ A ) )
      = ( A != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_106_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_107_ex__in__conv,axiom,
    ! [A: set_o] :
      ( ( ? [X3: $o] : ( member_o @ X3 @ A ) )
      = ( A != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_108_ex__in__conv,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( ? [X3: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X3 @ A ) )
      = ( A != bot_bo4854962954004695426nnreal ) ) ).

% ex_in_conv
thf(fact_109_ex__in__conv,axiom,
    ! [A: set_set_nat] :
      ( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
      = ( A != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_110_ex__in__conv,axiom,
    ! [A: set_set_o] :
      ( ( ? [X3: set_o] : ( member_set_o @ X3 @ A ) )
      = ( A != bot_bot_set_set_o ) ) ).

% ex_in_conv
thf(fact_111_ex__in__conv,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ( ? [X3: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X3 @ A ) )
      = ( A != bot_bo2988155216863113784nnreal ) ) ).

% ex_in_conv
thf(fact_112_ex__in__conv,axiom,
    ! [A: set_set_real] :
      ( ( ? [X3: set_real] : ( member_set_real @ X3 @ A ) )
      = ( A != bot_bot_set_set_real ) ) ).

% ex_in_conv
thf(fact_113_ex__in__conv,axiom,
    ! [A: set_real_a] :
      ( ( ? [X3: real > a] : ( member_real_a @ X3 @ A ) )
      = ( A != bot_bot_set_real_a ) ) ).

% ex_in_conv
thf(fact_114_ex__in__conv,axiom,
    ! [A: set_o_real] :
      ( ( ? [X3: $o > real] : ( member_o_real @ X3 @ A ) )
      = ( A != bot_bot_set_o_real ) ) ).

% ex_in_conv
thf(fact_115_equals0I,axiom,
    ! [A: set_real] :
      ( ! [Y3: real] :
          ~ ( member_real @ Y3 @ A )
     => ( A = bot_bot_set_real ) ) ).

% equals0I
thf(fact_116_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_117_equals0I,axiom,
    ! [A: set_o] :
      ( ! [Y3: $o] :
          ~ ( member_o @ Y3 @ A )
     => ( A = bot_bot_set_o ) ) ).

% equals0I
thf(fact_118_equals0I,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ! [Y3: extend8495563244428889912nnreal] :
          ~ ( member7908768830364227535nnreal @ Y3 @ A )
     => ( A = bot_bo4854962954004695426nnreal ) ) ).

% equals0I
thf(fact_119_equals0I,axiom,
    ! [A: set_set_nat] :
      ( ! [Y3: set_nat] :
          ~ ( member_set_nat @ Y3 @ A )
     => ( A = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_120_equals0I,axiom,
    ! [A: set_set_o] :
      ( ! [Y3: set_o] :
          ~ ( member_set_o @ Y3 @ A )
     => ( A = bot_bot_set_set_o ) ) ).

% equals0I
thf(fact_121_equals0I,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ! [Y3: set_Ex3793607809372303086nnreal] :
          ~ ( member603777416030116741nnreal @ Y3 @ A )
     => ( A = bot_bo2988155216863113784nnreal ) ) ).

% equals0I
thf(fact_122_equals0I,axiom,
    ! [A: set_set_real] :
      ( ! [Y3: set_real] :
          ~ ( member_set_real @ Y3 @ A )
     => ( A = bot_bot_set_set_real ) ) ).

% equals0I
thf(fact_123_equals0I,axiom,
    ! [A: set_real_a] :
      ( ! [Y3: real > a] :
          ~ ( member_real_a @ Y3 @ A )
     => ( A = bot_bot_set_real_a ) ) ).

% equals0I
thf(fact_124_equals0I,axiom,
    ! [A: set_o_real] :
      ( ! [Y3: $o > real] :
          ~ ( member_o_real @ Y3 @ A )
     => ( A = bot_bot_set_o_real ) ) ).

% equals0I
thf(fact_125_equals0D,axiom,
    ! [A: set_real,A2: real] :
      ( ( A = bot_bot_set_real )
     => ~ ( member_real @ A2 @ A ) ) ).

% equals0D
thf(fact_126_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_127_equals0D,axiom,
    ! [A: set_o,A2: $o] :
      ( ( A = bot_bot_set_o )
     => ~ ( member_o @ A2 @ A ) ) ).

% equals0D
thf(fact_128_equals0D,axiom,
    ! [A: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
      ( ( A = bot_bo4854962954004695426nnreal )
     => ~ ( member7908768830364227535nnreal @ A2 @ A ) ) ).

% equals0D
thf(fact_129_equals0D,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( A = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_130_equals0D,axiom,
    ! [A: set_set_o,A2: set_o] :
      ( ( A = bot_bot_set_set_o )
     => ~ ( member_set_o @ A2 @ A ) ) ).

% equals0D
thf(fact_131_equals0D,axiom,
    ! [A: set_se4580700918925141924nnreal,A2: set_Ex3793607809372303086nnreal] :
      ( ( A = bot_bo2988155216863113784nnreal )
     => ~ ( member603777416030116741nnreal @ A2 @ A ) ) ).

% equals0D
thf(fact_132_equals0D,axiom,
    ! [A: set_set_real,A2: set_real] :
      ( ( A = bot_bot_set_set_real )
     => ~ ( member_set_real @ A2 @ A ) ) ).

% equals0D
thf(fact_133_equals0D,axiom,
    ! [A: set_real_a,A2: real > a] :
      ( ( A = bot_bot_set_real_a )
     => ~ ( member_real_a @ A2 @ A ) ) ).

% equals0D
thf(fact_134_equals0D,axiom,
    ! [A: set_o_real,A2: $o > real] :
      ( ( A = bot_bot_set_o_real )
     => ~ ( member_o_real @ A2 @ A ) ) ).

% equals0D
thf(fact_135_emptyE,axiom,
    ! [A2: real] :
      ~ ( member_real @ A2 @ bot_bot_set_real ) ).

% emptyE
thf(fact_136_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_137_emptyE,axiom,
    ! [A2: $o] :
      ~ ( member_o @ A2 @ bot_bot_set_o ) ).

% emptyE
thf(fact_138_emptyE,axiom,
    ! [A2: extend8495563244428889912nnreal] :
      ~ ( member7908768830364227535nnreal @ A2 @ bot_bo4854962954004695426nnreal ) ).

% emptyE
thf(fact_139_emptyE,axiom,
    ! [A2: set_nat] :
      ~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_140_emptyE,axiom,
    ! [A2: set_o] :
      ~ ( member_set_o @ A2 @ bot_bot_set_set_o ) ).

% emptyE
thf(fact_141_emptyE,axiom,
    ! [A2: set_Ex3793607809372303086nnreal] :
      ~ ( member603777416030116741nnreal @ A2 @ bot_bo2988155216863113784nnreal ) ).

% emptyE
thf(fact_142_emptyE,axiom,
    ! [A2: set_real] :
      ~ ( member_set_real @ A2 @ bot_bot_set_set_real ) ).

% emptyE
thf(fact_143_emptyE,axiom,
    ! [A2: real > a] :
      ~ ( member_real_a @ A2 @ bot_bot_set_real_a ) ).

% emptyE
thf(fact_144_emptyE,axiom,
    ! [A2: $o > real] :
      ~ ( member_o_real @ A2 @ bot_bot_set_o_real ) ).

% emptyE
thf(fact_145_Inr__inject,axiom,
    ! [X2: c,Y: c] :
      ( ( ( sum_Inr_c_a @ X2 )
        = ( sum_Inr_c_a @ Y ) )
     => ( X2 = Y ) ) ).

% Inr_inject
thf(fact_146_Inr__inject,axiom,
    ! [X2: d,Y: d] :
      ( ( ( sum_Inr_d_b @ X2 )
        = ( sum_Inr_d_b @ Y ) )
     => ( X2 = Y ) ) ).

% Inr_inject
thf(fact_147_Inl__inject,axiom,
    ! [X2: a,Y: a] :
      ( ( ( sum_Inl_a_c @ X2 )
        = ( sum_Inl_a_c @ Y ) )
     => ( X2 = Y ) ) ).

% Inl_inject
thf(fact_148_Inl__inject,axiom,
    ! [X2: b,Y: b] :
      ( ( ( sum_Inl_b_d @ X2 )
        = ( sum_Inl_b_d @ Y ) )
     => ( X2 = Y ) ) ).

% Inl_inject
thf(fact_149_empty__not__UNIV,axiom,
    bot_bot_set_real != top_top_set_real ).

% empty_not_UNIV
thf(fact_150_empty__not__UNIV,axiom,
    bot_bot_set_o != top_top_set_o ).

% empty_not_UNIV
thf(fact_151_empty__not__UNIV,axiom,
    bot_bot_set_nat != top_top_set_nat ).

% empty_not_UNIV
thf(fact_152_empty__not__UNIV,axiom,
    bot_bo4854962954004695426nnreal != top_to7994903218803871134nnreal ).

% empty_not_UNIV
thf(fact_153_split__sum__all,axiom,
    ( ( ^ [P2: sum_sum_a_c > $o] :
        ! [X4: sum_sum_a_c] : ( P2 @ X4 ) )
    = ( ^ [P3: sum_sum_a_c > $o] :
          ( ! [X3: a] : ( P3 @ ( sum_Inl_a_c @ X3 ) )
          & ! [X3: c] : ( P3 @ ( sum_Inr_c_a @ X3 ) ) ) ) ) ).

% split_sum_all
thf(fact_154_split__sum__all,axiom,
    ( ( ^ [P2: sum_sum_b_d > $o] :
        ! [X4: sum_sum_b_d] : ( P2 @ X4 ) )
    = ( ^ [P3: sum_sum_b_d > $o] :
          ( ! [X3: b] : ( P3 @ ( sum_Inl_b_d @ X3 ) )
          & ! [X3: d] : ( P3 @ ( sum_Inr_d_b @ X3 ) ) ) ) ) ).

% split_sum_all
thf(fact_155_split__sum__ex,axiom,
    ( ( ^ [P2: sum_sum_a_c > $o] :
        ? [X4: sum_sum_a_c] : ( P2 @ X4 ) )
    = ( ^ [P3: sum_sum_a_c > $o] :
          ( ? [X3: a] : ( P3 @ ( sum_Inl_a_c @ X3 ) )
          | ? [X3: c] : ( P3 @ ( sum_Inr_c_a @ X3 ) ) ) ) ) ).

% split_sum_ex
thf(fact_156_split__sum__ex,axiom,
    ( ( ^ [P2: sum_sum_b_d > $o] :
        ? [X4: sum_sum_b_d] : ( P2 @ X4 ) )
    = ( ^ [P3: sum_sum_b_d > $o] :
          ( ? [X3: b] : ( P3 @ ( sum_Inl_b_d @ X3 ) )
          | ? [X3: d] : ( P3 @ ( sum_Inr_d_b @ X3 ) ) ) ) ) ).

% split_sum_ex
thf(fact_157_Inr__not__Inl,axiom,
    ! [B: c,A2: a] :
      ( ( sum_Inr_c_a @ B )
     != ( sum_Inl_a_c @ A2 ) ) ).

% Inr_not_Inl
thf(fact_158_Inr__not__Inl,axiom,
    ! [B: d,A2: b] :
      ( ( sum_Inr_d_b @ B )
     != ( sum_Inl_b_d @ A2 ) ) ).

% Inr_not_Inl
thf(fact_159_mem__Collect__eq,axiom,
    ! [A2: real > a,P: ( real > a ) > $o] :
      ( ( member_real_a @ A2 @ ( collect_real_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
    ! [A2: $o > real,P: ( $o > real ) > $o] :
      ( ( member_o_real @ A2 @ ( collect_o_real @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_161_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_162_mem__Collect__eq,axiom,
    ! [A2: c > d,P: ( c > d ) > $o] :
      ( ( member_c_d @ A2 @ ( collect_c_d @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_163_mem__Collect__eq,axiom,
    ! [A2: a > b,P: ( a > b ) > $o] :
      ( ( member_a_b @ A2 @ ( collect_a_b @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_164_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_165_mem__Collect__eq,axiom,
    ! [A2: set_o,P: set_o > $o] :
      ( ( member_set_o @ A2 @ ( collect_set_o @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_166_mem__Collect__eq,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,P: set_Ex3793607809372303086nnreal > $o] :
      ( ( member603777416030116741nnreal @ A2 @ ( collec4858231573021281987nnreal @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_167_mem__Collect__eq,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( member_set_real @ A2 @ ( collect_set_real @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_168_Collect__mem__eq,axiom,
    ! [A: set_real_a] :
      ( ( collect_real_a
        @ ^ [X3: real > a] : ( member_real_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
    ! [A: set_o_real] :
      ( ( collect_o_real
        @ ^ [X3: $o > real] : ( member_o_real @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
    ! [A: set_nat_real] :
      ( ( collect_nat_real
        @ ^ [X3: nat > real] : ( member_nat_real @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_171_Collect__mem__eq,axiom,
    ! [A: set_c_d] :
      ( ( collect_c_d
        @ ^ [X3: c > d] : ( member_c_d @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_172_Collect__mem__eq,axiom,
    ! [A: set_a_b] :
      ( ( collect_a_b
        @ ^ [X3: a > b] : ( member_a_b @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_173_Collect__mem__eq,axiom,
    ! [A: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_174_Collect__mem__eq,axiom,
    ! [A: set_set_o] :
      ( ( collect_set_o
        @ ^ [X3: set_o] : ( member_set_o @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_175_Collect__mem__eq,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ( collec4858231573021281987nnreal
        @ ^ [X3: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_176_Collect__mem__eq,axiom,
    ! [A: set_set_real] :
      ( ( collect_set_real
        @ ^ [X3: set_real] : ( member_set_real @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_177_sumE,axiom,
    ! [S2: sum_sum_a_c] :
      ( ! [X: a] :
          ( S2
         != ( sum_Inl_a_c @ X ) )
     => ~ ! [Y3: c] :
            ( S2
           != ( sum_Inr_c_a @ Y3 ) ) ) ).

% sumE
thf(fact_178_sumE,axiom,
    ! [S2: sum_sum_b_d] :
      ( ! [X: b] :
          ( S2
         != ( sum_Inl_b_d @ X ) )
     => ~ ! [Y3: d] :
            ( S2
           != ( sum_Inr_d_b @ Y3 ) ) ) ).

% sumE
thf(fact_179_old_Osum_Oexhaust,axiom,
    ! [Y: sum_sum_a_c] :
      ( ! [A4: a] :
          ( Y
         != ( sum_Inl_a_c @ A4 ) )
     => ~ ! [B3: c] :
            ( Y
           != ( sum_Inr_c_a @ B3 ) ) ) ).

% old.sum.exhaust
thf(fact_180_old_Osum_Oexhaust,axiom,
    ! [Y: sum_sum_b_d] :
      ( ! [A4: b] :
          ( Y
         != ( sum_Inl_b_d @ A4 ) )
     => ~ ! [B3: d] :
            ( Y
           != ( sum_Inr_d_b @ B3 ) ) ) ).

% old.sum.exhaust
thf(fact_181_old_Osum_Odistinct_I1_J,axiom,
    ! [A2: a,B2: c] :
      ( ( sum_Inl_a_c @ A2 )
     != ( sum_Inr_c_a @ B2 ) ) ).

% old.sum.distinct(1)
thf(fact_182_old_Osum_Odistinct_I1_J,axiom,
    ! [A2: b,B2: d] :
      ( ( sum_Inl_b_d @ A2 )
     != ( sum_Inr_d_b @ B2 ) ) ).

% old.sum.distinct(1)
thf(fact_183_old_Osum_Odistinct_I2_J,axiom,
    ! [B2: c,A2: a] :
      ( ( sum_Inr_c_a @ B2 )
     != ( sum_Inl_a_c @ A2 ) ) ).

% old.sum.distinct(2)
thf(fact_184_old_Osum_Odistinct_I2_J,axiom,
    ! [B2: d,A2: b] :
      ( ( sum_Inr_d_b @ B2 )
     != ( sum_Inl_b_d @ A2 ) ) ).

% old.sum.distinct(2)
thf(fact_185_sum_Odistinct_I1_J,axiom,
    ! [X1: a,X22: c] :
      ( ( sum_Inl_a_c @ X1 )
     != ( sum_Inr_c_a @ X22 ) ) ).

% sum.distinct(1)
thf(fact_186_sum_Odistinct_I1_J,axiom,
    ! [X1: b,X22: d] :
      ( ( sum_Inl_b_d @ X1 )
     != ( sum_Inr_d_b @ X22 ) ) ).

% sum.distinct(1)
thf(fact_187_iso__tuple__UNIV__I,axiom,
    ! [X2: real] : ( member_real @ X2 @ top_top_set_real ) ).

% iso_tuple_UNIV_I
thf(fact_188_iso__tuple__UNIV__I,axiom,
    ! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).

% iso_tuple_UNIV_I
thf(fact_189_iso__tuple__UNIV__I,axiom,
    ! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_190_iso__tuple__UNIV__I,axiom,
    ! [X2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X2 @ top_to7994903218803871134nnreal ) ).

% iso_tuple_UNIV_I
thf(fact_191_iso__tuple__UNIV__I,axiom,
    ! [X2: set_nat] : ( member_set_nat @ X2 @ top_top_set_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_192_iso__tuple__UNIV__I,axiom,
    ! [X2: set_o] : ( member_set_o @ X2 @ top_top_set_set_o ) ).

% iso_tuple_UNIV_I
thf(fact_193_iso__tuple__UNIV__I,axiom,
    ! [X2: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X2 @ top_to3356475028079756884nnreal ) ).

% iso_tuple_UNIV_I
thf(fact_194_iso__tuple__UNIV__I,axiom,
    ! [X2: set_real] : ( member_set_real @ X2 @ top_top_set_set_real ) ).

% iso_tuple_UNIV_I
thf(fact_195_iso__tuple__UNIV__I,axiom,
    ! [X2: real > a] : ( member_real_a @ X2 @ top_top_set_real_a ) ).

% iso_tuple_UNIV_I
thf(fact_196_iso__tuple__UNIV__I,axiom,
    ! [X2: $o > real] : ( member_o_real @ X2 @ top_top_set_o_real ) ).

% iso_tuple_UNIV_I
thf(fact_197_obj__sumE,axiom,
    ! [S2: sum_sum_a_c] :
      ( ! [X: a] :
          ( S2
         != ( sum_Inl_a_c @ X ) )
     => ~ ! [X: c] :
            ( S2
           != ( sum_Inr_c_a @ X ) ) ) ).

% obj_sumE
thf(fact_198_obj__sumE,axiom,
    ! [S2: sum_sum_b_d] :
      ( ! [X: b] :
          ( S2
         != ( sum_Inl_b_d @ X ) )
     => ~ ! [X: d] :
            ( S2
           != ( sum_Inr_d_b @ X ) ) ) ).

% obj_sumE
thf(fact_199_copair__qbs__Mx__equiv,axiom,
    binary8286901584692334522Mx_a_c = binary6242423198552412156x2_a_c ).

% copair_qbs_Mx_equiv
thf(fact_200_assms_I1_J,axiom,
    member_a_b @ f @ ( qbs_morphism_a_b @ x @ y ) ).

% assms(1)
thf(fact_201_assms_I2_J,axiom,
    member_c_d @ g @ ( qbs_morphism_c_d @ x2 @ y2 ) ).

% assms(2)
thf(fact_202_Set_Ois__empty__def,axiom,
    ( is_empty_real
    = ( ^ [A5: set_real] : ( A5 = bot_bot_set_real ) ) ) ).

% Set.is_empty_def
thf(fact_203_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A5: set_nat] : ( A5 = bot_bot_set_nat ) ) ) ).

% Set.is_empty_def
thf(fact_204_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A5: set_o] : ( A5 = bot_bot_set_o ) ) ) ).

% Set.is_empty_def
thf(fact_205_Set_Ois__empty__def,axiom,
    ( is_emp182806100662350310nnreal
    = ( ^ [A5: set_Ex3793607809372303086nnreal] : ( A5 = bot_bo4854962954004695426nnreal ) ) ) ).

% Set.is_empty_def
thf(fact_206_qbs__eqI,axiom,
    ! [X5: quasi_borel_a,Y4: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X5 )
        = ( qbs_Mx_a @ Y4 ) )
     => ( X5 = Y4 ) ) ).

% qbs_eqI
thf(fact_207_qbs__eqI,axiom,
    ! [X5: quasi_borel_c,Y4: quasi_borel_c] :
      ( ( ( qbs_Mx_c @ X5 )
        = ( qbs_Mx_c @ Y4 ) )
     => ( X5 = Y4 ) ) ).

% qbs_eqI
thf(fact_208_qbs__eqI,axiom,
    ! [X5: quasi_borel_b,Y4: quasi_borel_b] :
      ( ( ( qbs_Mx_b @ X5 )
        = ( qbs_Mx_b @ Y4 ) )
     => ( X5 = Y4 ) ) ).

% qbs_eqI
thf(fact_209_qbs__eqI,axiom,
    ! [X5: quasi_borel_d,Y4: quasi_borel_d] :
      ( ( ( qbs_Mx_d @ X5 )
        = ( qbs_Mx_d @ Y4 ) )
     => ( X5 = Y4 ) ) ).

% qbs_eqI
thf(fact_210_calculation_I1_J,axiom,
    member_real_b @ f2 @ ( qbs_Mx_b @ y ) ).

% calculation(1)
thf(fact_211_calculation_I2_J,axiom,
    member_real_d @ g2 @ ( qbs_Mx_d @ y2 ) ).

% calculation(2)
thf(fact_212_top__set__def,axiom,
    ( top_top_set_real
    = ( collect_real @ top_top_real_o ) ) ).

% top_set_def
thf(fact_213_top__set__def,axiom,
    ( top_top_set_o
    = ( collect_o @ top_top_o_o ) ) ).

% top_set_def
thf(fact_214_top__set__def,axiom,
    ( top_top_set_nat
    = ( collect_nat @ top_top_nat_o ) ) ).

% top_set_def
thf(fact_215_top__set__def,axiom,
    ( top_to7994903218803871134nnreal
    = ( collec6648975593938027277nnreal @ top_to5118619752887738471real_o ) ) ).

% top_set_def
thf(fact_216_not__arg__cong__Inr,axiom,
    ! [X2: c,Y: c] :
      ( ( X2 != Y )
     => ( ( sum_Inr_c_a @ X2 )
       != ( sum_Inr_c_a @ Y ) ) ) ).

% not_arg_cong_Inr
thf(fact_217_not__arg__cong__Inr,axiom,
    ! [X2: d,Y: d] :
      ( ( X2 != Y )
     => ( ( sum_Inr_d_b @ X2 )
       != ( sum_Inr_d_b @ Y ) ) ) ).

% not_arg_cong_Inr
thf(fact_218_f_H__def,axiom,
    ( f2
    = ( comp_a_b_real @ f @ alpha_1 ) ) ).

% f'_def
thf(fact_219_g_H__def,axiom,
    ( g2
    = ( comp_c_d_real @ g @ alpha_2 ) ) ).

% g'_def
thf(fact_220_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X3: real] : ( member_real @ X3 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_221_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_222_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X3: $o] : ( member_o @ X3 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_223_bot__empty__eq,axiom,
    ( bot_bo412624608084785539real_o
    = ( ^ [X3: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) ) ).

% bot_empty_eq
thf(fact_224_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_225_bot__empty__eq,axiom,
    ( bot_bot_set_o_o
    = ( ^ [X3: set_o] : ( member_set_o @ X3 @ bot_bot_set_set_o ) ) ) ).

% bot_empty_eq
thf(fact_226_bot__empty__eq,axiom,
    ( bot_bo5002694753204610125real_o
    = ( ^ [X3: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X3 @ bot_bo2988155216863113784nnreal ) ) ) ).

% bot_empty_eq
thf(fact_227_bot__empty__eq,axiom,
    ( bot_bot_set_real_o
    = ( ^ [X3: set_real] : ( member_set_real @ X3 @ bot_bot_set_set_real ) ) ) ).

% bot_empty_eq
thf(fact_228_bot__empty__eq,axiom,
    ( bot_bot_real_a_o
    = ( ^ [X3: real > a] : ( member_real_a @ X3 @ bot_bot_set_real_a ) ) ) ).

% bot_empty_eq
thf(fact_229_bot__empty__eq,axiom,
    ( bot_bot_o_real_o
    = ( ^ [X3: $o > real] : ( member_o_real @ X3 @ bot_bot_set_o_real ) ) ) ).

% bot_empty_eq
thf(fact_230_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_231_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_232_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_233_Collect__empty__eq__bot,axiom,
    ! [P: extend8495563244428889912nnreal > $o] :
      ( ( ( collec6648975593938027277nnreal @ P )
        = bot_bo4854962954004695426nnreal )
      = ( P = bot_bo412624608084785539real_o ) ) ).

% Collect_empty_eq_bot
thf(fact_234_copair__qbs__Mx,axiom,
    ! [X5: quasi_borel_a,Y4: quasi_borel_c] :
      ( ( qbs_Mx_Sum_sum_a_c @ ( binary8555328655094383375bs_a_c @ X5 @ Y4 ) )
      = ( binary8286901584692334522Mx_a_c @ X5 @ Y4 ) ) ).

% copair_qbs_Mx
thf(fact_235_eqb__Mx,axiom,
    ( ( qbs_Mx_a @ empty_quasi_borel_a )
    = bot_bot_set_real_a ) ).

% eqb_Mx
thf(fact_236_eqb__Mx,axiom,
    ( ( qbs_Mx_c @ empty_quasi_borel_c )
    = bot_bot_set_real_c ) ).

% eqb_Mx
thf(fact_237_eqb__Mx,axiom,
    ( ( qbs_Mx_b @ empty_quasi_borel_b )
    = bot_bot_set_real_b ) ).

% eqb_Mx
thf(fact_238_eqb__Mx,axiom,
    ( ( qbs_Mx_d @ empty_quasi_borel_d )
    = bot_bot_set_real_d ) ).

% eqb_Mx
thf(fact_239_calculation_I3_J,axiom,
    ( ( comp_S4176151853342328607d_real @ ( sum_map_sum_a_b_c_d @ f @ g ) @ alpha )
    = ( ^ [R: real] : ( if_Sum_sum_b_d @ ( member_real @ R @ s ) @ ( sum_Inl_b_d @ ( f2 @ R ) ) @ ( sum_Inr_d_b @ ( g2 @ R ) ) ) ) ) ).

% calculation(3)
thf(fact_240_sum__set__simps_I3_J,axiom,
    ! [X2: a] :
      ( ( basic_setr_a_c @ ( sum_Inl_a_c @ X2 ) )
      = bot_bot_set_c ) ).

% sum_set_simps(3)
thf(fact_241_sum__set__simps_I3_J,axiom,
    ! [X2: b] :
      ( ( basic_setr_b_d @ ( sum_Inl_b_d @ X2 ) )
      = bot_bot_set_d ) ).

% sum_set_simps(3)
thf(fact_242_sum__set__simps_I2_J,axiom,
    ! [X2: c] :
      ( ( basic_setl_a_c @ ( sum_Inr_c_a @ X2 ) )
      = bot_bot_set_a ) ).

% sum_set_simps(2)
thf(fact_243_sum__set__simps_I2_J,axiom,
    ! [X2: d] :
      ( ( basic_setl_b_d @ ( sum_Inr_d_b @ X2 ) )
      = bot_bot_set_b ) ).

% sum_set_simps(2)
thf(fact_244_top__empty__eq,axiom,
    ( top_top_real_o
    = ( ^ [X3: real] : ( member_real @ X3 @ top_top_set_real ) ) ) ).

% top_empty_eq
thf(fact_245_top__empty__eq,axiom,
    ( top_top_o_o
    = ( ^ [X3: $o] : ( member_o @ X3 @ top_top_set_o ) ) ) ).

% top_empty_eq
thf(fact_246_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_247_top__empty__eq,axiom,
    ( top_to5118619752887738471real_o
    = ( ^ [X3: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ X3 @ top_to7994903218803871134nnreal ) ) ) ).

% top_empty_eq
thf(fact_248_top__empty__eq,axiom,
    ( top_top_set_nat_o
    = ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ top_top_set_set_nat ) ) ) ).

% top_empty_eq
thf(fact_249_top__empty__eq,axiom,
    ( top_top_set_o_o
    = ( ^ [X3: set_o] : ( member_set_o @ X3 @ top_top_set_set_o ) ) ) ).

% top_empty_eq
thf(fact_250_top__empty__eq,axiom,
    ( top_to5272770551662541617real_o
    = ( ^ [X3: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ X3 @ top_to3356475028079756884nnreal ) ) ) ).

% top_empty_eq
thf(fact_251_top__empty__eq,axiom,
    ( top_top_set_real_o
    = ( ^ [X3: set_real] : ( member_set_real @ X3 @ top_top_set_set_real ) ) ) ).

% top_empty_eq
thf(fact_252_top__empty__eq,axiom,
    ( top_top_real_a_o
    = ( ^ [X3: real > a] : ( member_real_a @ X3 @ top_top_set_real_a ) ) ) ).

% top_empty_eq
thf(fact_253_top__empty__eq,axiom,
    ( top_top_o_real_o
    = ( ^ [X3: $o > real] : ( member_o_real @ X3 @ top_top_set_o_real ) ) ) ).

% top_empty_eq
thf(fact_254_map__sum_Ocomp,axiom,
    ! [F2: real > real,G: real > real,H: real > real,I2: real > real] :
      ( ( comp_S4948336051826200842l_real @ ( sum_ma9028575376852974268l_real @ F2 @ G ) @ ( sum_ma9028575376852974268l_real @ H @ I2 ) )
      = ( sum_ma9028575376852974268l_real @ ( comp_real_real_real @ F2 @ H ) @ ( comp_real_real_real @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_255_map__sum_Ocomp,axiom,
    ! [F2: real > real,G: real > $o,H: real > real,I2: $o > real] :
      ( ( comp_S5772965672812085466real_o @ ( sum_ma4609325004622031752real_o @ F2 @ G ) @ ( sum_ma3100260629580862178o_real @ H @ I2 ) )
      = ( sum_ma5398380231479159522al_o_o @ ( comp_real_real_real @ F2 @ H ) @ ( comp_real_o_o @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_256_map__sum_Ocomp,axiom,
    ! [F2: real > real,G: real > nat,H: real > real,I2: nat > real] :
      ( ( comp_S1631106420198528594al_nat @ ( sum_ma8618515792258342880al_nat @ F2 @ G ) @ ( sum_ma4051207219129444192t_real @ H @ I2 ) )
      = ( sum_ma5138984332203957892at_nat @ ( comp_real_real_real @ F2 @ H ) @ ( comp_real_nat_nat @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_257_map__sum_Ocomp,axiom,
    ! [F2: real > real,G: c > d,H: real > real,I2: real > c] :
      ( ( comp_S8379285214102871771l_real @ ( sum_ma5455772731263491683al_c_d @ F2 @ G ) @ ( sum_ma1303030344635120368real_c @ H @ I2 ) )
      = ( sum_ma1303030344635120369real_d @ ( comp_real_real_real @ F2 @ H ) @ ( comp_c_d_real @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_258_map__sum_Ocomp,axiom,
    ! [F2: real > real,G: a > b,H: real > real,I2: real > a] :
      ( ( comp_S5849561226221799003l_real @ ( sum_ma1807311858354067939al_a_b @ F2 @ G ) @ ( sum_ma1303030344635120366real_a @ H @ I2 ) )
      = ( sum_ma1303030344635120367real_b @ ( comp_real_real_real @ F2 @ H ) @ ( comp_a_b_real @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_259_map__sum_Ocomp,axiom,
    ! [F2: real > $o,G: real > real,H: $o > real,I2: real > real] :
      ( ( comp_S4972280585966916698o_real @ ( sum_ma5402501128500593980l_real @ F2 @ G ) @ ( sum_ma7581390967160423062l_real @ H @ I2 ) )
      = ( sum_ma4692674304805643746l_real @ ( comp_real_o_o @ F2 @ H ) @ ( comp_real_real_real @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_260_map__sum_Ocomp,axiom,
    ! [F2: real > $o,G: real > $o,H: $o > real,I2: $o > real] :
      ( ( comp_S7546904374943229414um_o_o @ ( sum_ma8868230608733716744real_o @ F2 @ G ) @ ( sum_ma2390433805885911944o_real @ H @ I2 ) )
      = ( sum_map_sum_o_o_o_o @ ( comp_real_o_o @ F2 @ H ) @ ( comp_real_o_o @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_261_map__sum_Ocomp,axiom,
    ! [F2: real > $o,G: real > nat,H: $o > real,I2: nat > real] :
      ( ( comp_S4704009564200261978_o_nat @ ( sum_ma3912033132148697696al_nat @ F2 @ G ) @ ( sum_ma9203121963192613946t_real @ H @ I2 ) )
      = ( sum_ma344318232541114282at_nat @ ( comp_real_o_o @ F2 @ H ) @ ( comp_real_nat_nat @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_262_map__sum_Ocomp,axiom,
    ! [F2: real > $o,G: c > d,H: $o > real,I2: real > c] :
      ( ( comp_S8369131124857326909o_real @ ( sum_ma8893391999227668707_o_c_d @ F2 @ G ) @ ( sum_ma6566039659836893334real_c @ H @ I2 ) )
      = ( sum_ma2721287238701537867real_d @ ( comp_real_o_o @ F2 @ H ) @ ( comp_c_d_real @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_263_map__sum_Ocomp,axiom,
    ! [F2: real > $o,G: a > b,H: $o > real,I2: real > a] :
      ( ( comp_S1583458553065613633o_real @ ( sum_ma5244931126318244963_o_a_b @ F2 @ G ) @ ( sum_ma6566039659836893332real_a @ H @ I2 ) )
      = ( sum_ma2721287238701537865real_b @ ( comp_real_o_o @ F2 @ H ) @ ( comp_a_b_real @ G @ I2 ) ) ) ).

% map_sum.comp
thf(fact_264_map__sum_Ocompositionality,axiom,
    ! [F2: real > real,G: real > real,H: real > real,I2: real > real,Sum: sum_sum_real_real] :
      ( ( sum_ma9028575376852974268l_real @ F2 @ G @ ( sum_ma9028575376852974268l_real @ H @ I2 @ Sum ) )
      = ( sum_ma9028575376852974268l_real @ ( comp_real_real_real @ F2 @ H ) @ ( comp_real_real_real @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_265_map__sum_Ocompositionality,axiom,
    ! [F2: real > real,G: real > $o,H: real > real,I2: $o > real,Sum: sum_sum_real_o] :
      ( ( sum_ma4609325004622031752real_o @ F2 @ G @ ( sum_ma3100260629580862178o_real @ H @ I2 @ Sum ) )
      = ( sum_ma5398380231479159522al_o_o @ ( comp_real_real_real @ F2 @ H ) @ ( comp_real_o_o @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_266_map__sum_Ocompositionality,axiom,
    ! [F2: real > real,G: real > nat,H: real > real,I2: nat > real,Sum: sum_sum_real_nat] :
      ( ( sum_ma8618515792258342880al_nat @ F2 @ G @ ( sum_ma4051207219129444192t_real @ H @ I2 @ Sum ) )
      = ( sum_ma5138984332203957892at_nat @ ( comp_real_real_real @ F2 @ H ) @ ( comp_real_nat_nat @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_267_map__sum_Ocompositionality,axiom,
    ! [F2: real > real,G: c > d,H: real > real,I2: real > c,Sum: sum_sum_real_real] :
      ( ( sum_ma5455772731263491683al_c_d @ F2 @ G @ ( sum_ma1303030344635120368real_c @ H @ I2 @ Sum ) )
      = ( sum_ma1303030344635120369real_d @ ( comp_real_real_real @ F2 @ H ) @ ( comp_c_d_real @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_268_map__sum_Ocompositionality,axiom,
    ! [F2: real > real,G: a > b,H: real > real,I2: real > a,Sum: sum_sum_real_real] :
      ( ( sum_ma1807311858354067939al_a_b @ F2 @ G @ ( sum_ma1303030344635120366real_a @ H @ I2 @ Sum ) )
      = ( sum_ma1303030344635120367real_b @ ( comp_real_real_real @ F2 @ H ) @ ( comp_a_b_real @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_269_map__sum_Ocompositionality,axiom,
    ! [F2: real > $o,G: real > real,H: $o > real,I2: real > real,Sum: sum_sum_o_real] :
      ( ( sum_ma5402501128500593980l_real @ F2 @ G @ ( sum_ma7581390967160423062l_real @ H @ I2 @ Sum ) )
      = ( sum_ma4692674304805643746l_real @ ( comp_real_o_o @ F2 @ H ) @ ( comp_real_real_real @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_270_map__sum_Ocompositionality,axiom,
    ! [F2: real > $o,G: real > $o,H: $o > real,I2: $o > real,Sum: sum_sum_o_o] :
      ( ( sum_ma8868230608733716744real_o @ F2 @ G @ ( sum_ma2390433805885911944o_real @ H @ I2 @ Sum ) )
      = ( sum_map_sum_o_o_o_o @ ( comp_real_o_o @ F2 @ H ) @ ( comp_real_o_o @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_271_map__sum_Ocompositionality,axiom,
    ! [F2: real > $o,G: real > nat,H: $o > real,I2: nat > real,Sum: sum_sum_o_nat] :
      ( ( sum_ma3912033132148697696al_nat @ F2 @ G @ ( sum_ma9203121963192613946t_real @ H @ I2 @ Sum ) )
      = ( sum_ma344318232541114282at_nat @ ( comp_real_o_o @ F2 @ H ) @ ( comp_real_nat_nat @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_272_map__sum_Ocompositionality,axiom,
    ! [F2: real > $o,G: c > d,H: $o > real,I2: real > c,Sum: sum_sum_o_real] :
      ( ( sum_ma8893391999227668707_o_c_d @ F2 @ G @ ( sum_ma6566039659836893334real_c @ H @ I2 @ Sum ) )
      = ( sum_ma2721287238701537867real_d @ ( comp_real_o_o @ F2 @ H ) @ ( comp_c_d_real @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_273_map__sum_Ocompositionality,axiom,
    ! [F2: real > $o,G: a > b,H: $o > real,I2: real > a,Sum: sum_sum_o_real] :
      ( ( sum_ma5244931126318244963_o_a_b @ F2 @ G @ ( sum_ma6566039659836893332real_a @ H @ I2 @ Sum ) )
      = ( sum_ma2721287238701537865real_b @ ( comp_real_o_o @ F2 @ H ) @ ( comp_a_b_real @ G @ I2 ) @ Sum ) ) ).

% map_sum.compositionality
thf(fact_274_rewriteL__comp__comp,axiom,
    ! [F2: $o > $o,G: real > $o,L: real > $o,H: $o > real] :
      ( ( ( comp_o_o_real @ F2 @ G )
        = L )
     => ( ( comp_o_o_o @ F2 @ ( comp_real_o_o @ G @ H ) )
        = ( comp_real_o_o @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_275_rewriteL__comp__comp,axiom,
    ! [F2: nat > nat,G: real > nat,L: real > nat,H: nat > real] :
      ( ( ( comp_nat_nat_real @ F2 @ G )
        = L )
     => ( ( comp_nat_nat_nat @ F2 @ ( comp_real_nat_nat @ G @ H ) )
        = ( comp_real_nat_nat @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_276_rewriteL__comp__comp,axiom,
    ! [F2: d > d,G: c > d,L: c > d,H: real > c] :
      ( ( ( comp_d_d_c @ F2 @ G )
        = L )
     => ( ( comp_d_d_real @ F2 @ ( comp_c_d_real @ G @ H ) )
        = ( comp_c_d_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_277_rewriteL__comp__comp,axiom,
    ! [F2: b > b,G: a > b,L: a > b,H: real > a] :
      ( ( ( comp_b_b_a @ F2 @ G )
        = L )
     => ( ( comp_b_b_real @ F2 @ ( comp_a_b_real @ G @ H ) )
        = ( comp_a_b_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_278_rewriteL__comp__comp,axiom,
    ! [F2: real > $o,G: real > real,L: real > $o,H: $o > real] :
      ( ( ( comp_real_o_real @ F2 @ G )
        = L )
     => ( ( comp_real_o_o @ F2 @ ( comp_real_real_o @ G @ H ) )
        = ( comp_real_o_o @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_279_rewriteL__comp__comp,axiom,
    ! [F2: real > nat,G: real > real,L: real > nat,H: nat > real] :
      ( ( ( comp_real_nat_real @ F2 @ G )
        = L )
     => ( ( comp_real_nat_nat @ F2 @ ( comp_real_real_nat @ G @ H ) )
        = ( comp_real_nat_nat @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_280_rewriteL__comp__comp,axiom,
    ! [F2: c > d,G: c > c,L: c > d,H: real > c] :
      ( ( ( comp_c_d_c @ F2 @ G )
        = L )
     => ( ( comp_c_d_real @ F2 @ ( comp_c_c_real @ G @ H ) )
        = ( comp_c_d_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_281_rewriteL__comp__comp,axiom,
    ! [F2: a > b,G: a > a,L: a > b,H: real > a] :
      ( ( ( comp_a_b_a @ F2 @ G )
        = L )
     => ( ( comp_a_b_real @ F2 @ ( comp_a_a_real @ G @ H ) )
        = ( comp_a_b_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_282_rewriteL__comp__comp,axiom,
    ! [F2: real > real,G: real > real,L: real > real,H: real > real] :
      ( ( ( comp_real_real_real @ F2 @ G )
        = L )
     => ( ( comp_real_real_real @ F2 @ ( comp_real_real_real @ G @ H ) )
        = ( comp_real_real_real @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_283_rewriteL__comp__comp,axiom,
    ! [F2: real > $o,G: $o > real,L: $o > $o,H: $o > $o] :
      ( ( ( comp_real_o_o @ F2 @ G )
        = L )
     => ( ( comp_real_o_o @ F2 @ ( comp_o_real_o @ G @ H ) )
        = ( comp_o_o_o @ L @ H ) ) ) ).

% rewriteL_comp_comp
thf(fact_284_rewriteR__comp__comp,axiom,
    ! [G: $o > real,H: $o > $o,R2: $o > real,F2: real > $o] :
      ( ( ( comp_o_real_o @ G @ H )
        = R2 )
     => ( ( comp_o_o_o @ ( comp_real_o_o @ F2 @ G ) @ H )
        = ( comp_real_o_o @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_285_rewriteR__comp__comp,axiom,
    ! [G: nat > real,H: nat > nat,R2: nat > real,F2: real > nat] :
      ( ( ( comp_nat_real_nat @ G @ H )
        = R2 )
     => ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F2 @ G ) @ H )
        = ( comp_real_nat_nat @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_286_rewriteR__comp__comp,axiom,
    ! [G: real > c,H: real > real,R2: real > c,F2: c > d] :
      ( ( ( comp_real_c_real @ G @ H )
        = R2 )
     => ( ( comp_real_d_real @ ( comp_c_d_real @ F2 @ G ) @ H )
        = ( comp_c_d_real @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_287_rewriteR__comp__comp,axiom,
    ! [G: real > a,H: real > real,R2: real > a,F2: a > b] :
      ( ( ( comp_real_a_real @ G @ H )
        = R2 )
     => ( ( comp_real_b_real @ ( comp_a_b_real @ F2 @ G ) @ H )
        = ( comp_a_b_real @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_288_rewriteR__comp__comp,axiom,
    ! [G: real > real,H: $o > real,R2: $o > real,F2: real > $o] :
      ( ( ( comp_real_real_o @ G @ H )
        = R2 )
     => ( ( comp_real_o_o @ ( comp_real_o_real @ F2 @ G ) @ H )
        = ( comp_real_o_o @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_289_rewriteR__comp__comp,axiom,
    ! [G: real > real,H: nat > real,R2: nat > real,F2: real > nat] :
      ( ( ( comp_real_real_nat @ G @ H )
        = R2 )
     => ( ( comp_real_nat_nat @ ( comp_real_nat_real @ F2 @ G ) @ H )
        = ( comp_real_nat_nat @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_290_rewriteR__comp__comp,axiom,
    ! [G: c > c,H: real > c,R2: real > c,F2: c > d] :
      ( ( ( comp_c_c_real @ G @ H )
        = R2 )
     => ( ( comp_c_d_real @ ( comp_c_d_c @ F2 @ G ) @ H )
        = ( comp_c_d_real @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_291_rewriteR__comp__comp,axiom,
    ! [G: a > a,H: real > a,R2: real > a,F2: a > b] :
      ( ( ( comp_a_a_real @ G @ H )
        = R2 )
     => ( ( comp_a_b_real @ ( comp_a_b_a @ F2 @ G ) @ H )
        = ( comp_a_b_real @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_292_rewriteR__comp__comp,axiom,
    ! [G: real > real,H: real > real,R2: real > real,F2: real > real] :
      ( ( ( comp_real_real_real @ G @ H )
        = R2 )
     => ( ( comp_real_real_real @ ( comp_real_real_real @ F2 @ G ) @ H )
        = ( comp_real_real_real @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_293_rewriteR__comp__comp,axiom,
    ! [G: real > $o,H: $o > real,R2: $o > $o,F2: $o > $o] :
      ( ( ( comp_real_o_o @ G @ H )
        = R2 )
     => ( ( comp_real_o_o @ ( comp_o_o_real @ F2 @ G ) @ H )
        = ( comp_o_o_o @ F2 @ R2 ) ) ) ).

% rewriteR_comp_comp
thf(fact_294_rewriteL__comp__comp2,axiom,
    ! [F2: $o > $o,G: real > $o,L1: real > $o,L2: real > real,H: $o > real,R2: $o > real] :
      ( ( ( comp_o_o_real @ F2 @ G )
        = ( comp_real_o_real @ L1 @ L2 ) )
     => ( ( ( comp_real_real_o @ L2 @ H )
          = R2 )
       => ( ( comp_o_o_o @ F2 @ ( comp_real_o_o @ G @ H ) )
          = ( comp_real_o_o @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_295_rewriteL__comp__comp2,axiom,
    ! [F2: nat > nat,G: real > nat,L1: real > nat,L2: real > real,H: nat > real,R2: nat > real] :
      ( ( ( comp_nat_nat_real @ F2 @ G )
        = ( comp_real_nat_real @ L1 @ L2 ) )
     => ( ( ( comp_real_real_nat @ L2 @ H )
          = R2 )
       => ( ( comp_nat_nat_nat @ F2 @ ( comp_real_nat_nat @ G @ H ) )
          = ( comp_real_nat_nat @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_296_rewriteL__comp__comp2,axiom,
    ! [F2: d > real,G: c > d,L1: real > real,L2: c > real,H: real > c,R2: real > real] :
      ( ( ( comp_d_real_c @ F2 @ G )
        = ( comp_real_real_c @ L1 @ L2 ) )
     => ( ( ( comp_c_real_real @ L2 @ H )
          = R2 )
       => ( ( comp_d_real_real @ F2 @ ( comp_c_d_real @ G @ H ) )
          = ( comp_real_real_real @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_297_rewriteL__comp__comp2,axiom,
    ! [F2: d > d,G: c > d,L1: c > d,L2: c > c,H: real > c,R2: real > c] :
      ( ( ( comp_d_d_c @ F2 @ G )
        = ( comp_c_d_c @ L1 @ L2 ) )
     => ( ( ( comp_c_c_real @ L2 @ H )
          = R2 )
       => ( ( comp_d_d_real @ F2 @ ( comp_c_d_real @ G @ H ) )
          = ( comp_c_d_real @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_298_rewriteL__comp__comp2,axiom,
    ! [F2: d > b,G: c > d,L1: a > b,L2: c > a,H: real > c,R2: real > a] :
      ( ( ( comp_d_b_c @ F2 @ G )
        = ( comp_a_b_c @ L1 @ L2 ) )
     => ( ( ( comp_c_a_real @ L2 @ H )
          = R2 )
       => ( ( comp_d_b_real @ F2 @ ( comp_c_d_real @ G @ H ) )
          = ( comp_a_b_real @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_299_rewriteL__comp__comp2,axiom,
    ! [F2: b > real,G: a > b,L1: real > real,L2: a > real,H: real > a,R2: real > real] :
      ( ( ( comp_b_real_a @ F2 @ G )
        = ( comp_real_real_a @ L1 @ L2 ) )
     => ( ( ( comp_a_real_real @ L2 @ H )
          = R2 )
       => ( ( comp_b_real_real @ F2 @ ( comp_a_b_real @ G @ H ) )
          = ( comp_real_real_real @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_300_rewriteL__comp__comp2,axiom,
    ! [F2: b > d,G: a > b,L1: c > d,L2: a > c,H: real > a,R2: real > c] :
      ( ( ( comp_b_d_a @ F2 @ G )
        = ( comp_c_d_a @ L1 @ L2 ) )
     => ( ( ( comp_a_c_real @ L2 @ H )
          = R2 )
       => ( ( comp_b_d_real @ F2 @ ( comp_a_b_real @ G @ H ) )
          = ( comp_c_d_real @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_301_rewriteL__comp__comp2,axiom,
    ! [F2: b > b,G: a > b,L1: a > b,L2: a > a,H: real > a,R2: real > a] :
      ( ( ( comp_b_b_a @ F2 @ G )
        = ( comp_a_b_a @ L1 @ L2 ) )
     => ( ( ( comp_a_a_real @ L2 @ H )
          = R2 )
       => ( ( comp_b_b_real @ F2 @ ( comp_a_b_real @ G @ H ) )
          = ( comp_a_b_real @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_302_rewriteL__comp__comp2,axiom,
    ! [F2: real > $o,G: real > real,L1: $o > $o,L2: real > $o,H: $o > real,R2: $o > $o] :
      ( ( ( comp_real_o_real @ F2 @ G )
        = ( comp_o_o_real @ L1 @ L2 ) )
     => ( ( ( comp_real_o_o @ L2 @ H )
          = R2 )
       => ( ( comp_real_o_o @ F2 @ ( comp_real_real_o @ G @ H ) )
          = ( comp_o_o_o @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_303_rewriteL__comp__comp2,axiom,
    ! [F2: real > nat,G: real > real,L1: nat > nat,L2: real > nat,H: nat > real,R2: nat > nat] :
      ( ( ( comp_real_nat_real @ F2 @ G )
        = ( comp_nat_nat_real @ L1 @ L2 ) )
     => ( ( ( comp_real_nat_nat @ L2 @ H )
          = R2 )
       => ( ( comp_real_nat_nat @ F2 @ ( comp_real_real_nat @ G @ H ) )
          = ( comp_nat_nat_nat @ L1 @ R2 ) ) ) ) ).

% rewriteL_comp_comp2
thf(fact_304_rewriteR__comp__comp2,axiom,
    ! [G: $o > real,H: $o > $o,R1: real > real,R22: $o > real,F2: real > $o,L: real > $o] :
      ( ( ( comp_o_real_o @ G @ H )
        = ( comp_real_real_o @ R1 @ R22 ) )
     => ( ( ( comp_real_o_real @ F2 @ R1 )
          = L )
       => ( ( comp_o_o_o @ ( comp_real_o_o @ F2 @ G ) @ H )
          = ( comp_real_o_o @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_305_rewriteR__comp__comp2,axiom,
    ! [G: nat > real,H: nat > nat,R1: real > real,R22: nat > real,F2: real > nat,L: real > nat] :
      ( ( ( comp_nat_real_nat @ G @ H )
        = ( comp_real_real_nat @ R1 @ R22 ) )
     => ( ( ( comp_real_nat_real @ F2 @ R1 )
          = L )
       => ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F2 @ G ) @ H )
          = ( comp_real_nat_nat @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_306_rewriteR__comp__comp2,axiom,
    ! [G: real > c,H: real > real,R1: c > c,R22: real > c,F2: c > d,L: c > d] :
      ( ( ( comp_real_c_real @ G @ H )
        = ( comp_c_c_real @ R1 @ R22 ) )
     => ( ( ( comp_c_d_c @ F2 @ R1 )
          = L )
       => ( ( comp_real_d_real @ ( comp_c_d_real @ F2 @ G ) @ H )
          = ( comp_c_d_real @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_307_rewriteR__comp__comp2,axiom,
    ! [G: real > a,H: real > real,R1: a > a,R22: real > a,F2: a > b,L: a > b] :
      ( ( ( comp_real_a_real @ G @ H )
        = ( comp_a_a_real @ R1 @ R22 ) )
     => ( ( ( comp_a_b_a @ F2 @ R1 )
          = L )
       => ( ( comp_real_b_real @ ( comp_a_b_real @ F2 @ G ) @ H )
          = ( comp_a_b_real @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_308_rewriteR__comp__comp2,axiom,
    ! [G: real > real,H: $o > real,R1: $o > real,R22: $o > $o,F2: real > $o,L: $o > $o] :
      ( ( ( comp_real_real_o @ G @ H )
        = ( comp_o_real_o @ R1 @ R22 ) )
     => ( ( ( comp_real_o_o @ F2 @ R1 )
          = L )
       => ( ( comp_real_o_o @ ( comp_real_o_real @ F2 @ G ) @ H )
          = ( comp_o_o_o @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_309_rewriteR__comp__comp2,axiom,
    ! [G: real > real,H: nat > real,R1: nat > real,R22: nat > nat,F2: real > nat,L: nat > nat] :
      ( ( ( comp_real_real_nat @ G @ H )
        = ( comp_nat_real_nat @ R1 @ R22 ) )
     => ( ( ( comp_real_nat_nat @ F2 @ R1 )
          = L )
       => ( ( comp_real_nat_nat @ ( comp_real_nat_real @ F2 @ G ) @ H )
          = ( comp_nat_nat_nat @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_310_rewriteR__comp__comp2,axiom,
    ! [G: c > c,H: real > c,R1: real > c,R22: real > real,F2: c > d,L: real > d] :
      ( ( ( comp_c_c_real @ G @ H )
        = ( comp_real_c_real @ R1 @ R22 ) )
     => ( ( ( comp_c_d_real @ F2 @ R1 )
          = L )
       => ( ( comp_c_d_real @ ( comp_c_d_c @ F2 @ G ) @ H )
          = ( comp_real_d_real @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_311_rewriteR__comp__comp2,axiom,
    ! [G: a > a,H: real > a,R1: real > a,R22: real > real,F2: a > b,L: real > b] :
      ( ( ( comp_a_a_real @ G @ H )
        = ( comp_real_a_real @ R1 @ R22 ) )
     => ( ( ( comp_a_b_real @ F2 @ R1 )
          = L )
       => ( ( comp_a_b_real @ ( comp_a_b_a @ F2 @ G ) @ H )
          = ( comp_real_b_real @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_312_rewriteR__comp__comp2,axiom,
    ! [G: $o > real,H: real > $o,R1: real > real,R22: real > real,F2: real > $o,L: real > $o] :
      ( ( ( comp_o_real_real @ G @ H )
        = ( comp_real_real_real @ R1 @ R22 ) )
     => ( ( ( comp_real_o_real @ F2 @ R1 )
          = L )
       => ( ( comp_o_o_real @ ( comp_real_o_o @ F2 @ G ) @ H )
          = ( comp_real_o_real @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_313_rewriteR__comp__comp2,axiom,
    ! [G: nat > real,H: real > nat,R1: real > real,R22: real > real,F2: real > nat,L: real > nat] :
      ( ( ( comp_nat_real_real @ G @ H )
        = ( comp_real_real_real @ R1 @ R22 ) )
     => ( ( ( comp_real_nat_real @ F2 @ R1 )
          = L )
       => ( ( comp_nat_nat_real @ ( comp_real_nat_nat @ F2 @ G ) @ H )
          = ( comp_real_nat_real @ L @ R22 ) ) ) ) ).

% rewriteR_comp_comp2
thf(fact_314_qbs__morphism__comp,axiom,
    ! [F2: real > real,X5: quasi_borel_real,Y4: quasi_borel_real,G: real > real,Z: quasi_borel_real] :
      ( ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X5 @ Y4 ) )
     => ( ( member_real_real @ G @ ( qbs_mo5229770564518008146l_real @ Y4 @ Z ) )
       => ( member_real_real @ ( comp_real_real_real @ G @ F2 ) @ ( qbs_mo5229770564518008146l_real @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_315_qbs__morphism__comp,axiom,
    ! [F2: real > real,X5: quasi_borel_real,Y4: quasi_borel_real,G: real > a,Z: quasi_borel_a] :
      ( ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X5 @ Y4 ) )
     => ( ( member_real_a @ G @ ( qbs_morphism_real_a @ Y4 @ Z ) )
       => ( member_real_a @ ( comp_real_a_real @ G @ F2 ) @ ( qbs_morphism_real_a @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_316_qbs__morphism__comp,axiom,
    ! [F2: $o > $o,X5: quasi_borel_o,Y4: quasi_borel_o,G: $o > real,Z: quasi_borel_real] :
      ( ( member_o_o @ F2 @ ( qbs_morphism_o_o @ X5 @ Y4 ) )
     => ( ( member_o_real @ G @ ( qbs_morphism_o_real @ Y4 @ Z ) )
       => ( member_o_real @ ( comp_o_real_o @ G @ F2 ) @ ( qbs_morphism_o_real @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_317_qbs__morphism__comp,axiom,
    ! [F2: nat > $o,X5: quasi_borel_nat,Y4: quasi_borel_o,G: $o > real,Z: quasi_borel_real] :
      ( ( member_nat_o @ F2 @ ( qbs_morphism_nat_o @ X5 @ Y4 ) )
     => ( ( member_o_real @ G @ ( qbs_morphism_o_real @ Y4 @ Z ) )
       => ( member_nat_real @ ( comp_o_real_nat @ G @ F2 ) @ ( qbs_mo2000642995705457910t_real @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_318_qbs__morphism__comp,axiom,
    ! [F2: $o > nat,X5: quasi_borel_o,Y4: quasi_borel_nat,G: nat > real,Z: quasi_borel_real] :
      ( ( member_o_nat @ F2 @ ( qbs_morphism_o_nat @ X5 @ Y4 ) )
     => ( ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z ) )
       => ( member_o_real @ ( comp_nat_real_o @ G @ F2 ) @ ( qbs_morphism_o_real @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_319_qbs__morphism__comp,axiom,
    ! [F2: nat > nat,X5: quasi_borel_nat,Y4: quasi_borel_nat,G: nat > real,Z: quasi_borel_real] :
      ( ( member_nat_nat @ F2 @ ( qbs_morphism_nat_nat @ X5 @ Y4 ) )
     => ( ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z ) )
       => ( member_nat_real @ ( comp_nat_real_nat @ G @ F2 ) @ ( qbs_mo2000642995705457910t_real @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_320_qbs__morphism__comp,axiom,
    ! [F2: real > a,X5: quasi_borel_real,Y4: quasi_borel_a,G: a > a,Z: quasi_borel_a] :
      ( ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X5 @ Y4 ) )
     => ( ( member_a_a @ G @ ( qbs_morphism_a_a @ Y4 @ Z ) )
       => ( member_real_a @ ( comp_a_a_real @ G @ F2 ) @ ( qbs_morphism_real_a @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_321_qbs__morphism__comp,axiom,
    ! [F2: $o > real,X5: quasi_borel_o,Y4: quasi_borel_real,G: real > $o,Z: quasi_borel_o] :
      ( ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X5 @ Y4 ) )
     => ( ( member_real_o @ G @ ( qbs_morphism_real_o @ Y4 @ Z ) )
       => ( member_o_o @ ( comp_real_o_o @ G @ F2 ) @ ( qbs_morphism_o_o @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_322_qbs__morphism__comp,axiom,
    ! [F2: $o > real,X5: quasi_borel_o,Y4: quasi_borel_real,G: real > real,Z: quasi_borel_real] :
      ( ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X5 @ Y4 ) )
     => ( ( member_real_real @ G @ ( qbs_mo5229770564518008146l_real @ Y4 @ Z ) )
       => ( member_o_real @ ( comp_real_real_o @ G @ F2 ) @ ( qbs_morphism_o_real @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_323_qbs__morphism__comp,axiom,
    ! [F2: $o > real,X5: quasi_borel_o,Y4: quasi_borel_real,G: real > a,Z: quasi_borel_a] :
      ( ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X5 @ Y4 ) )
     => ( ( member_real_a @ G @ ( qbs_morphism_real_a @ Y4 @ Z ) )
       => ( member_o_a @ ( comp_real_a_o @ G @ F2 ) @ ( qbs_morphism_o_a @ X5 @ Z ) ) ) ) ).

% qbs_morphism_comp
thf(fact_324_map__sum__if__distrib__then_I2_J,axiom,
    ! [E: $o,F2: a > a,G: c > c,X2: c,Y: sum_sum_a_c] :
      ( ( E
       => ( ( sum_map_sum_a_a_c_c @ F2 @ G @ ( if_Sum_sum_a_c @ E @ ( sum_Inr_c_a @ X2 ) @ Y ) )
          = ( sum_Inr_c_a @ ( G @ X2 ) ) ) )
      & ( ~ E
       => ( ( sum_map_sum_a_a_c_c @ F2 @ G @ ( if_Sum_sum_a_c @ E @ ( sum_Inr_c_a @ X2 ) @ Y ) )
          = ( sum_map_sum_a_a_c_c @ F2 @ G @ Y ) ) ) ) ).

% map_sum_if_distrib_then(2)
thf(fact_325_map__sum__if__distrib__then_I2_J,axiom,
    ! [E: $o,F2: b > a,G: d > c,X2: d,Y: sum_sum_b_d] :
      ( ( E
       => ( ( sum_map_sum_b_a_d_c @ F2 @ G @ ( if_Sum_sum_b_d @ E @ ( sum_Inr_d_b @ X2 ) @ Y ) )
          = ( sum_Inr_c_a @ ( G @ X2 ) ) ) )
      & ( ~ E
       => ( ( sum_map_sum_b_a_d_c @ F2 @ G @ ( if_Sum_sum_b_d @ E @ ( sum_Inr_d_b @ X2 ) @ Y ) )
          = ( sum_map_sum_b_a_d_c @ F2 @ G @ Y ) ) ) ) ).

% map_sum_if_distrib_then(2)
thf(fact_326_map__sum__if__distrib__then_I2_J,axiom,
    ! [E: $o,F2: b > b,G: d > d,X2: d,Y: sum_sum_b_d] :
      ( ( E
       => ( ( sum_map_sum_b_b_d_d @ F2 @ G @ ( if_Sum_sum_b_d @ E @ ( sum_Inr_d_b @ X2 ) @ Y ) )
          = ( sum_Inr_d_b @ ( G @ X2 ) ) ) )
      & ( ~ E
       => ( ( sum_map_sum_b_b_d_d @ F2 @ G @ ( if_Sum_sum_b_d @ E @ ( sum_Inr_d_b @ X2 ) @ Y ) )
          = ( sum_map_sum_b_b_d_d @ F2 @ G @ Y ) ) ) ) ).

% map_sum_if_distrib_then(2)
thf(fact_327_map__sum__if__distrib__then_I2_J,axiom,
    ! [E: $o,F2: a > b,G: c > d,X2: c,Y: sum_sum_a_c] :
      ( ( E
       => ( ( sum_map_sum_a_b_c_d @ F2 @ G @ ( if_Sum_sum_a_c @ E @ ( sum_Inr_c_a @ X2 ) @ Y ) )
          = ( sum_Inr_d_b @ ( G @ X2 ) ) ) )
      & ( ~ E
       => ( ( sum_map_sum_a_b_c_d @ F2 @ G @ ( if_Sum_sum_a_c @ E @ ( sum_Inr_c_a @ X2 ) @ Y ) )
          = ( sum_map_sum_a_b_c_d @ F2 @ G @ Y ) ) ) ) ).

% map_sum_if_distrib_then(2)
thf(fact_328_map__sum__if__distrib__else_I2_J,axiom,
    ! [E: $o,F2: a > a,G: c > c,X2: sum_sum_a_c,Y: c] :
      ( ( E
       => ( ( sum_map_sum_a_a_c_c @ F2 @ G @ ( if_Sum_sum_a_c @ E @ X2 @ ( sum_Inr_c_a @ Y ) ) )
          = ( sum_map_sum_a_a_c_c @ F2 @ G @ X2 ) ) )
      & ( ~ E
       => ( ( sum_map_sum_a_a_c_c @ F2 @ G @ ( if_Sum_sum_a_c @ E @ X2 @ ( sum_Inr_c_a @ Y ) ) )
          = ( sum_Inr_c_a @ ( G @ Y ) ) ) ) ) ).

% map_sum_if_distrib_else(2)
thf(fact_329_map__sum__if__distrib__else_I2_J,axiom,
    ! [E: $o,F2: b > a,G: d > c,X2: sum_sum_b_d,Y: d] :
      ( ( E
       => ( ( sum_map_sum_b_a_d_c @ F2 @ G @ ( if_Sum_sum_b_d @ E @ X2 @ ( sum_Inr_d_b @ Y ) ) )
          = ( sum_map_sum_b_a_d_c @ F2 @ G @ X2 ) ) )
      & ( ~ E
       => ( ( sum_map_sum_b_a_d_c @ F2 @ G @ ( if_Sum_sum_b_d @ E @ X2 @ ( sum_Inr_d_b @ Y ) ) )
          = ( sum_Inr_c_a @ ( G @ Y ) ) ) ) ) ).

% map_sum_if_distrib_else(2)
thf(fact_330_map__sum__if__distrib__else_I2_J,axiom,
    ! [E: $o,F2: b > b,G: d > d,X2: sum_sum_b_d,Y: d] :
      ( ( E
       => ( ( sum_map_sum_b_b_d_d @ F2 @ G @ ( if_Sum_sum_b_d @ E @ X2 @ ( sum_Inr_d_b @ Y ) ) )
          = ( sum_map_sum_b_b_d_d @ F2 @ G @ X2 ) ) )
      & ( ~ E
       => ( ( sum_map_sum_b_b_d_d @ F2 @ G @ ( if_Sum_sum_b_d @ E @ X2 @ ( sum_Inr_d_b @ Y ) ) )
          = ( sum_Inr_d_b @ ( G @ Y ) ) ) ) ) ).

% map_sum_if_distrib_else(2)
thf(fact_331_map__sum__if__distrib__else_I2_J,axiom,
    ! [E: $o,F2: a > b,G: c > d,X2: sum_sum_a_c,Y: c] :
      ( ( E
       => ( ( sum_map_sum_a_b_c_d @ F2 @ G @ ( if_Sum_sum_a_c @ E @ X2 @ ( sum_Inr_c_a @ Y ) ) )
          = ( sum_map_sum_a_b_c_d @ F2 @ G @ X2 ) ) )
      & ( ~ E
       => ( ( sum_map_sum_a_b_c_d @ F2 @ G @ ( if_Sum_sum_a_c @ E @ X2 @ ( sum_Inr_c_a @ Y ) ) )
          = ( sum_Inr_d_b @ ( G @ Y ) ) ) ) ) ).

% map_sum_if_distrib_else(2)
thf(fact_332_map__sum_Osimps_I2_J,axiom,
    ! [F1: a > a,F22: c > c,A2: c] :
      ( ( sum_map_sum_a_a_c_c @ F1 @ F22 @ ( sum_Inr_c_a @ A2 ) )
      = ( sum_Inr_c_a @ ( F22 @ A2 ) ) ) ).

% map_sum.simps(2)
thf(fact_333_map__sum_Osimps_I2_J,axiom,
    ! [F1: b > a,F22: d > c,A2: d] :
      ( ( sum_map_sum_b_a_d_c @ F1 @ F22 @ ( sum_Inr_d_b @ A2 ) )
      = ( sum_Inr_c_a @ ( F22 @ A2 ) ) ) ).

% map_sum.simps(2)
thf(fact_334_map__sum_Osimps_I2_J,axiom,
    ! [F1: b > b,F22: d > d,A2: d] :
      ( ( sum_map_sum_b_b_d_d @ F1 @ F22 @ ( sum_Inr_d_b @ A2 ) )
      = ( sum_Inr_d_b @ ( F22 @ A2 ) ) ) ).

% map_sum.simps(2)
thf(fact_335_map__sum_Osimps_I2_J,axiom,
    ! [F1: a > b,F22: c > d,A2: c] :
      ( ( sum_map_sum_a_b_c_d @ F1 @ F22 @ ( sum_Inr_c_a @ A2 ) )
      = ( sum_Inr_d_b @ ( F22 @ A2 ) ) ) ).

% map_sum.simps(2)
thf(fact_336_map__sum__if__distrib__then_I1_J,axiom,
    ! [E: $o,F2: a > a,G: c > c,X2: a,Y: sum_sum_a_c] :
      ( ( E
       => ( ( sum_map_sum_a_a_c_c @ F2 @ G @ ( if_Sum_sum_a_c @ E @ ( sum_Inl_a_c @ X2 ) @ Y ) )
          = ( sum_Inl_a_c @ ( F2 @ X2 ) ) ) )
      & ( ~ E
       => ( ( sum_map_sum_a_a_c_c @ F2 @ G @ ( if_Sum_sum_a_c @ E @ ( sum_Inl_a_c @ X2 ) @ Y ) )
          = ( sum_map_sum_a_a_c_c @ F2 @ G @ Y ) ) ) ) ).

% map_sum_if_distrib_then(1)
thf(fact_337_map__sum__if__distrib__then_I1_J,axiom,
    ! [E: $o,F2: b > a,G: d > c,X2: b,Y: sum_sum_b_d] :
      ( ( E
       => ( ( sum_map_sum_b_a_d_c @ F2 @ G @ ( if_Sum_sum_b_d @ E @ ( sum_Inl_b_d @ X2 ) @ Y ) )
          = ( sum_Inl_a_c @ ( F2 @ X2 ) ) ) )
      & ( ~ E
       => ( ( sum_map_sum_b_a_d_c @ F2 @ G @ ( if_Sum_sum_b_d @ E @ ( sum_Inl_b_d @ X2 ) @ Y ) )
          = ( sum_map_sum_b_a_d_c @ F2 @ G @ Y ) ) ) ) ).

% map_sum_if_distrib_then(1)
thf(fact_338_map__sum__if__distrib__then_I1_J,axiom,
    ! [E: $o,F2: b > b,G: d > d,X2: b,Y: sum_sum_b_d] :
      ( ( E
       => ( ( sum_map_sum_b_b_d_d @ F2 @ G @ ( if_Sum_sum_b_d @ E @ ( sum_Inl_b_d @ X2 ) @ Y ) )
          = ( sum_Inl_b_d @ ( F2 @ X2 ) ) ) )
      & ( ~ E
       => ( ( sum_map_sum_b_b_d_d @ F2 @ G @ ( if_Sum_sum_b_d @ E @ ( sum_Inl_b_d @ X2 ) @ Y ) )
          = ( sum_map_sum_b_b_d_d @ F2 @ G @ Y ) ) ) ) ).

% map_sum_if_distrib_then(1)
thf(fact_339_map__sum__if__distrib__then_I1_J,axiom,
    ! [E: $o,F2: a > b,G: c > d,X2: a,Y: sum_sum_a_c] :
      ( ( E
       => ( ( sum_map_sum_a_b_c_d @ F2 @ G @ ( if_Sum_sum_a_c @ E @ ( sum_Inl_a_c @ X2 ) @ Y ) )
          = ( sum_Inl_b_d @ ( F2 @ X2 ) ) ) )
      & ( ~ E
       => ( ( sum_map_sum_a_b_c_d @ F2 @ G @ ( if_Sum_sum_a_c @ E @ ( sum_Inl_a_c @ X2 ) @ Y ) )
          = ( sum_map_sum_a_b_c_d @ F2 @ G @ Y ) ) ) ) ).

% map_sum_if_distrib_then(1)
thf(fact_340_map__sum__if__distrib__else_I1_J,axiom,
    ! [E: $o,F2: a > a,G: c > c,X2: sum_sum_a_c,Y: a] :
      ( ( E
       => ( ( sum_map_sum_a_a_c_c @ F2 @ G @ ( if_Sum_sum_a_c @ E @ X2 @ ( sum_Inl_a_c @ Y ) ) )
          = ( sum_map_sum_a_a_c_c @ F2 @ G @ X2 ) ) )
      & ( ~ E
       => ( ( sum_map_sum_a_a_c_c @ F2 @ G @ ( if_Sum_sum_a_c @ E @ X2 @ ( sum_Inl_a_c @ Y ) ) )
          = ( sum_Inl_a_c @ ( F2 @ Y ) ) ) ) ) ).

% map_sum_if_distrib_else(1)
thf(fact_341_map__sum__if__distrib__else_I1_J,axiom,
    ! [E: $o,F2: b > a,G: d > c,X2: sum_sum_b_d,Y: b] :
      ( ( E
       => ( ( sum_map_sum_b_a_d_c @ F2 @ G @ ( if_Sum_sum_b_d @ E @ X2 @ ( sum_Inl_b_d @ Y ) ) )
          = ( sum_map_sum_b_a_d_c @ F2 @ G @ X2 ) ) )
      & ( ~ E
       => ( ( sum_map_sum_b_a_d_c @ F2 @ G @ ( if_Sum_sum_b_d @ E @ X2 @ ( sum_Inl_b_d @ Y ) ) )
          = ( sum_Inl_a_c @ ( F2 @ Y ) ) ) ) ) ).

% map_sum_if_distrib_else(1)
thf(fact_342_map__sum__if__distrib__else_I1_J,axiom,
    ! [E: $o,F2: b > b,G: d > d,X2: sum_sum_b_d,Y: b] :
      ( ( E
       => ( ( sum_map_sum_b_b_d_d @ F2 @ G @ ( if_Sum_sum_b_d @ E @ X2 @ ( sum_Inl_b_d @ Y ) ) )
          = ( sum_map_sum_b_b_d_d @ F2 @ G @ X2 ) ) )
      & ( ~ E
       => ( ( sum_map_sum_b_b_d_d @ F2 @ G @ ( if_Sum_sum_b_d @ E @ X2 @ ( sum_Inl_b_d @ Y ) ) )
          = ( sum_Inl_b_d @ ( F2 @ Y ) ) ) ) ) ).

% map_sum_if_distrib_else(1)
thf(fact_343_map__sum__if__distrib__else_I1_J,axiom,
    ! [E: $o,F2: a > b,G: c > d,X2: sum_sum_a_c,Y: a] :
      ( ( E
       => ( ( sum_map_sum_a_b_c_d @ F2 @ G @ ( if_Sum_sum_a_c @ E @ X2 @ ( sum_Inl_a_c @ Y ) ) )
          = ( sum_map_sum_a_b_c_d @ F2 @ G @ X2 ) ) )
      & ( ~ E
       => ( ( sum_map_sum_a_b_c_d @ F2 @ G @ ( if_Sum_sum_a_c @ E @ X2 @ ( sum_Inl_a_c @ Y ) ) )
          = ( sum_Inl_b_d @ ( F2 @ Y ) ) ) ) ) ).

% map_sum_if_distrib_else(1)
thf(fact_344_map__sum_Osimps_I1_J,axiom,
    ! [F1: a > a,F22: c > c,A2: a] :
      ( ( sum_map_sum_a_a_c_c @ F1 @ F22 @ ( sum_Inl_a_c @ A2 ) )
      = ( sum_Inl_a_c @ ( F1 @ A2 ) ) ) ).

% map_sum.simps(1)
thf(fact_345_map__sum_Osimps_I1_J,axiom,
    ! [F1: b > a,F22: d > c,A2: b] :
      ( ( sum_map_sum_b_a_d_c @ F1 @ F22 @ ( sum_Inl_b_d @ A2 ) )
      = ( sum_Inl_a_c @ ( F1 @ A2 ) ) ) ).

% map_sum.simps(1)
thf(fact_346_map__sum_Osimps_I1_J,axiom,
    ! [F1: b > b,F22: d > d,A2: b] :
      ( ( sum_map_sum_b_b_d_d @ F1 @ F22 @ ( sum_Inl_b_d @ A2 ) )
      = ( sum_Inl_b_d @ ( F1 @ A2 ) ) ) ).

% map_sum.simps(1)
thf(fact_347_map__sum_Osimps_I1_J,axiom,
    ! [F1: a > b,F22: c > d,A2: a] :
      ( ( sum_map_sum_a_b_c_d @ F1 @ F22 @ ( sum_Inl_a_c @ A2 ) )
      = ( sum_Inl_b_d @ ( F1 @ A2 ) ) ) ).

% map_sum.simps(1)
thf(fact_348_qbs__morphismE_I3_J,axiom,
    ! [F2: real > real,X5: quasi_borel_real,Y4: quasi_borel_real,Alpha: real > real] :
      ( ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X5 @ Y4 ) )
     => ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X5 ) )
       => ( member_real_real @ ( comp_real_real_real @ F2 @ Alpha ) @ ( qbs_Mx_real @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_349_qbs__morphismE_I3_J,axiom,
    ! [F2: $o > real,X5: quasi_borel_o,Y4: quasi_borel_real,Alpha: real > $o] :
      ( ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X5 @ Y4 ) )
     => ( ( member_real_o @ Alpha @ ( qbs_Mx_o @ X5 ) )
       => ( member_real_real @ ( comp_o_real_real @ F2 @ Alpha ) @ ( qbs_Mx_real @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_350_qbs__morphismE_I3_J,axiom,
    ! [F2: nat > real,X5: quasi_borel_nat,Y4: quasi_borel_real,Alpha: real > nat] :
      ( ( member_nat_real @ F2 @ ( qbs_mo2000642995705457910t_real @ X5 @ Y4 ) )
     => ( ( member_real_nat @ Alpha @ ( qbs_Mx_nat @ X5 ) )
       => ( member_real_real @ ( comp_nat_real_real @ F2 @ Alpha ) @ ( qbs_Mx_real @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_351_qbs__morphismE_I3_J,axiom,
    ! [F2: real > a,X5: quasi_borel_real,Y4: quasi_borel_a,Alpha: real > real] :
      ( ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X5 @ Y4 ) )
     => ( ( member_real_real @ Alpha @ ( qbs_Mx_real @ X5 ) )
       => ( member_real_a @ ( comp_real_a_real @ F2 @ Alpha ) @ ( qbs_Mx_a @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_352_qbs__morphismE_I3_J,axiom,
    ! [F2: a > a,X5: quasi_borel_a,Y4: quasi_borel_a,Alpha: real > a] :
      ( ( member_a_a @ F2 @ ( qbs_morphism_a_a @ X5 @ Y4 ) )
     => ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X5 ) )
       => ( member_real_a @ ( comp_a_a_real @ F2 @ Alpha ) @ ( qbs_Mx_a @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_353_qbs__morphismE_I3_J,axiom,
    ! [F2: a > c,X5: quasi_borel_a,Y4: quasi_borel_c,Alpha: real > a] :
      ( ( member_a_c @ F2 @ ( qbs_morphism_a_c @ X5 @ Y4 ) )
     => ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X5 ) )
       => ( member_real_c @ ( comp_a_c_real @ F2 @ Alpha ) @ ( qbs_Mx_c @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_354_qbs__morphismE_I3_J,axiom,
    ! [F2: a > d,X5: quasi_borel_a,Y4: quasi_borel_d,Alpha: real > a] :
      ( ( member_a_d @ F2 @ ( qbs_morphism_a_d @ X5 @ Y4 ) )
     => ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X5 ) )
       => ( member_real_d @ ( comp_a_d_real @ F2 @ Alpha ) @ ( qbs_Mx_d @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_355_qbs__morphismE_I3_J,axiom,
    ! [F2: c > a,X5: quasi_borel_c,Y4: quasi_borel_a,Alpha: real > c] :
      ( ( member_c_a @ F2 @ ( qbs_morphism_c_a @ X5 @ Y4 ) )
     => ( ( member_real_c @ Alpha @ ( qbs_Mx_c @ X5 ) )
       => ( member_real_a @ ( comp_c_a_real @ F2 @ Alpha ) @ ( qbs_Mx_a @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_356_qbs__morphismE_I3_J,axiom,
    ! [F2: c > c,X5: quasi_borel_c,Y4: quasi_borel_c,Alpha: real > c] :
      ( ( member_c_c @ F2 @ ( qbs_morphism_c_c @ X5 @ Y4 ) )
     => ( ( member_real_c @ Alpha @ ( qbs_Mx_c @ X5 ) )
       => ( member_real_c @ ( comp_c_c_real @ F2 @ Alpha ) @ ( qbs_Mx_c @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_357_qbs__morphismE_I3_J,axiom,
    ! [F2: c > b,X5: quasi_borel_c,Y4: quasi_borel_b,Alpha: real > c] :
      ( ( member_c_b @ F2 @ ( qbs_morphism_c_b @ X5 @ Y4 ) )
     => ( ( member_real_c @ Alpha @ ( qbs_Mx_c @ X5 ) )
       => ( member_real_b @ ( comp_c_b_real @ F2 @ Alpha ) @ ( qbs_Mx_b @ Y4 ) ) ) ) ).

% qbs_morphismE(3)
thf(fact_358_qbs__morphismI,axiom,
    ! [X5: quasi_borel_real,F2: real > real,Y4: quasi_borel_real] :
      ( ! [Alpha2: real > real] :
          ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X5 ) )
         => ( member_real_real @ ( comp_real_real_real @ F2 @ Alpha2 ) @ ( qbs_Mx_real @ Y4 ) ) )
     => ( member_real_real @ F2 @ ( qbs_mo5229770564518008146l_real @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_359_qbs__morphismI,axiom,
    ! [X5: quasi_borel_o,F2: $o > real,Y4: quasi_borel_real] :
      ( ! [Alpha2: real > $o] :
          ( ( member_real_o @ Alpha2 @ ( qbs_Mx_o @ X5 ) )
         => ( member_real_real @ ( comp_o_real_real @ F2 @ Alpha2 ) @ ( qbs_Mx_real @ Y4 ) ) )
     => ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_360_qbs__morphismI,axiom,
    ! [X5: quasi_borel_nat,F2: nat > real,Y4: quasi_borel_real] :
      ( ! [Alpha2: real > nat] :
          ( ( member_real_nat @ Alpha2 @ ( qbs_Mx_nat @ X5 ) )
         => ( member_real_real @ ( comp_nat_real_real @ F2 @ Alpha2 ) @ ( qbs_Mx_real @ Y4 ) ) )
     => ( member_nat_real @ F2 @ ( qbs_mo2000642995705457910t_real @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_361_qbs__morphismI,axiom,
    ! [X5: quasi_borel_real,F2: real > a,Y4: quasi_borel_a] :
      ( ! [Alpha2: real > real] :
          ( ( member_real_real @ Alpha2 @ ( qbs_Mx_real @ X5 ) )
         => ( member_real_a @ ( comp_real_a_real @ F2 @ Alpha2 ) @ ( qbs_Mx_a @ Y4 ) ) )
     => ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_362_qbs__morphismI,axiom,
    ! [X5: quasi_borel_a,F2: a > a,Y4: quasi_borel_a] :
      ( ! [Alpha2: real > a] :
          ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X5 ) )
         => ( member_real_a @ ( comp_a_a_real @ F2 @ Alpha2 ) @ ( qbs_Mx_a @ Y4 ) ) )
     => ( member_a_a @ F2 @ ( qbs_morphism_a_a @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_363_qbs__morphismI,axiom,
    ! [X5: quasi_borel_a,F2: a > c,Y4: quasi_borel_c] :
      ( ! [Alpha2: real > a] :
          ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X5 ) )
         => ( member_real_c @ ( comp_a_c_real @ F2 @ Alpha2 ) @ ( qbs_Mx_c @ Y4 ) ) )
     => ( member_a_c @ F2 @ ( qbs_morphism_a_c @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_364_qbs__morphismI,axiom,
    ! [X5: quasi_borel_a,F2: a > d,Y4: quasi_borel_d] :
      ( ! [Alpha2: real > a] :
          ( ( member_real_a @ Alpha2 @ ( qbs_Mx_a @ X5 ) )
         => ( member_real_d @ ( comp_a_d_real @ F2 @ Alpha2 ) @ ( qbs_Mx_d @ Y4 ) ) )
     => ( member_a_d @ F2 @ ( qbs_morphism_a_d @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_365_qbs__morphismI,axiom,
    ! [X5: quasi_borel_c,F2: c > a,Y4: quasi_borel_a] :
      ( ! [Alpha2: real > c] :
          ( ( member_real_c @ Alpha2 @ ( qbs_Mx_c @ X5 ) )
         => ( member_real_a @ ( comp_c_a_real @ F2 @ Alpha2 ) @ ( qbs_Mx_a @ Y4 ) ) )
     => ( member_c_a @ F2 @ ( qbs_morphism_c_a @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_366_qbs__morphismI,axiom,
    ! [X5: quasi_borel_c,F2: c > c,Y4: quasi_borel_c] :
      ( ! [Alpha2: real > c] :
          ( ( member_real_c @ Alpha2 @ ( qbs_Mx_c @ X5 ) )
         => ( member_real_c @ ( comp_c_c_real @ F2 @ Alpha2 ) @ ( qbs_Mx_c @ Y4 ) ) )
     => ( member_c_c @ F2 @ ( qbs_morphism_c_c @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_367_qbs__morphismI,axiom,
    ! [X5: quasi_borel_c,F2: c > b,Y4: quasi_borel_b] :
      ( ! [Alpha2: real > c] :
          ( ( member_real_c @ Alpha2 @ ( qbs_Mx_c @ X5 ) )
         => ( member_real_b @ ( comp_c_b_real @ F2 @ Alpha2 ) @ ( qbs_Mx_b @ Y4 ) ) )
     => ( member_c_b @ F2 @ ( qbs_morphism_c_b @ X5 @ Y4 ) ) ) ).

% qbs_morphismI
thf(fact_368_map__sum__o__inj_I1_J,axiom,
    ! [F2: a > a,G: c > c] :
      ( ( comp_S7118176607350915467_a_c_a @ ( sum_map_sum_a_a_c_c @ F2 @ G ) @ sum_Inl_a_c )
      = ( comp_a_Sum_sum_a_c_a @ sum_Inl_a_c @ F2 ) ) ).

% map_sum_o_inj(1)
thf(fact_369_map__sum__o__inj_I1_J,axiom,
    ! [F2: b > a,G: d > c] :
      ( ( comp_S1474653431308885964_a_c_b @ ( sum_map_sum_b_a_d_c @ F2 @ G ) @ sum_Inl_b_d )
      = ( comp_a_Sum_sum_a_c_b @ sum_Inl_a_c @ F2 ) ) ).

% map_sum_o_inj(1)
thf(fact_370_map__sum__o__inj_I1_J,axiom,
    ! [F2: b > b,G: d > d] :
      ( ( comp_S6435223045892029964_b_d_b @ ( sum_map_sum_b_b_d_d @ F2 @ G ) @ sum_Inl_b_d )
      = ( comp_b_Sum_sum_b_d_b @ sum_Inl_b_d @ F2 ) ) ).

% map_sum_o_inj(1)
thf(fact_371_map__sum__o__inj_I1_J,axiom,
    ! [F2: a > b,G: c > d] :
      ( ( comp_S2855374185079283659_b_d_a @ ( sum_map_sum_a_b_c_d @ F2 @ G ) @ sum_Inl_a_c )
      = ( comp_b_Sum_sum_b_d_a @ sum_Inl_b_d @ F2 ) ) ).

% map_sum_o_inj(1)
thf(fact_372_map__sum__o__inj_I2_J,axiom,
    ! [F2: a > a,G: c > c] :
      ( ( comp_S7118176607350915469_a_c_c @ ( sum_map_sum_a_a_c_c @ F2 @ G ) @ sum_Inr_c_a )
      = ( comp_c_Sum_sum_a_c_c @ sum_Inr_c_a @ G ) ) ).

% map_sum_o_inj(2)
thf(fact_373_map__sum__o__inj_I2_J,axiom,
    ! [F2: b > a,G: d > c] :
      ( ( comp_S1474653431308885966_a_c_d @ ( sum_map_sum_b_a_d_c @ F2 @ G ) @ sum_Inr_d_b )
      = ( comp_c_Sum_sum_a_c_d @ sum_Inr_c_a @ G ) ) ).

% map_sum_o_inj(2)
thf(fact_374_map__sum__o__inj_I2_J,axiom,
    ! [F2: b > b,G: d > d] :
      ( ( comp_S6435223045892029966_b_d_d @ ( sum_map_sum_b_b_d_d @ F2 @ G ) @ sum_Inr_d_b )
      = ( comp_d_Sum_sum_b_d_d @ sum_Inr_d_b @ G ) ) ).

% map_sum_o_inj(2)
thf(fact_375_map__sum__o__inj_I2_J,axiom,
    ! [F2: a > b,G: c > d] :
      ( ( comp_S2855374185079283661_b_d_c @ ( sum_map_sum_a_b_c_d @ F2 @ G ) @ sum_Inr_c_a )
      = ( comp_d_Sum_sum_b_d_c @ sum_Inr_d_b @ G ) ) ).

% map_sum_o_inj(2)
thf(fact_376_sum_Omap__ident__strong,axiom,
    ! [T: sum_su6357119772783438699et_nat,F1: set_nat > set_nat,F22: set_nat > set_nat] :
      ( ! [Z1: set_nat] :
          ( ( member_set_nat @ Z1 @ ( basic_3910105186785813299et_nat @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_nat] :
            ( ( member_set_nat @ Z2 @ ( basic_7585541094054265657et_nat @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma5866040581394992932et_nat @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_377_sum_Omap__ident__strong,axiom,
    ! [T: sum_su180556917423924329_set_o,F1: set_nat > set_nat,F22: set_o > set_o] :
      ( ! [Z1: set_nat] :
          ( ( member_set_nat @ Z1 @ ( basic_650432819501328779_set_o @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_o] :
            ( ( member_set_o @ Z2 @ ( basic_6010508404979096581_set_o @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma5689907174169415190_set_o @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_378_sum_Omap__ident__strong,axiom,
    ! [T: sum_su7445071152329184979nnreal,F1: set_nat > set_nat,F22: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal] :
      ( ! [Z1: set_nat] :
          ( ( member_set_nat @ Z1 @ ( basic_842870997940628123nnreal @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_Ex3793607809372303086nnreal] :
            ( ( member603777416030116741nnreal @ Z2 @ ( basic_8760871286204816033nnreal @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma4567893330926093812nnreal @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_379_sum_Omap__ident__strong,axiom,
    ! [T: sum_su6439591028793565127t_real,F1: set_nat > set_nat,F22: set_real > set_real] :
      ( ! [Z1: set_nat] :
          ( ( member_set_nat @ Z1 @ ( basic_4495003616656931855t_real @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_real] :
            ( ( member_set_real @ Z2 @ ( basic_4563158366440306965t_real @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma2514991182478280028t_real @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_380_sum_Omap__ident__strong,axiom,
    ! [T: sum_su7609443801303718063et_nat,F1: set_o > set_o,F22: set_nat > set_nat] :
      ( ! [Z1: set_o] :
          ( ( member_set_o @ Z1 @ ( basic_112841166889062809et_nat @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_nat] :
            ( ( member_set_nat @ Z2 @ ( basic_5472916752366830611et_nat @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma5195006407206924054et_nat @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_381_sum_Omap__ident__strong,axiom,
    ! [T: sum_sum_set_o_set_o,F1: set_o > set_o,F22: set_o > set_o] :
      ( ! [Z1: set_o] :
          ( ( member_set_o @ Z1 @ ( basic_2831768515958928805_set_o @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_o] :
            ( ( member_set_o @ Z2 @ ( basic_1118969371388924075_set_o @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma5536731787355361032_set_o @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_382_sum_Omap__ident__strong,axiom,
    ! [T: sum_su4962168936726258711nnreal,F1: set_o > set_o,F22: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal] :
      ( ! [Z1: set_o] :
          ( ( member_set_o @ Z1 @ ( basic_127669078395480321nnreal @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_Ex3793607809372303086nnreal] :
            ( ( member603777416030116741nnreal @ Z2 @ ( basic_5549938302119316347nnreal @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma1877970594666251750nnreal @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_383_sum_Omap__ident__strong,axiom,
    ! [T: sum_su2944506142428048139t_real,F1: set_o > set_o,F22: set_real > set_real] :
      ( ! [Z1: set_o] :
          ( ( member_set_o @ Z1 @ ( basic_5135449112562407285t_real @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_real] :
            ( ( member_set_real @ Z2 @ ( basic_3640605392351816943t_real @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma1644795144998597454t_real @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_384_sum_Omap__ident__strong,axiom,
    ! [T: sum_su1570082439829250771et_nat,F1: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,F22: set_nat > set_nat] :
      ( ! [Z1: set_Ex3793607809372303086nnreal] :
          ( ( member603777416030116741nnreal @ Z1 @ ( basic_981975278752359067et_nat @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_nat] :
            ( ( member_set_nat @ Z2 @ ( basic_8899975567016546977et_nat @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma204727134861501940et_nat @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_385_sum_Omap__ident__strong,axiom,
    ! [T: sum_su145664975714009857_set_o,F1: set_Ex3793607809372303086nnreal > set_Ex3793607809372303086nnreal,F22: set_o > set_o] :
      ( ! [Z1: set_Ex3793607809372303086nnreal] :
          ( ( member603777416030116741nnreal @ Z1 @ ( basic_68364610332180515_set_o @ T ) )
         => ( ( F1 @ Z1 )
            = Z1 ) )
     => ( ! [Z2: set_o] :
            ( ( member_set_o @ Z2 @ ( basic_5490633834056016541_set_o @ T ) )
           => ( ( F22 @ Z2 )
              = Z2 ) )
       => ( ( sum_ma2058262178995162342_set_o @ F1 @ F22 @ T )
          = T ) ) ) ).

% sum.map_ident_strong
thf(fact_386_sum_Oinj__map__strong,axiom,
    ! [X2: sum_sum_a_c,Xa2: sum_sum_a_c,F1: a > b,F1a: a > b,F22: c > d,F2a: c > d] :
      ( ! [Z1: a,Z1a: a] :
          ( ( member_a @ Z1 @ ( basic_setl_a_c @ X2 ) )
         => ( ( member_a @ Z1a @ ( basic_setl_a_c @ Xa2 ) )
           => ( ( ( F1 @ Z1 )
                = ( F1a @ Z1a ) )
             => ( Z1 = Z1a ) ) ) )
     => ( ! [Z2: c,Z2a: c] :
            ( ( member_c @ Z2 @ ( basic_setr_a_c @ X2 ) )
           => ( ( member_c @ Z2a @ ( basic_setr_a_c @ Xa2 ) )
             => ( ( ( F22 @ Z2 )
                  = ( F2a @ Z2a ) )
               => ( Z2 = Z2a ) ) ) )
       => ( ( ( sum_map_sum_a_b_c_d @ F1 @ F22 @ X2 )
            = ( sum_map_sum_a_b_c_d @ F1a @ F2a @ Xa2 ) )
         => ( X2 = Xa2 ) ) ) ) ).

% sum.inj_map_strong
thf(fact_387_sum_Omap__cong0,axiom,
    ! [X2: sum_sum_a_c,F1: a > b,G1: a > b,F22: c > d,G2: c > d] :
      ( ! [Z1: a] :
          ( ( member_a @ Z1 @ ( basic_setl_a_c @ X2 ) )
         => ( ( F1 @ Z1 )
            = ( G1 @ Z1 ) ) )
     => ( ! [Z2: c] :
            ( ( member_c @ Z2 @ ( basic_setr_a_c @ X2 ) )
           => ( ( F22 @ Z2 )
              = ( G2 @ Z2 ) ) )
       => ( ( sum_map_sum_a_b_c_d @ F1 @ F22 @ X2 )
          = ( sum_map_sum_a_b_c_d @ G1 @ G2 @ X2 ) ) ) ) ).

% sum.map_cong0
thf(fact_388_sum_Omap__cong,axiom,
    ! [X2: sum_sum_a_c,Ya: sum_sum_a_c,F1: a > b,G1: a > b,F22: c > d,G2: c > d] :
      ( ( X2 = Ya )
     => ( ! [Z1: a] :
            ( ( member_a @ Z1 @ ( basic_setl_a_c @ Ya ) )
           => ( ( F1 @ Z1 )
              = ( G1 @ Z1 ) ) )
       => ( ! [Z2: c] :
              ( ( member_c @ Z2 @ ( basic_setr_a_c @ Ya ) )
             => ( ( F22 @ Z2 )
                = ( G2 @ Z2 ) ) )
         => ( ( sum_map_sum_a_b_c_d @ F1 @ F22 @ X2 )
            = ( sum_map_sum_a_b_c_d @ G1 @ G2 @ Ya ) ) ) ) ) ).

% sum.map_cong
thf(fact_389_comp__apply,axiom,
    ( comp_real_real_real
    = ( ^ [F3: real > real,G3: real > real,X3: real] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_apply
thf(fact_390_comp__apply,axiom,
    ( comp_real_o_o
    = ( ^ [F3: real > $o,G3: $o > real,X3: $o] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_apply
thf(fact_391_comp__apply,axiom,
    ( comp_real_nat_nat
    = ( ^ [F3: real > nat,G3: nat > real,X3: nat] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_apply
thf(fact_392_comp__apply,axiom,
    ( comp_c_d_real
    = ( ^ [F3: c > d,G3: real > c,X3: real] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_apply
thf(fact_393_comp__apply,axiom,
    ( comp_a_b_real
    = ( ^ [F3: a > b,G3: real > a,X3: real] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_apply
thf(fact_394_setl_Ointros,axiom,
    ! [S2: sum_sum_a_c,X2: a] :
      ( ( S2
        = ( sum_Inl_a_c @ X2 ) )
     => ( member_a @ X2 @ ( basic_setl_a_c @ S2 ) ) ) ).

% setl.intros
thf(fact_395_setl_Ointros,axiom,
    ! [S2: sum_sum_b_d,X2: b] :
      ( ( S2
        = ( sum_Inl_b_d @ X2 ) )
     => ( member_b @ X2 @ ( basic_setl_b_d @ S2 ) ) ) ).

% setl.intros
thf(fact_396_setl_Osimps,axiom,
    ! [A2: a,S2: sum_sum_a_c] :
      ( ( member_a @ A2 @ ( basic_setl_a_c @ S2 ) )
      = ( ? [X3: a] :
            ( ( A2 = X3 )
            & ( S2
              = ( sum_Inl_a_c @ X3 ) ) ) ) ) ).

% setl.simps
thf(fact_397_setl_Osimps,axiom,
    ! [A2: b,S2: sum_sum_b_d] :
      ( ( member_b @ A2 @ ( basic_setl_b_d @ S2 ) )
      = ( ? [X3: b] :
            ( ( A2 = X3 )
            & ( S2
              = ( sum_Inl_b_d @ X3 ) ) ) ) ) ).

% setl.simps
thf(fact_398_fun_Omap__comp,axiom,
    ! [G: $o > $o,F2: real > $o,V: $o > real] :
      ( ( comp_o_o_o @ G @ ( comp_real_o_o @ F2 @ V ) )
      = ( comp_real_o_o @ ( comp_o_o_real @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_399_fun_Omap__comp,axiom,
    ! [G: nat > nat,F2: real > nat,V: nat > real] :
      ( ( comp_nat_nat_nat @ G @ ( comp_real_nat_nat @ F2 @ V ) )
      = ( comp_real_nat_nat @ ( comp_nat_nat_real @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_400_fun_Omap__comp,axiom,
    ! [G: d > d,F2: c > d,V: real > c] :
      ( ( comp_d_d_real @ G @ ( comp_c_d_real @ F2 @ V ) )
      = ( comp_c_d_real @ ( comp_d_d_c @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_401_fun_Omap__comp,axiom,
    ! [G: b > b,F2: a > b,V: real > a] :
      ( ( comp_b_b_real @ G @ ( comp_a_b_real @ F2 @ V ) )
      = ( comp_a_b_real @ ( comp_b_b_a @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_402_fun_Omap__comp,axiom,
    ! [G: real > real,F2: real > real,V: real > real] :
      ( ( comp_real_real_real @ G @ ( comp_real_real_real @ F2 @ V ) )
      = ( comp_real_real_real @ ( comp_real_real_real @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_403_fun_Omap__comp,axiom,
    ! [G: real > $o,F2: $o > real,V: $o > $o] :
      ( ( comp_real_o_o @ G @ ( comp_o_real_o @ F2 @ V ) )
      = ( comp_o_o_o @ ( comp_real_o_o @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_404_fun_Omap__comp,axiom,
    ! [G: real > $o,F2: real > real,V: $o > real] :
      ( ( comp_real_o_o @ G @ ( comp_real_real_o @ F2 @ V ) )
      = ( comp_real_o_o @ ( comp_real_o_real @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_405_fun_Omap__comp,axiom,
    ! [G: real > nat,F2: nat > real,V: nat > nat] :
      ( ( comp_real_nat_nat @ G @ ( comp_nat_real_nat @ F2 @ V ) )
      = ( comp_nat_nat_nat @ ( comp_real_nat_nat @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_406_fun_Omap__comp,axiom,
    ! [G: real > nat,F2: real > real,V: nat > real] :
      ( ( comp_real_nat_nat @ G @ ( comp_real_real_nat @ F2 @ V ) )
      = ( comp_real_nat_nat @ ( comp_real_nat_real @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_407_fun_Omap__comp,axiom,
    ! [G: c > d,F2: real > c,V: real > real] :
      ( ( comp_c_d_real @ G @ ( comp_real_c_real @ F2 @ V ) )
      = ( comp_real_d_real @ ( comp_c_d_real @ G @ F2 ) @ V ) ) ).

% fun.map_comp
thf(fact_408_comp__def,axiom,
    ( comp_real_real_real
    = ( ^ [F3: real > real,G3: real > real,X3: real] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_def
thf(fact_409_comp__def,axiom,
    ( comp_real_o_o
    = ( ^ [F3: real > $o,G3: $o > real,X3: $o] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_def
thf(fact_410_comp__def,axiom,
    ( comp_real_nat_nat
    = ( ^ [F3: real > nat,G3: nat > real,X3: nat] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_def
thf(fact_411_comp__def,axiom,
    ( comp_c_d_real
    = ( ^ [F3: c > d,G3: real > c,X3: real] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_def
thf(fact_412_comp__def,axiom,
    ( comp_a_b_real
    = ( ^ [F3: a > b,G3: real > a,X3: real] : ( F3 @ ( G3 @ X3 ) ) ) ) ).

% comp_def
thf(fact_413_comp__assoc,axiom,
    ! [F2: real > $o,G: $o > real,H: $o > $o] :
      ( ( comp_o_o_o @ ( comp_real_o_o @ F2 @ G ) @ H )
      = ( comp_real_o_o @ F2 @ ( comp_o_real_o @ G @ H ) ) ) ).

% comp_assoc
thf(fact_414_comp__assoc,axiom,
    ! [F2: real > nat,G: nat > real,H: nat > nat] :
      ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F2 @ G ) @ H )
      = ( comp_real_nat_nat @ F2 @ ( comp_nat_real_nat @ G @ H ) ) ) ).

% comp_assoc
thf(fact_415_comp__assoc,axiom,
    ! [F2: c > d,G: real > c,H: real > real] :
      ( ( comp_real_d_real @ ( comp_c_d_real @ F2 @ G ) @ H )
      = ( comp_c_d_real @ F2 @ ( comp_real_c_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_416_comp__assoc,axiom,
    ! [F2: a > b,G: real > a,H: real > real] :
      ( ( comp_real_b_real @ ( comp_a_b_real @ F2 @ G ) @ H )
      = ( comp_a_b_real @ F2 @ ( comp_real_a_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_417_comp__assoc,axiom,
    ! [F2: real > real,G: real > real,H: real > real] :
      ( ( comp_real_real_real @ ( comp_real_real_real @ F2 @ G ) @ H )
      = ( comp_real_real_real @ F2 @ ( comp_real_real_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_418_comp__assoc,axiom,
    ! [F2: $o > $o,G: real > $o,H: $o > real] :
      ( ( comp_real_o_o @ ( comp_o_o_real @ F2 @ G ) @ H )
      = ( comp_o_o_o @ F2 @ ( comp_real_o_o @ G @ H ) ) ) ).

% comp_assoc
thf(fact_419_comp__assoc,axiom,
    ! [F2: real > $o,G: real > real,H: $o > real] :
      ( ( comp_real_o_o @ ( comp_real_o_real @ F2 @ G ) @ H )
      = ( comp_real_o_o @ F2 @ ( comp_real_real_o @ G @ H ) ) ) ).

% comp_assoc
thf(fact_420_comp__assoc,axiom,
    ! [F2: nat > nat,G: real > nat,H: nat > real] :
      ( ( comp_real_nat_nat @ ( comp_nat_nat_real @ F2 @ G ) @ H )
      = ( comp_nat_nat_nat @ F2 @ ( comp_real_nat_nat @ G @ H ) ) ) ).

% comp_assoc
thf(fact_421_comp__assoc,axiom,
    ! [F2: real > nat,G: real > real,H: nat > real] :
      ( ( comp_real_nat_nat @ ( comp_real_nat_real @ F2 @ G ) @ H )
      = ( comp_real_nat_nat @ F2 @ ( comp_real_real_nat @ G @ H ) ) ) ).

% comp_assoc
thf(fact_422_comp__assoc,axiom,
    ! [F2: d > d,G: c > d,H: real > c] :
      ( ( comp_c_d_real @ ( comp_d_d_c @ F2 @ G ) @ H )
      = ( comp_d_d_real @ F2 @ ( comp_c_d_real @ G @ H ) ) ) ).

% comp_assoc
thf(fact_423_comp__eq__dest,axiom,
    ! [A2: real > real,B: real > real,C: real > real,D: real > real,V: real] :
      ( ( ( comp_real_real_real @ A2 @ B )
        = ( comp_real_real_real @ C @ D ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_424_comp__eq__dest,axiom,
    ! [A2: real > $o,B: $o > real,C: real > $o,D: $o > real,V: $o] :
      ( ( ( comp_real_o_o @ A2 @ B )
        = ( comp_real_o_o @ C @ D ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_425_comp__eq__dest,axiom,
    ! [A2: real > nat,B: nat > real,C: real > nat,D: nat > real,V: nat] :
      ( ( ( comp_real_nat_nat @ A2 @ B )
        = ( comp_real_nat_nat @ C @ D ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_426_comp__eq__dest,axiom,
    ! [A2: c > d,B: real > c,C: c > d,D: real > c,V: real] :
      ( ( ( comp_c_d_real @ A2 @ B )
        = ( comp_c_d_real @ C @ D ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_427_comp__eq__dest,axiom,
    ! [A2: a > b,B: real > a,C: a > b,D: real > a,V: real] :
      ( ( ( comp_a_b_real @ A2 @ B )
        = ( comp_a_b_real @ C @ D ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ ( D @ V ) ) ) ) ).

% comp_eq_dest
thf(fact_428_comp__eq__elim,axiom,
    ! [A2: real > real,B: real > real,C: real > real,D: real > real] :
      ( ( ( comp_real_real_real @ A2 @ B )
        = ( comp_real_real_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A2 @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_429_comp__eq__elim,axiom,
    ! [A2: real > $o,B: $o > real,C: real > $o,D: $o > real] :
      ( ( ( comp_real_o_o @ A2 @ B )
        = ( comp_real_o_o @ C @ D ) )
     => ! [V2: $o] :
          ( ( A2 @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_430_comp__eq__elim,axiom,
    ! [A2: real > nat,B: nat > real,C: real > nat,D: nat > real] :
      ( ( ( comp_real_nat_nat @ A2 @ B )
        = ( comp_real_nat_nat @ C @ D ) )
     => ! [V2: nat] :
          ( ( A2 @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_431_comp__eq__elim,axiom,
    ! [A2: c > d,B: real > c,C: c > d,D: real > c] :
      ( ( ( comp_c_d_real @ A2 @ B )
        = ( comp_c_d_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A2 @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_432_comp__eq__elim,axiom,
    ! [A2: a > b,B: real > a,C: a > b,D: real > a] :
      ( ( ( comp_a_b_real @ A2 @ B )
        = ( comp_a_b_real @ C @ D ) )
     => ! [V2: real] :
          ( ( A2 @ ( B @ V2 ) )
          = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_elim
thf(fact_433_comp__eq__dest__lhs,axiom,
    ! [A2: real > real,B: real > real,C: real > real,V: real] :
      ( ( ( comp_real_real_real @ A2 @ B )
        = C )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_434_comp__eq__dest__lhs,axiom,
    ! [A2: real > $o,B: $o > real,C: $o > $o,V: $o] :
      ( ( ( comp_real_o_o @ A2 @ B )
        = C )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_435_comp__eq__dest__lhs,axiom,
    ! [A2: real > nat,B: nat > real,C: nat > nat,V: nat] :
      ( ( ( comp_real_nat_nat @ A2 @ B )
        = C )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_436_comp__eq__dest__lhs,axiom,
    ! [A2: c > d,B: real > c,C: real > d,V: real] :
      ( ( ( comp_c_d_real @ A2 @ B )
        = C )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_437_comp__eq__dest__lhs,axiom,
    ! [A2: a > b,B: real > a,C: real > b,V: real] :
      ( ( ( comp_a_b_real @ A2 @ B )
        = C )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_dest_lhs
thf(fact_438_comp__apply__eq,axiom,
    ! [F2: real > real,G: real > real,X2: real,H: real > real,K: real > real] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_real_real_real @ F2 @ G @ X2 )
        = ( comp_real_real_real @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_439_comp__apply__eq,axiom,
    ! [F2: real > $o,G: $o > real,X2: $o,H: real > $o,K: $o > real] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_real_o_o @ F2 @ G @ X2 )
        = ( comp_real_o_o @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_440_comp__apply__eq,axiom,
    ! [F2: real > nat,G: nat > real,X2: nat,H: real > nat,K: nat > real] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_real_nat_nat @ F2 @ G @ X2 )
        = ( comp_real_nat_nat @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_441_comp__apply__eq,axiom,
    ! [F2: c > d,G: real > c,X2: real,H: c > d,K: real > c] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_c_d_real @ F2 @ G @ X2 )
        = ( comp_c_d_real @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_442_comp__apply__eq,axiom,
    ! [F2: a > b,G: real > a,X2: real,H: a > b,K: real > a] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( H @ ( K @ X2 ) ) )
     => ( ( comp_a_b_real @ F2 @ G @ X2 )
        = ( comp_a_b_real @ H @ K @ X2 ) ) ) ).

% comp_apply_eq
thf(fact_443_Inr__qbs__morphism,axiom,
    ! [Y4: quasi_borel_c,X5: quasi_borel_a] : ( member_c_Sum_sum_a_c @ sum_Inr_c_a @ ( qbs_mo5084992033439934511um_a_c @ Y4 @ ( binary8555328655094383375bs_a_c @ X5 @ Y4 ) ) ) ).

% Inr_qbs_morphism
thf(fact_444_Inr__qbs__morphism,axiom,
    ! [Y4: quasi_borel_d,X5: quasi_borel_b] : ( member_d_Sum_sum_b_d @ sum_Inr_d_b @ ( qbs_mo1214661810789969904um_b_d @ Y4 @ ( binary5767873073121707343bs_b_d @ X5 @ Y4 ) ) ) ).

% Inr_qbs_morphism
thf(fact_445_Inl__qbs__morphism,axiom,
    ! [X5: quasi_borel_a,Y4: quasi_borel_c] : ( member_a_Sum_sum_a_c @ sum_Inl_a_c @ ( qbs_mo7250741323400969261um_a_c @ X5 @ ( binary8555328655094383375bs_a_c @ X5 @ Y4 ) ) ) ).

% Inl_qbs_morphism
thf(fact_446_Inl__qbs__morphism,axiom,
    ! [X5: quasi_borel_b,Y4: quasi_borel_d] : ( member_b_Sum_sum_b_d @ sum_Inl_b_d @ ( qbs_mo3380411100751004654um_b_d @ X5 @ ( binary5767873073121707343bs_b_d @ X5 @ Y4 ) ) ) ).

% Inl_qbs_morphism
thf(fact_447_sum_Omap__comp,axiom,
    ! [G1: real > real,G2: real > real,F1: real > real,F22: real > real,V: sum_sum_real_real] :
      ( ( sum_ma9028575376852974268l_real @ G1 @ G2 @ ( sum_ma9028575376852974268l_real @ F1 @ F22 @ V ) )
      = ( sum_ma9028575376852974268l_real @ ( comp_real_real_real @ G1 @ F1 ) @ ( comp_real_real_real @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_448_sum_Omap__comp,axiom,
    ! [G1: real > real,G2: real > $o,F1: real > real,F22: $o > real,V: sum_sum_real_o] :
      ( ( sum_ma4609325004622031752real_o @ G1 @ G2 @ ( sum_ma3100260629580862178o_real @ F1 @ F22 @ V ) )
      = ( sum_ma5398380231479159522al_o_o @ ( comp_real_real_real @ G1 @ F1 ) @ ( comp_real_o_o @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_449_sum_Omap__comp,axiom,
    ! [G1: real > real,G2: real > nat,F1: real > real,F22: nat > real,V: sum_sum_real_nat] :
      ( ( sum_ma8618515792258342880al_nat @ G1 @ G2 @ ( sum_ma4051207219129444192t_real @ F1 @ F22 @ V ) )
      = ( sum_ma5138984332203957892at_nat @ ( comp_real_real_real @ G1 @ F1 ) @ ( comp_real_nat_nat @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_450_sum_Omap__comp,axiom,
    ! [G1: real > real,G2: c > d,F1: real > real,F22: real > c,V: sum_sum_real_real] :
      ( ( sum_ma5455772731263491683al_c_d @ G1 @ G2 @ ( sum_ma1303030344635120368real_c @ F1 @ F22 @ V ) )
      = ( sum_ma1303030344635120369real_d @ ( comp_real_real_real @ G1 @ F1 ) @ ( comp_c_d_real @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_451_sum_Omap__comp,axiom,
    ! [G1: real > real,G2: a > b,F1: real > real,F22: real > a,V: sum_sum_real_real] :
      ( ( sum_ma1807311858354067939al_a_b @ G1 @ G2 @ ( sum_ma1303030344635120366real_a @ F1 @ F22 @ V ) )
      = ( sum_ma1303030344635120367real_b @ ( comp_real_real_real @ G1 @ F1 ) @ ( comp_a_b_real @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_452_sum_Omap__comp,axiom,
    ! [G1: real > $o,G2: real > real,F1: $o > real,F22: real > real,V: sum_sum_o_real] :
      ( ( sum_ma5402501128500593980l_real @ G1 @ G2 @ ( sum_ma7581390967160423062l_real @ F1 @ F22 @ V ) )
      = ( sum_ma4692674304805643746l_real @ ( comp_real_o_o @ G1 @ F1 ) @ ( comp_real_real_real @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_453_sum_Omap__comp,axiom,
    ! [G1: real > $o,G2: real > $o,F1: $o > real,F22: $o > real,V: sum_sum_o_o] :
      ( ( sum_ma8868230608733716744real_o @ G1 @ G2 @ ( sum_ma2390433805885911944o_real @ F1 @ F22 @ V ) )
      = ( sum_map_sum_o_o_o_o @ ( comp_real_o_o @ G1 @ F1 ) @ ( comp_real_o_o @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_454_sum_Omap__comp,axiom,
    ! [G1: real > $o,G2: real > nat,F1: $o > real,F22: nat > real,V: sum_sum_o_nat] :
      ( ( sum_ma3912033132148697696al_nat @ G1 @ G2 @ ( sum_ma9203121963192613946t_real @ F1 @ F22 @ V ) )
      = ( sum_ma344318232541114282at_nat @ ( comp_real_o_o @ G1 @ F1 ) @ ( comp_real_nat_nat @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_455_sum_Omap__comp,axiom,
    ! [G1: real > $o,G2: c > d,F1: $o > real,F22: real > c,V: sum_sum_o_real] :
      ( ( sum_ma8893391999227668707_o_c_d @ G1 @ G2 @ ( sum_ma6566039659836893334real_c @ F1 @ F22 @ V ) )
      = ( sum_ma2721287238701537867real_d @ ( comp_real_o_o @ G1 @ F1 ) @ ( comp_c_d_real @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_456_sum_Omap__comp,axiom,
    ! [G1: real > $o,G2: a > b,F1: $o > real,F22: real > a,V: sum_sum_o_real] :
      ( ( sum_ma5244931126318244963_o_a_b @ G1 @ G2 @ ( sum_ma6566039659836893332real_a @ F1 @ F22 @ V ) )
      = ( sum_ma2721287238701537865real_b @ ( comp_real_o_o @ G1 @ F1 ) @ ( comp_a_b_real @ G2 @ F22 ) @ V ) ) ).

% sum.map_comp
thf(fact_457_setr_Ocases,axiom,
    ! [A2: c,S2: sum_sum_a_c] :
      ( ( member_c @ A2 @ ( basic_setr_a_c @ S2 ) )
     => ( S2
        = ( sum_Inr_c_a @ A2 ) ) ) ).

% setr.cases
thf(fact_458_setr_Ocases,axiom,
    ! [A2: d,S2: sum_sum_b_d] :
      ( ( member_d @ A2 @ ( basic_setr_b_d @ S2 ) )
     => ( S2
        = ( sum_Inr_d_b @ A2 ) ) ) ).

% setr.cases
thf(fact_459_setr_Osimps,axiom,
    ! [A2: c,S2: sum_sum_a_c] :
      ( ( member_c @ A2 @ ( basic_setr_a_c @ S2 ) )
      = ( ? [X3: c] :
            ( ( A2 = X3 )
            & ( S2
              = ( sum_Inr_c_a @ X3 ) ) ) ) ) ).

% setr.simps
thf(fact_460_setr_Osimps,axiom,
    ! [A2: d,S2: sum_sum_b_d] :
      ( ( member_d @ A2 @ ( basic_setr_b_d @ S2 ) )
      = ( ? [X3: d] :
            ( ( A2 = X3 )
            & ( S2
              = ( sum_Inr_d_b @ X3 ) ) ) ) ) ).

% setr.simps
thf(fact_461_setr_Ointros,axiom,
    ! [S2: sum_sum_a_c,X2: c] :
      ( ( S2
        = ( sum_Inr_c_a @ X2 ) )
     => ( member_c @ X2 @ ( basic_setr_a_c @ S2 ) ) ) ).

% setr.intros
thf(fact_462_setr_Ointros,axiom,
    ! [S2: sum_sum_b_d,X2: d] :
      ( ( S2
        = ( sum_Inr_d_b @ X2 ) )
     => ( member_d @ X2 @ ( basic_setr_b_d @ S2 ) ) ) ).

% setr.intros
thf(fact_463_setl_Ocases,axiom,
    ! [A2: a,S2: sum_sum_a_c] :
      ( ( member_a @ A2 @ ( basic_setl_a_c @ S2 ) )
     => ( S2
        = ( sum_Inl_a_c @ A2 ) ) ) ).

% setl.cases
thf(fact_464_setl_Ocases,axiom,
    ! [A2: b,S2: sum_sum_b_d] :
      ( ( member_b @ A2 @ ( basic_setl_b_d @ S2 ) )
     => ( S2
        = ( sum_Inl_b_d @ A2 ) ) ) ).

% setl.cases
thf(fact_465_case__sum__o__map__sum,axiom,
    ! [F2: real > real,G: real > real,H1: real > real,H2: real > real] :
      ( ( comp_S1658032853064855039l_real @ ( sum_ca8732840427581260704l_real @ F2 @ G ) @ ( sum_ma9028575376852974268l_real @ H1 @ H2 ) )
      = ( sum_ca8732840427581260704l_real @ ( comp_real_real_real @ F2 @ H1 ) @ ( comp_real_real_real @ G @ H2 ) ) ) ).

% case_sum_o_map_sum
thf(fact_466_case__sum__o__map__sum,axiom,
    ! [F2: real > $o,G: real > $o,H1: $o > real,H2: $o > real] :
      ( ( comp_S7834891836943127203um_o_o @ ( sum_ca7497528202123613950o_real @ F2 @ G ) @ ( sum_ma2390433805885911944o_real @ H1 @ H2 ) )
      = ( sum_case_sum_o_o_o @ ( comp_real_o_o @ F2 @ H1 ) @ ( comp_real_o_o @ G @ H2 ) ) ) ).

% case_sum_o_map_sum
thf(fact_467_case__sum__o__map__sum,axiom,
    ! [F2: real > nat,G: real > nat,H1: nat > real,H2: nat > real] :
      ( ( comp_S193255956842892907at_nat @ ( sum_ca2049687194608230980t_real @ F2 @ G ) @ ( sum_ma3776711750342522116t_real @ H1 @ H2 ) )
      = ( sum_ca6763686470577984908at_nat @ ( comp_real_nat_nat @ F2 @ H1 ) @ ( comp_real_nat_nat @ G @ H2 ) ) ) ).

% case_sum_o_map_sum
thf(fact_468_case__sum__o__map__sum,axiom,
    ! [F2: c > d,G: c > d,H1: real > c,H2: real > c] :
      ( ( comp_S5966200288902075824l_real @ ( sum_case_sum_c_d_c @ F2 @ G ) @ ( sum_ma393449481388452872real_c @ H1 @ H2 ) )
      = ( sum_ca2032921040086735207d_real @ ( comp_c_d_real @ F2 @ H1 ) @ ( comp_c_d_real @ G @ H2 ) ) ) ).

% case_sum_o_map_sum
thf(fact_469_case__sum__o__map__sum,axiom,
    ! [F2: a > b,G: a > b,H1: real > a,H2: real > a] :
      ( ( comp_S8357179028215158318l_real @ ( sum_case_sum_a_b_a @ F2 @ G ) @ ( sum_ma6640897443653631624real_a @ H1 @ H2 ) )
      = ( sum_ca63855846565249637b_real @ ( comp_a_b_real @ F2 @ H1 ) @ ( comp_a_b_real @ G @ H2 ) ) ) ).

% case_sum_o_map_sum
thf(fact_470_sum__set__simps_I1_J,axiom,
    ! [X2: a] :
      ( ( basic_setl_a_c @ ( sum_Inl_a_c @ X2 ) )
      = ( insert_a @ X2 @ bot_bot_set_a ) ) ).

% sum_set_simps(1)
thf(fact_471_sum__set__simps_I1_J,axiom,
    ! [X2: b] :
      ( ( basic_setl_b_d @ ( sum_Inl_b_d @ X2 ) )
      = ( insert_b @ X2 @ bot_bot_set_b ) ) ).

% sum_set_simps(1)
thf(fact_472_sum__set__simps_I4_J,axiom,
    ! [X2: c] :
      ( ( basic_setr_a_c @ ( sum_Inr_c_a @ X2 ) )
      = ( insert_c @ X2 @ bot_bot_set_c ) ) ).

% sum_set_simps(4)
thf(fact_473_sum__set__simps_I4_J,axiom,
    ! [X2: d] :
      ( ( basic_setr_b_d @ ( sum_Inr_d_b @ X2 ) )
      = ( insert_d @ X2 @ bot_bot_set_d ) ) ).

% sum_set_simps(4)
thf(fact_474_qbs__empty__equiv,axiom,
    ! [X5: quasi_borel_a] :
      ( ( ( qbs_space_a @ X5 )
        = bot_bot_set_a )
      = ( ( qbs_Mx_a @ X5 )
        = bot_bot_set_real_a ) ) ).

% qbs_empty_equiv
thf(fact_475_qbs__empty__equiv,axiom,
    ! [X5: quasi_borel_c] :
      ( ( ( qbs_space_c @ X5 )
        = bot_bot_set_c )
      = ( ( qbs_Mx_c @ X5 )
        = bot_bot_set_real_c ) ) ).

% qbs_empty_equiv
thf(fact_476_qbs__empty__equiv,axiom,
    ! [X5: quasi_borel_b] :
      ( ( ( qbs_space_b @ X5 )
        = bot_bot_set_b )
      = ( ( qbs_Mx_b @ X5 )
        = bot_bot_set_real_b ) ) ).

% qbs_empty_equiv
thf(fact_477_qbs__empty__equiv,axiom,
    ! [X5: quasi_borel_d] :
      ( ( ( qbs_space_d @ X5 )
        = bot_bot_set_d )
      = ( ( qbs_Mx_d @ X5 )
        = bot_bot_set_real_d ) ) ).

% qbs_empty_equiv
thf(fact_478_qbs__empty__equiv,axiom,
    ! [X5: quasi_borel_real] :
      ( ( ( qbs_space_real @ X5 )
        = bot_bot_set_real )
      = ( ( qbs_Mx_real @ X5 )
        = bot_bo6767488733719836353l_real ) ) ).

% qbs_empty_equiv
thf(fact_479_qbs__empty__equiv,axiom,
    ! [X5: quasi_borel_nat] :
      ( ( ( qbs_space_nat @ X5 )
        = bot_bot_set_nat )
      = ( ( qbs_Mx_nat @ X5 )
        = bot_bot_set_real_nat ) ) ).

% qbs_empty_equiv
thf(fact_480_qbs__empty__equiv,axiom,
    ! [X5: quasi_borel_o] :
      ( ( ( qbs_space_o @ X5 )
        = bot_bot_set_o )
      = ( ( qbs_Mx_o @ X5 )
        = bot_bot_set_real_o2 ) ) ).

% qbs_empty_equiv
thf(fact_481_qbs__empty__equiv,axiom,
    ! [X5: quasi_9015997321629101608nnreal] :
      ( ( ( qbs_sp175953267596557954nnreal @ X5 )
        = bot_bo4854962954004695426nnreal )
      = ( ( qbs_Mx6523938229262583809nnreal @ X5 )
        = bot_bo6037503491064675021nnreal ) ) ).

% qbs_empty_equiv
thf(fact_482_comp__cong,axiom,
    ! [F2: real > real,G: real > real,X2: real,F4: real > real,G4: real > real,X6: real] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F4 @ ( G4 @ X6 ) ) )
     => ( ( comp_real_real_real @ F2 @ G @ X2 )
        = ( comp_real_real_real @ F4 @ G4 @ X6 ) ) ) ).

% comp_cong
thf(fact_483_comp__cong,axiom,
    ! [F2: real > $o,G: $o > real,X2: $o,F4: real > $o,G4: $o > real,X6: $o] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F4 @ ( G4 @ X6 ) ) )
     => ( ( comp_real_o_o @ F2 @ G @ X2 )
        = ( comp_real_o_o @ F4 @ G4 @ X6 ) ) ) ).

% comp_cong
thf(fact_484_comp__cong,axiom,
    ! [F2: real > nat,G: nat > real,X2: nat,F4: real > nat,G4: nat > real,X6: nat] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F4 @ ( G4 @ X6 ) ) )
     => ( ( comp_real_nat_nat @ F2 @ G @ X2 )
        = ( comp_real_nat_nat @ F4 @ G4 @ X6 ) ) ) ).

% comp_cong
thf(fact_485_comp__cong,axiom,
    ! [F2: c > d,G: real > c,X2: real,F4: c > d,G4: real > c,X6: real] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F4 @ ( G4 @ X6 ) ) )
     => ( ( comp_c_d_real @ F2 @ G @ X2 )
        = ( comp_c_d_real @ F4 @ G4 @ X6 ) ) ) ).

% comp_cong
thf(fact_486_comp__cong,axiom,
    ! [F2: a > b,G: real > a,X2: real,F4: a > b,G4: real > a,X6: real] :
      ( ( ( F2 @ ( G @ X2 ) )
        = ( F4 @ ( G4 @ X6 ) ) )
     => ( ( comp_a_b_real @ F2 @ G @ X2 )
        = ( comp_a_b_real @ F4 @ G4 @ X6 ) ) ) ).

% comp_cong
thf(fact_487_function__factors__left,axiom,
    ! [G: real > real,F2: real > real] :
      ( ( ! [X3: real,Y5: real] :
            ( ( ( G @ X3 )
              = ( G @ Y5 ) )
           => ( ( F2 @ X3 )
              = ( F2 @ Y5 ) ) ) )
      = ( ? [H3: real > real] :
            ( F2
            = ( comp_real_real_real @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_488_function__factors__left,axiom,
    ! [G: $o > real,F2: $o > $o] :
      ( ( ! [X3: $o,Y5: $o] :
            ( ( ( G @ X3 )
              = ( G @ Y5 ) )
           => ( ( F2 @ X3 )
              = ( F2 @ Y5 ) ) ) )
      = ( ? [H3: real > $o] :
            ( F2
            = ( comp_real_o_o @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_489_function__factors__left,axiom,
    ! [G: nat > real,F2: nat > nat] :
      ( ( ! [X3: nat,Y5: nat] :
            ( ( ( G @ X3 )
              = ( G @ Y5 ) )
           => ( ( F2 @ X3 )
              = ( F2 @ Y5 ) ) ) )
      = ( ? [H3: real > nat] :
            ( F2
            = ( comp_real_nat_nat @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_490_function__factors__left,axiom,
    ! [G: real > c,F2: real > d] :
      ( ( ! [X3: real,Y5: real] :
            ( ( ( G @ X3 )
              = ( G @ Y5 ) )
           => ( ( F2 @ X3 )
              = ( F2 @ Y5 ) ) ) )
      = ( ? [H3: c > d] :
            ( F2
            = ( comp_c_d_real @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_491_function__factors__left,axiom,
    ! [G: real > a,F2: real > b] :
      ( ( ! [X3: real,Y5: real] :
            ( ( ( G @ X3 )
              = ( G @ Y5 ) )
           => ( ( F2 @ X3 )
              = ( F2 @ Y5 ) ) ) )
      = ( ? [H3: a > b] :
            ( F2
            = ( comp_a_b_real @ H3 @ G ) ) ) ) ).

% function_factors_left
thf(fact_492_insertCI,axiom,
    ! [A2: real,B4: set_real,B: real] :
      ( ( ~ ( member_real @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_real @ A2 @ ( insert_real @ B @ B4 ) ) ) ).

% insertCI
thf(fact_493_insertCI,axiom,
    ! [A2: nat,B4: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_494_insertCI,axiom,
    ! [A2: $o,B4: set_o,B: $o] :
      ( ( ~ ( member_o @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_o @ A2 @ ( insert_o @ B @ B4 ) ) ) ).

% insertCI
thf(fact_495_insertCI,axiom,
    ! [A2: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( ~ ( member7908768830364227535nnreal @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member7908768830364227535nnreal @ A2 @ ( insert7407984058720857448nnreal @ B @ B4 ) ) ) ).

% insertCI
thf(fact_496_insertCI,axiom,
    ! [A2: set_nat,B4: set_set_nat,B: set_nat] :
      ( ( ~ ( member_set_nat @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_497_insertCI,axiom,
    ! [A2: set_o,B4: set_set_o,B: set_o] :
      ( ( ~ ( member_set_o @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_set_o @ A2 @ ( insert_set_o @ B @ B4 ) ) ) ).

% insertCI
thf(fact_498_insertCI,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,B4: set_se4580700918925141924nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( ~ ( member603777416030116741nnreal @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ B @ B4 ) ) ) ).

% insertCI
thf(fact_499_insertCI,axiom,
    ! [A2: set_real,B4: set_set_real,B: set_real] :
      ( ( ~ ( member_set_real @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_set_real @ A2 @ ( insert_set_real @ B @ B4 ) ) ) ).

% insertCI
thf(fact_500_insertCI,axiom,
    ! [A2: real > a,B4: set_real_a,B: real > a] :
      ( ( ~ ( member_real_a @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_real_a @ A2 @ ( insert_real_a @ B @ B4 ) ) ) ).

% insertCI
thf(fact_501_insertCI,axiom,
    ! [A2: $o > real,B4: set_o_real,B: $o > real] :
      ( ( ~ ( member_o_real @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_o_real @ A2 @ ( insert_o_real @ B @ B4 ) ) ) ).

% insertCI
thf(fact_502_insert__iff,axiom,
    ! [A2: real,B: real,A: set_real] :
      ( ( member_real @ A2 @ ( insert_real @ B @ A ) )
      = ( ( A2 = B )
        | ( member_real @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_503_insert__iff,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
      = ( ( A2 = B )
        | ( member_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_504_insert__iff,axiom,
    ! [A2: $o,B: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B @ A ) )
      = ( ( A2 = B )
        | ( member_o @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_505_insert__iff,axiom,
    ! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A2 @ ( insert7407984058720857448nnreal @ B @ A ) )
      = ( ( A2 = B )
        | ( member7908768830364227535nnreal @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_506_insert__iff,axiom,
    ! [A2: set_nat,B: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
      = ( ( A2 = B )
        | ( member_set_nat @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_507_insert__iff,axiom,
    ! [A2: set_o,B: set_o,A: set_set_o] :
      ( ( member_set_o @ A2 @ ( insert_set_o @ B @ A ) )
      = ( ( A2 = B )
        | ( member_set_o @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_508_insert__iff,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
      ( ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ B @ A ) )
      = ( ( A2 = B )
        | ( member603777416030116741nnreal @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_509_insert__iff,axiom,
    ! [A2: set_real,B: set_real,A: set_set_real] :
      ( ( member_set_real @ A2 @ ( insert_set_real @ B @ A ) )
      = ( ( A2 = B )
        | ( member_set_real @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_510_insert__iff,axiom,
    ! [A2: real > a,B: real > a,A: set_real_a] :
      ( ( member_real_a @ A2 @ ( insert_real_a @ B @ A ) )
      = ( ( A2 = B )
        | ( member_real_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_511_insert__iff,axiom,
    ! [A2: $o > real,B: $o > real,A: set_o_real] :
      ( ( member_o_real @ A2 @ ( insert_o_real @ B @ A ) )
      = ( ( A2 = B )
        | ( member_o_real @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_512_insert__absorb2,axiom,
    ! [X2: real,A: set_real] :
      ( ( insert_real @ X2 @ ( insert_real @ X2 @ A ) )
      = ( insert_real @ X2 @ A ) ) ).

% insert_absorb2
thf(fact_513_insert__absorb2,axiom,
    ! [X2: nat,A: set_nat] :
      ( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A ) )
      = ( insert_nat @ X2 @ A ) ) ).

% insert_absorb2
thf(fact_514_insert__absorb2,axiom,
    ! [X2: $o,A: set_o] :
      ( ( insert_o @ X2 @ ( insert_o @ X2 @ A ) )
      = ( insert_o @ X2 @ A ) ) ).

% insert_absorb2
thf(fact_515_insert__absorb2,axiom,
    ! [X2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ X2 @ ( insert7407984058720857448nnreal @ X2 @ A ) )
      = ( insert7407984058720857448nnreal @ X2 @ A ) ) ).

% insert_absorb2
thf(fact_516_singletonI,axiom,
    ! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_517_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_518_singletonI,axiom,
    ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_519_singletonI,axiom,
    ! [A2: extend8495563244428889912nnreal] : ( member7908768830364227535nnreal @ A2 @ ( insert7407984058720857448nnreal @ A2 @ bot_bo4854962954004695426nnreal ) ) ).

% singletonI
thf(fact_520_singletonI,axiom,
    ! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_521_singletonI,axiom,
    ! [A2: set_o] : ( member_set_o @ A2 @ ( insert_set_o @ A2 @ bot_bot_set_set_o ) ) ).

% singletonI
thf(fact_522_singletonI,axiom,
    ! [A2: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ A2 @ bot_bo2988155216863113784nnreal ) ) ).

% singletonI
thf(fact_523_singletonI,axiom,
    ! [A2: set_real] : ( member_set_real @ A2 @ ( insert_set_real @ A2 @ bot_bot_set_set_real ) ) ).

% singletonI
thf(fact_524_singletonI,axiom,
    ! [A2: real > a] : ( member_real_a @ A2 @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) ) ).

% singletonI
thf(fact_525_singletonI,axiom,
    ! [A2: $o > real] : ( member_o_real @ A2 @ ( insert_o_real @ A2 @ bot_bot_set_o_real ) ) ).

% singletonI
thf(fact_526_eqb__space,axiom,
    ( ( qbs_space_real @ empty_1876425439295802446l_real )
    = bot_bot_set_real ) ).

% eqb_space
thf(fact_527_eqb__space,axiom,
    ( ( qbs_space_nat @ empty_8278123436611590770el_nat )
    = bot_bot_set_nat ) ).

% eqb_space
thf(fact_528_eqb__space,axiom,
    ( ( qbs_space_o @ empty_quasi_borel_o )
    = bot_bot_set_o ) ).

% eqb_space
thf(fact_529_eqb__space,axiom,
    ( ( qbs_sp175953267596557954nnreal @ empty_1788085430566700506nnreal )
    = bot_bo4854962954004695426nnreal ) ).

% eqb_space
thf(fact_530_insertE,axiom,
    ! [A2: real,B: real,A: set_real] :
      ( ( member_real @ A2 @ ( insert_real @ B @ A ) )
     => ( ( A2 != B )
       => ( member_real @ A2 @ A ) ) ) ).

% insertE
thf(fact_531_insertE,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
     => ( ( A2 != B )
       => ( member_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_532_insertE,axiom,
    ! [A2: $o,B: $o,A: set_o] :
      ( ( member_o @ A2 @ ( insert_o @ B @ A ) )
     => ( ( A2 = ~ B )
       => ( member_o @ A2 @ A ) ) ) ).

% insertE
thf(fact_533_insertE,axiom,
    ! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A2 @ ( insert7407984058720857448nnreal @ B @ A ) )
     => ( ( A2 != B )
       => ( member7908768830364227535nnreal @ A2 @ A ) ) ) ).

% insertE
thf(fact_534_insertE,axiom,
    ! [A2: set_nat,B: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
     => ( ( A2 != B )
       => ( member_set_nat @ A2 @ A ) ) ) ).

% insertE
thf(fact_535_insertE,axiom,
    ! [A2: set_o,B: set_o,A: set_set_o] :
      ( ( member_set_o @ A2 @ ( insert_set_o @ B @ A ) )
     => ( ( A2 != B )
       => ( member_set_o @ A2 @ A ) ) ) ).

% insertE
thf(fact_536_insertE,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,B: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
      ( ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ B @ A ) )
     => ( ( A2 != B )
       => ( member603777416030116741nnreal @ A2 @ A ) ) ) ).

% insertE
thf(fact_537_insertE,axiom,
    ! [A2: set_real,B: set_real,A: set_set_real] :
      ( ( member_set_real @ A2 @ ( insert_set_real @ B @ A ) )
     => ( ( A2 != B )
       => ( member_set_real @ A2 @ A ) ) ) ).

% insertE
thf(fact_538_insertE,axiom,
    ! [A2: real > a,B: real > a,A: set_real_a] :
      ( ( member_real_a @ A2 @ ( insert_real_a @ B @ A ) )
     => ( ( A2 != B )
       => ( member_real_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_539_insertE,axiom,
    ! [A2: $o > real,B: $o > real,A: set_o_real] :
      ( ( member_o_real @ A2 @ ( insert_o_real @ B @ A ) )
     => ( ( A2 != B )
       => ( member_o_real @ A2 @ A ) ) ) ).

% insertE
thf(fact_540_insertI1,axiom,
    ! [A2: real,B4: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B4 ) ) ).

% insertI1
thf(fact_541_insertI1,axiom,
    ! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).

% insertI1
thf(fact_542_insertI1,axiom,
    ! [A2: $o,B4: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B4 ) ) ).

% insertI1
thf(fact_543_insertI1,axiom,
    ! [A2: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] : ( member7908768830364227535nnreal @ A2 @ ( insert7407984058720857448nnreal @ A2 @ B4 ) ) ).

% insertI1
thf(fact_544_insertI1,axiom,
    ! [A2: set_nat,B4: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B4 ) ) ).

% insertI1
thf(fact_545_insertI1,axiom,
    ! [A2: set_o,B4: set_set_o] : ( member_set_o @ A2 @ ( insert_set_o @ A2 @ B4 ) ) ).

% insertI1
thf(fact_546_insertI1,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,B4: set_se4580700918925141924nnreal] : ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ A2 @ B4 ) ) ).

% insertI1
thf(fact_547_insertI1,axiom,
    ! [A2: set_real,B4: set_set_real] : ( member_set_real @ A2 @ ( insert_set_real @ A2 @ B4 ) ) ).

% insertI1
thf(fact_548_insertI1,axiom,
    ! [A2: real > a,B4: set_real_a] : ( member_real_a @ A2 @ ( insert_real_a @ A2 @ B4 ) ) ).

% insertI1
thf(fact_549_insertI1,axiom,
    ! [A2: $o > real,B4: set_o_real] : ( member_o_real @ A2 @ ( insert_o_real @ A2 @ B4 ) ) ).

% insertI1
thf(fact_550_insertI2,axiom,
    ! [A2: real,B4: set_real,B: real] :
      ( ( member_real @ A2 @ B4 )
     => ( member_real @ A2 @ ( insert_real @ B @ B4 ) ) ) ).

% insertI2
thf(fact_551_insertI2,axiom,
    ! [A2: nat,B4: set_nat,B: nat] :
      ( ( member_nat @ A2 @ B4 )
     => ( member_nat @ A2 @ ( insert_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_552_insertI2,axiom,
    ! [A2: $o,B4: set_o,B: $o] :
      ( ( member_o @ A2 @ B4 )
     => ( member_o @ A2 @ ( insert_o @ B @ B4 ) ) ) ).

% insertI2
thf(fact_553_insertI2,axiom,
    ! [A2: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ A2 @ B4 )
     => ( member7908768830364227535nnreal @ A2 @ ( insert7407984058720857448nnreal @ B @ B4 ) ) ) ).

% insertI2
thf(fact_554_insertI2,axiom,
    ! [A2: set_nat,B4: set_set_nat,B: set_nat] :
      ( ( member_set_nat @ A2 @ B4 )
     => ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_555_insertI2,axiom,
    ! [A2: set_o,B4: set_set_o,B: set_o] :
      ( ( member_set_o @ A2 @ B4 )
     => ( member_set_o @ A2 @ ( insert_set_o @ B @ B4 ) ) ) ).

% insertI2
thf(fact_556_insertI2,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,B4: set_se4580700918925141924nnreal,B: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ A2 @ B4 )
     => ( member603777416030116741nnreal @ A2 @ ( insert1343806209672318238nnreal @ B @ B4 ) ) ) ).

% insertI2
thf(fact_557_insertI2,axiom,
    ! [A2: set_real,B4: set_set_real,B: set_real] :
      ( ( member_set_real @ A2 @ B4 )
     => ( member_set_real @ A2 @ ( insert_set_real @ B @ B4 ) ) ) ).

% insertI2
thf(fact_558_insertI2,axiom,
    ! [A2: real > a,B4: set_real_a,B: real > a] :
      ( ( member_real_a @ A2 @ B4 )
     => ( member_real_a @ A2 @ ( insert_real_a @ B @ B4 ) ) ) ).

% insertI2
thf(fact_559_insertI2,axiom,
    ! [A2: $o > real,B4: set_o_real,B: $o > real] :
      ( ( member_o_real @ A2 @ B4 )
     => ( member_o_real @ A2 @ ( insert_o_real @ B @ B4 ) ) ) ).

% insertI2
thf(fact_560_Set_Oset__insert,axiom,
    ! [X2: real,A: set_real] :
      ( ( member_real @ X2 @ A )
     => ~ ! [B5: set_real] :
            ( ( A
              = ( insert_real @ X2 @ B5 ) )
           => ( member_real @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_561_Set_Oset__insert,axiom,
    ! [X2: nat,A: set_nat] :
      ( ( member_nat @ X2 @ A )
     => ~ ! [B5: set_nat] :
            ( ( A
              = ( insert_nat @ X2 @ B5 ) )
           => ( member_nat @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_562_Set_Oset__insert,axiom,
    ! [X2: $o,A: set_o] :
      ( ( member_o @ X2 @ A )
     => ~ ! [B5: set_o] :
            ( ( A
              = ( insert_o @ X2 @ B5 ) )
           => ( member_o @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_563_Set_Oset__insert,axiom,
    ! [X2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ X2 @ A )
     => ~ ! [B5: set_Ex3793607809372303086nnreal] :
            ( ( A
              = ( insert7407984058720857448nnreal @ X2 @ B5 ) )
           => ( member7908768830364227535nnreal @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_564_Set_Oset__insert,axiom,
    ! [X2: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ X2 @ A )
     => ~ ! [B5: set_set_nat] :
            ( ( A
              = ( insert_set_nat @ X2 @ B5 ) )
           => ( member_set_nat @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_565_Set_Oset__insert,axiom,
    ! [X2: set_o,A: set_set_o] :
      ( ( member_set_o @ X2 @ A )
     => ~ ! [B5: set_set_o] :
            ( ( A
              = ( insert_set_o @ X2 @ B5 ) )
           => ( member_set_o @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_566_Set_Oset__insert,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
      ( ( member603777416030116741nnreal @ X2 @ A )
     => ~ ! [B5: set_se4580700918925141924nnreal] :
            ( ( A
              = ( insert1343806209672318238nnreal @ X2 @ B5 ) )
           => ( member603777416030116741nnreal @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_567_Set_Oset__insert,axiom,
    ! [X2: set_real,A: set_set_real] :
      ( ( member_set_real @ X2 @ A )
     => ~ ! [B5: set_set_real] :
            ( ( A
              = ( insert_set_real @ X2 @ B5 ) )
           => ( member_set_real @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_568_Set_Oset__insert,axiom,
    ! [X2: real > a,A: set_real_a] :
      ( ( member_real_a @ X2 @ A )
     => ~ ! [B5: set_real_a] :
            ( ( A
              = ( insert_real_a @ X2 @ B5 ) )
           => ( member_real_a @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_569_Set_Oset__insert,axiom,
    ! [X2: $o > real,A: set_o_real] :
      ( ( member_o_real @ X2 @ A )
     => ~ ! [B5: set_o_real] :
            ( ( A
              = ( insert_o_real @ X2 @ B5 ) )
           => ( member_o_real @ X2 @ B5 ) ) ) ).

% Set.set_insert
thf(fact_570_insert__ident,axiom,
    ! [X2: real,A: set_real,B4: set_real] :
      ( ~ ( member_real @ X2 @ A )
     => ( ~ ( member_real @ X2 @ B4 )
       => ( ( ( insert_real @ X2 @ A )
            = ( insert_real @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_571_insert__ident,axiom,
    ! [X2: nat,A: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X2 @ A )
     => ( ~ ( member_nat @ X2 @ B4 )
       => ( ( ( insert_nat @ X2 @ A )
            = ( insert_nat @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_572_insert__ident,axiom,
    ! [X2: $o,A: set_o,B4: set_o] :
      ( ~ ( member_o @ X2 @ A )
     => ( ~ ( member_o @ X2 @ B4 )
       => ( ( ( insert_o @ X2 @ A )
            = ( insert_o @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_573_insert__ident,axiom,
    ! [X2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ X2 @ A )
     => ( ~ ( member7908768830364227535nnreal @ X2 @ B4 )
       => ( ( ( insert7407984058720857448nnreal @ X2 @ A )
            = ( insert7407984058720857448nnreal @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_574_insert__ident,axiom,
    ! [X2: set_nat,A: set_set_nat,B4: set_set_nat] :
      ( ~ ( member_set_nat @ X2 @ A )
     => ( ~ ( member_set_nat @ X2 @ B4 )
       => ( ( ( insert_set_nat @ X2 @ A )
            = ( insert_set_nat @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_575_insert__ident,axiom,
    ! [X2: set_o,A: set_set_o,B4: set_set_o] :
      ( ~ ( member_set_o @ X2 @ A )
     => ( ~ ( member_set_o @ X2 @ B4 )
       => ( ( ( insert_set_o @ X2 @ A )
            = ( insert_set_o @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_576_insert__ident,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal] :
      ( ~ ( member603777416030116741nnreal @ X2 @ A )
     => ( ~ ( member603777416030116741nnreal @ X2 @ B4 )
       => ( ( ( insert1343806209672318238nnreal @ X2 @ A )
            = ( insert1343806209672318238nnreal @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_577_insert__ident,axiom,
    ! [X2: set_real,A: set_set_real,B4: set_set_real] :
      ( ~ ( member_set_real @ X2 @ A )
     => ( ~ ( member_set_real @ X2 @ B4 )
       => ( ( ( insert_set_real @ X2 @ A )
            = ( insert_set_real @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_578_insert__ident,axiom,
    ! [X2: real > a,A: set_real_a,B4: set_real_a] :
      ( ~ ( member_real_a @ X2 @ A )
     => ( ~ ( member_real_a @ X2 @ B4 )
       => ( ( ( insert_real_a @ X2 @ A )
            = ( insert_real_a @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_579_insert__ident,axiom,
    ! [X2: $o > real,A: set_o_real,B4: set_o_real] :
      ( ~ ( member_o_real @ X2 @ A )
     => ( ~ ( member_o_real @ X2 @ B4 )
       => ( ( ( insert_o_real @ X2 @ A )
            = ( insert_o_real @ X2 @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_580_insert__absorb,axiom,
    ! [A2: real,A: set_real] :
      ( ( member_real @ A2 @ A )
     => ( ( insert_real @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_581_insert__absorb,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_582_insert__absorb,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ( ( insert_o @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_583_insert__absorb,axiom,
    ! [A2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A2 @ A )
     => ( ( insert7407984058720857448nnreal @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_584_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_585_insert__absorb,axiom,
    ! [A2: set_o,A: set_set_o] :
      ( ( member_set_o @ A2 @ A )
     => ( ( insert_set_o @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_586_insert__absorb,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
      ( ( member603777416030116741nnreal @ A2 @ A )
     => ( ( insert1343806209672318238nnreal @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_587_insert__absorb,axiom,
    ! [A2: set_real,A: set_set_real] :
      ( ( member_set_real @ A2 @ A )
     => ( ( insert_set_real @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_588_insert__absorb,axiom,
    ! [A2: real > a,A: set_real_a] :
      ( ( member_real_a @ A2 @ A )
     => ( ( insert_real_a @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_589_insert__absorb,axiom,
    ! [A2: $o > real,A: set_o_real] :
      ( ( member_o_real @ A2 @ A )
     => ( ( insert_o_real @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_590_insert__eq__iff,axiom,
    ! [A2: real,A: set_real,B: real,B4: set_real] :
      ( ~ ( member_real @ A2 @ A )
     => ( ~ ( member_real @ B @ B4 )
       => ( ( ( insert_real @ A2 @ A )
            = ( insert_real @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_real] :
                  ( ( A
                    = ( insert_real @ B @ C2 ) )
                  & ~ ( member_real @ B @ C2 )
                  & ( B4
                    = ( insert_real @ A2 @ C2 ) )
                  & ~ ( member_real @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_591_insert__eq__iff,axiom,
    ! [A2: nat,A: set_nat,B: nat,B4: set_nat] :
      ( ~ ( member_nat @ A2 @ A )
     => ( ~ ( member_nat @ B @ B4 )
       => ( ( ( insert_nat @ A2 @ A )
            = ( insert_nat @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_nat] :
                  ( ( A
                    = ( insert_nat @ B @ C2 ) )
                  & ~ ( member_nat @ B @ C2 )
                  & ( B4
                    = ( insert_nat @ A2 @ C2 ) )
                  & ~ ( member_nat @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_592_insert__eq__iff,axiom,
    ! [A2: $o,A: set_o,B: $o,B4: set_o] :
      ( ~ ( member_o @ A2 @ A )
     => ( ~ ( member_o @ B @ B4 )
       => ( ( ( insert_o @ A2 @ A )
            = ( insert_o @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 = ~ B )
             => ? [C2: set_o] :
                  ( ( A
                    = ( insert_o @ B @ C2 ) )
                  & ~ ( member_o @ B @ C2 )
                  & ( B4
                    = ( insert_o @ A2 @ C2 ) )
                  & ~ ( member_o @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_593_insert__eq__iff,axiom,
    ! [A2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ A2 @ A )
     => ( ~ ( member7908768830364227535nnreal @ B @ B4 )
       => ( ( ( insert7407984058720857448nnreal @ A2 @ A )
            = ( insert7407984058720857448nnreal @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_Ex3793607809372303086nnreal] :
                  ( ( A
                    = ( insert7407984058720857448nnreal @ B @ C2 ) )
                  & ~ ( member7908768830364227535nnreal @ B @ C2 )
                  & ( B4
                    = ( insert7407984058720857448nnreal @ A2 @ C2 ) )
                  & ~ ( member7908768830364227535nnreal @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_594_insert__eq__iff,axiom,
    ! [A2: set_nat,A: set_set_nat,B: set_nat,B4: set_set_nat] :
      ( ~ ( member_set_nat @ A2 @ A )
     => ( ~ ( member_set_nat @ B @ B4 )
       => ( ( ( insert_set_nat @ A2 @ A )
            = ( insert_set_nat @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_set_nat] :
                  ( ( A
                    = ( insert_set_nat @ B @ C2 ) )
                  & ~ ( member_set_nat @ B @ C2 )
                  & ( B4
                    = ( insert_set_nat @ A2 @ C2 ) )
                  & ~ ( member_set_nat @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_595_insert__eq__iff,axiom,
    ! [A2: set_o,A: set_set_o,B: set_o,B4: set_set_o] :
      ( ~ ( member_set_o @ A2 @ A )
     => ( ~ ( member_set_o @ B @ B4 )
       => ( ( ( insert_set_o @ A2 @ A )
            = ( insert_set_o @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_set_o] :
                  ( ( A
                    = ( insert_set_o @ B @ C2 ) )
                  & ~ ( member_set_o @ B @ C2 )
                  & ( B4
                    = ( insert_set_o @ A2 @ C2 ) )
                  & ~ ( member_set_o @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_596_insert__eq__iff,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B: set_Ex3793607809372303086nnreal,B4: set_se4580700918925141924nnreal] :
      ( ~ ( member603777416030116741nnreal @ A2 @ A )
     => ( ~ ( member603777416030116741nnreal @ B @ B4 )
       => ( ( ( insert1343806209672318238nnreal @ A2 @ A )
            = ( insert1343806209672318238nnreal @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_se4580700918925141924nnreal] :
                  ( ( A
                    = ( insert1343806209672318238nnreal @ B @ C2 ) )
                  & ~ ( member603777416030116741nnreal @ B @ C2 )
                  & ( B4
                    = ( insert1343806209672318238nnreal @ A2 @ C2 ) )
                  & ~ ( member603777416030116741nnreal @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_597_insert__eq__iff,axiom,
    ! [A2: set_real,A: set_set_real,B: set_real,B4: set_set_real] :
      ( ~ ( member_set_real @ A2 @ A )
     => ( ~ ( member_set_real @ B @ B4 )
       => ( ( ( insert_set_real @ A2 @ A )
            = ( insert_set_real @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_set_real] :
                  ( ( A
                    = ( insert_set_real @ B @ C2 ) )
                  & ~ ( member_set_real @ B @ C2 )
                  & ( B4
                    = ( insert_set_real @ A2 @ C2 ) )
                  & ~ ( member_set_real @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_598_insert__eq__iff,axiom,
    ! [A2: real > a,A: set_real_a,B: real > a,B4: set_real_a] :
      ( ~ ( member_real_a @ A2 @ A )
     => ( ~ ( member_real_a @ B @ B4 )
       => ( ( ( insert_real_a @ A2 @ A )
            = ( insert_real_a @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_real_a] :
                  ( ( A
                    = ( insert_real_a @ B @ C2 ) )
                  & ~ ( member_real_a @ B @ C2 )
                  & ( B4
                    = ( insert_real_a @ A2 @ C2 ) )
                  & ~ ( member_real_a @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_599_insert__eq__iff,axiom,
    ! [A2: $o > real,A: set_o_real,B: $o > real,B4: set_o_real] :
      ( ~ ( member_o_real @ A2 @ A )
     => ( ~ ( member_o_real @ B @ B4 )
       => ( ( ( insert_o_real @ A2 @ A )
            = ( insert_o_real @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_o_real] :
                  ( ( A
                    = ( insert_o_real @ B @ C2 ) )
                  & ~ ( member_o_real @ B @ C2 )
                  & ( B4
                    = ( insert_o_real @ A2 @ C2 ) )
                  & ~ ( member_o_real @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_600_insert__commute,axiom,
    ! [X2: real,Y: real,A: set_real] :
      ( ( insert_real @ X2 @ ( insert_real @ Y @ A ) )
      = ( insert_real @ Y @ ( insert_real @ X2 @ A ) ) ) ).

% insert_commute
thf(fact_601_insert__commute,axiom,
    ! [X2: nat,Y: nat,A: set_nat] :
      ( ( insert_nat @ X2 @ ( insert_nat @ Y @ A ) )
      = ( insert_nat @ Y @ ( insert_nat @ X2 @ A ) ) ) ).

% insert_commute
thf(fact_602_insert__commute,axiom,
    ! [X2: $o,Y: $o,A: set_o] :
      ( ( insert_o @ X2 @ ( insert_o @ Y @ A ) )
      = ( insert_o @ Y @ ( insert_o @ X2 @ A ) ) ) ).

% insert_commute
thf(fact_603_insert__commute,axiom,
    ! [X2: extend8495563244428889912nnreal,Y: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ X2 @ ( insert7407984058720857448nnreal @ Y @ A ) )
      = ( insert7407984058720857448nnreal @ Y @ ( insert7407984058720857448nnreal @ X2 @ A ) ) ) ).

% insert_commute
thf(fact_604_mk__disjoint__insert,axiom,
    ! [A2: real,A: set_real] :
      ( ( member_real @ A2 @ A )
     => ? [B5: set_real] :
          ( ( A
            = ( insert_real @ A2 @ B5 ) )
          & ~ ( member_real @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_605_mk__disjoint__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ? [B5: set_nat] :
          ( ( A
            = ( insert_nat @ A2 @ B5 ) )
          & ~ ( member_nat @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_606_mk__disjoint__insert,axiom,
    ! [A2: $o,A: set_o] :
      ( ( member_o @ A2 @ A )
     => ? [B5: set_o] :
          ( ( A
            = ( insert_o @ A2 @ B5 ) )
          & ~ ( member_o @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_607_mk__disjoint__insert,axiom,
    ! [A2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( member7908768830364227535nnreal @ A2 @ A )
     => ? [B5: set_Ex3793607809372303086nnreal] :
          ( ( A
            = ( insert7407984058720857448nnreal @ A2 @ B5 ) )
          & ~ ( member7908768830364227535nnreal @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_608_mk__disjoint__insert,axiom,
    ! [A2: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ A2 @ A )
     => ? [B5: set_set_nat] :
          ( ( A
            = ( insert_set_nat @ A2 @ B5 ) )
          & ~ ( member_set_nat @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_609_mk__disjoint__insert,axiom,
    ! [A2: set_o,A: set_set_o] :
      ( ( member_set_o @ A2 @ A )
     => ? [B5: set_set_o] :
          ( ( A
            = ( insert_set_o @ A2 @ B5 ) )
          & ~ ( member_set_o @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_610_mk__disjoint__insert,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] :
      ( ( member603777416030116741nnreal @ A2 @ A )
     => ? [B5: set_se4580700918925141924nnreal] :
          ( ( A
            = ( insert1343806209672318238nnreal @ A2 @ B5 ) )
          & ~ ( member603777416030116741nnreal @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_611_mk__disjoint__insert,axiom,
    ! [A2: set_real,A: set_set_real] :
      ( ( member_set_real @ A2 @ A )
     => ? [B5: set_set_real] :
          ( ( A
            = ( insert_set_real @ A2 @ B5 ) )
          & ~ ( member_set_real @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_612_mk__disjoint__insert,axiom,
    ! [A2: real > a,A: set_real_a] :
      ( ( member_real_a @ A2 @ A )
     => ? [B5: set_real_a] :
          ( ( A
            = ( insert_real_a @ A2 @ B5 ) )
          & ~ ( member_real_a @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_613_mk__disjoint__insert,axiom,
    ! [A2: $o > real,A: set_o_real] :
      ( ( member_o_real @ A2 @ A )
     => ? [B5: set_o_real] :
          ( ( A
            = ( insert_o_real @ A2 @ B5 ) )
          & ~ ( member_o_real @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_614_qbs__space__eq__Mx,axiom,
    ! [X5: quasi_borel_a,Y4: quasi_borel_a] :
      ( ( ( qbs_Mx_a @ X5 )
        = ( qbs_Mx_a @ Y4 ) )
     => ( ( qbs_space_a @ X5 )
        = ( qbs_space_a @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_615_qbs__space__eq__Mx,axiom,
    ! [X5: quasi_borel_c,Y4: quasi_borel_c] :
      ( ( ( qbs_Mx_c @ X5 )
        = ( qbs_Mx_c @ Y4 ) )
     => ( ( qbs_space_c @ X5 )
        = ( qbs_space_c @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_616_qbs__space__eq__Mx,axiom,
    ! [X5: quasi_borel_b,Y4: quasi_borel_b] :
      ( ( ( qbs_Mx_b @ X5 )
        = ( qbs_Mx_b @ Y4 ) )
     => ( ( qbs_space_b @ X5 )
        = ( qbs_space_b @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_617_qbs__space__eq__Mx,axiom,
    ! [X5: quasi_borel_d,Y4: quasi_borel_d] :
      ( ( ( qbs_Mx_d @ X5 )
        = ( qbs_Mx_d @ Y4 ) )
     => ( ( qbs_space_d @ X5 )
        = ( qbs_space_d @ Y4 ) ) ) ).

% qbs_space_eq_Mx
thf(fact_618_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > a,X5: quasi_borel_a,R2: real] :
      ( ( member_real_a @ Alpha @ ( qbs_Mx_a @ X5 ) )
     => ( member_a @ ( Alpha @ R2 ) @ ( qbs_space_a @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_619_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > c,X5: quasi_borel_c,R2: real] :
      ( ( member_real_c @ Alpha @ ( qbs_Mx_c @ X5 ) )
     => ( member_c @ ( Alpha @ R2 ) @ ( qbs_space_c @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_620_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > b,X5: quasi_borel_b,R2: real] :
      ( ( member_real_b @ Alpha @ ( qbs_Mx_b @ X5 ) )
     => ( member_b @ ( Alpha @ R2 ) @ ( qbs_space_b @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_621_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > d,X5: quasi_borel_d,R2: real] :
      ( ( member_real_d @ Alpha @ ( qbs_Mx_d @ X5 ) )
     => ( member_d @ ( Alpha @ R2 ) @ ( qbs_space_d @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_622_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > set_nat,X5: quasi_borel_set_nat,R2: real] :
      ( ( member_real_set_nat @ Alpha @ ( qbs_Mx_set_nat @ X5 ) )
     => ( member_set_nat @ ( Alpha @ R2 ) @ ( qbs_space_set_nat @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_623_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > set_o,X5: quasi_borel_set_o,R2: real] :
      ( ( member_real_set_o @ Alpha @ ( qbs_Mx_set_o @ X5 ) )
     => ( member_set_o @ ( Alpha @ R2 ) @ ( qbs_space_set_o @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_624_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > set_Ex3793607809372303086nnreal,X5: quasi_953260806197706462nnreal,R2: real] :
      ( ( member8689841359643572048nnreal @ Alpha @ ( qbs_Mx3319822389276443575nnreal @ X5 ) )
     => ( member603777416030116741nnreal @ ( Alpha @ R2 ) @ ( qbs_sp6328763913151411768nnreal @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_625_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > set_real,X5: quasi_borel_set_real,R2: real] :
      ( ( member_real_set_real @ Alpha @ ( qbs_Mx_set_real @ X5 ) )
     => ( member_set_real @ ( Alpha @ R2 ) @ ( qbs_space_set_real @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_626_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > real > a,X5: quasi_borel_real_a,R2: real] :
      ( ( member_real_real_a @ Alpha @ ( qbs_Mx_real_a @ X5 ) )
     => ( member_real_a @ ( Alpha @ R2 ) @ ( qbs_space_real_a @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_627_qbs__Mx__to__X_I2_J,axiom,
    ! [Alpha: real > $o > real,X5: quasi_borel_o_real,R2: real] :
      ( ( member_real_o_real @ Alpha @ ( qbs_Mx_o_real @ X5 ) )
     => ( member_o_real @ ( Alpha @ R2 ) @ ( qbs_space_o_real @ X5 ) ) ) ).

% qbs_Mx_to_X(2)
thf(fact_628_insert__UNIV,axiom,
    ! [X2: real] :
      ( ( insert_real @ X2 @ top_top_set_real )
      = top_top_set_real ) ).

% insert_UNIV
thf(fact_629_insert__UNIV,axiom,
    ! [X2: $o] :
      ( ( insert_o @ X2 @ top_top_set_o )
      = top_top_set_o ) ).

% insert_UNIV
thf(fact_630_insert__UNIV,axiom,
    ! [X2: nat] :
      ( ( insert_nat @ X2 @ top_top_set_nat )
      = top_top_set_nat ) ).

% insert_UNIV
thf(fact_631_insert__UNIV,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( insert7407984058720857448nnreal @ X2 @ top_to7994903218803871134nnreal )
      = top_to7994903218803871134nnreal ) ).

% insert_UNIV
thf(fact_632_singleton__inject,axiom,
    ! [A2: real,B: real] :
      ( ( ( insert_real @ A2 @ bot_bot_set_real )
        = ( insert_real @ B @ bot_bot_set_real ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_633_singleton__inject,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_634_singleton__inject,axiom,
    ! [A2: $o,B: $o] :
      ( ( ( insert_o @ A2 @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_635_singleton__inject,axiom,
    ! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( insert7407984058720857448nnreal @ A2 @ bot_bo4854962954004695426nnreal )
        = ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_636_insert__not__empty,axiom,
    ! [A2: real,A: set_real] :
      ( ( insert_real @ A2 @ A )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_637_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ A )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_638_insert__not__empty,axiom,
    ! [A2: $o,A: set_o] :
      ( ( insert_o @ A2 @ A )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_639_insert__not__empty,axiom,
    ! [A2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( insert7407984058720857448nnreal @ A2 @ A )
     != bot_bo4854962954004695426nnreal ) ).

% insert_not_empty
thf(fact_640_doubleton__eq__iff,axiom,
    ! [A2: real,B: real,C: real,D: real] :
      ( ( ( insert_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_641_doubleton__eq__iff,axiom,
    ! [A2: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_642_doubleton__eq__iff,axiom,
    ! [A2: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A2 @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_643_doubleton__eq__iff,axiom,
    ! [A2: extend8495563244428889912nnreal,B: extend8495563244428889912nnreal,C: extend8495563244428889912nnreal,D: extend8495563244428889912nnreal] :
      ( ( ( insert7407984058720857448nnreal @ A2 @ ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) )
        = ( insert7407984058720857448nnreal @ C @ ( insert7407984058720857448nnreal @ D @ bot_bo4854962954004695426nnreal ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_644_singleton__iff,axiom,
    ! [B: real,A2: real] :
      ( ( member_real @ B @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_645_singleton__iff,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_646_singleton__iff,axiom,
    ! [B: $o,A2: $o] :
      ( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_647_singleton__iff,axiom,
    ! [B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ B @ ( insert7407984058720857448nnreal @ A2 @ bot_bo4854962954004695426nnreal ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_648_singleton__iff,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_649_singleton__iff,axiom,
    ! [B: set_o,A2: set_o] :
      ( ( member_set_o @ B @ ( insert_set_o @ A2 @ bot_bot_set_set_o ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_650_singleton__iff,axiom,
    ! [B: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ B @ ( insert1343806209672318238nnreal @ A2 @ bot_bo2988155216863113784nnreal ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_651_singleton__iff,axiom,
    ! [B: set_real,A2: set_real] :
      ( ( member_set_real @ B @ ( insert_set_real @ A2 @ bot_bot_set_set_real ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_652_singleton__iff,axiom,
    ! [B: real > a,A2: real > a] :
      ( ( member_real_a @ B @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_653_singleton__iff,axiom,
    ! [B: $o > real,A2: $o > real] :
      ( ( member_o_real @ B @ ( insert_o_real @ A2 @ bot_bot_set_o_real ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_654_singletonD,axiom,
    ! [B: real,A2: real] :
      ( ( member_real @ B @ ( insert_real @ A2 @ bot_bot_set_real ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_655_singletonD,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_656_singletonD,axiom,
    ! [B: $o,A2: $o] :
      ( ( member_o @ B @ ( insert_o @ A2 @ bot_bot_set_o ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_657_singletonD,axiom,
    ! [B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ B @ ( insert7407984058720857448nnreal @ A2 @ bot_bo4854962954004695426nnreal ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_658_singletonD,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_659_singletonD,axiom,
    ! [B: set_o,A2: set_o] :
      ( ( member_set_o @ B @ ( insert_set_o @ A2 @ bot_bot_set_set_o ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_660_singletonD,axiom,
    ! [B: set_Ex3793607809372303086nnreal,A2: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ B @ ( insert1343806209672318238nnreal @ A2 @ bot_bo2988155216863113784nnreal ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_661_singletonD,axiom,
    ! [B: set_real,A2: set_real] :
      ( ( member_set_real @ B @ ( insert_set_real @ A2 @ bot_bot_set_set_real ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_662_singletonD,axiom,
    ! [B: real > a,A2: real > a] :
      ( ( member_real_a @ B @ ( insert_real_a @ A2 @ bot_bot_set_real_a ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_663_singletonD,axiom,
    ! [B: $o > real,A2: $o > real] :
      ( ( member_o_real @ B @ ( insert_o_real @ A2 @ bot_bot_set_o_real ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_664_qbs__morphism__cong,axiom,
    ! [X5: quasi_borel_real,F2: real > a,G: real > a,Y4: quasi_borel_a] :
      ( ! [X: real] :
          ( ( member_real @ X @ ( qbs_space_real @ X5 ) )
         => ( ( F2 @ X )
            = ( G @ X ) ) )
     => ( ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X5 @ Y4 ) )
       => ( member_real_a @ G @ ( qbs_morphism_real_a @ X5 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_665_qbs__morphism__cong,axiom,
    ! [X5: quasi_borel_o,F2: $o > real,G: $o > real,Y4: quasi_borel_real] :
      ( ! [X: $o] :
          ( ( member_o @ X @ ( qbs_space_o @ X5 ) )
         => ( ( F2 @ X )
            = ( G @ X ) ) )
     => ( ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X5 @ Y4 ) )
       => ( member_o_real @ G @ ( qbs_morphism_o_real @ X5 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_666_qbs__morphism__cong,axiom,
    ! [X5: quasi_borel_nat,F2: nat > real,G: nat > real,Y4: quasi_borel_real] :
      ( ! [X: nat] :
          ( ( member_nat @ X @ ( qbs_space_nat @ X5 ) )
         => ( ( F2 @ X )
            = ( G @ X ) ) )
     => ( ( member_nat_real @ F2 @ ( qbs_mo2000642995705457910t_real @ X5 @ Y4 ) )
       => ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ X5 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_667_qbs__morphism__cong,axiom,
    ! [X5: quasi_borel_a,F2: a > b,G: a > b,Y4: quasi_borel_b] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( qbs_space_a @ X5 ) )
         => ( ( F2 @ X )
            = ( G @ X ) ) )
     => ( ( member_a_b @ F2 @ ( qbs_morphism_a_b @ X5 @ Y4 ) )
       => ( member_a_b @ G @ ( qbs_morphism_a_b @ X5 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_668_qbs__morphism__cong,axiom,
    ! [X5: quasi_borel_c,F2: c > d,G: c > d,Y4: quasi_borel_d] :
      ( ! [X: c] :
          ( ( member_c @ X @ ( qbs_space_c @ X5 ) )
         => ( ( F2 @ X )
            = ( G @ X ) ) )
     => ( ( member_c_d @ F2 @ ( qbs_morphism_c_d @ X5 @ Y4 ) )
       => ( member_c_d @ G @ ( qbs_morphism_c_d @ X5 @ Y4 ) ) ) ) ).

% qbs_morphism_cong
thf(fact_669_qbs__morphismE_I2_J,axiom,
    ! [F2: real > a,X5: quasi_borel_real,Y4: quasi_borel_a,X2: real] :
      ( ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X5 @ Y4 ) )
     => ( ( member_real @ X2 @ ( qbs_space_real @ X5 ) )
       => ( member_a @ ( F2 @ X2 ) @ ( qbs_space_a @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_670_qbs__morphismE_I2_J,axiom,
    ! [F2: $o > real,X5: quasi_borel_o,Y4: quasi_borel_real,X2: $o] :
      ( ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X5 @ Y4 ) )
     => ( ( member_o @ X2 @ ( qbs_space_o @ X5 ) )
       => ( member_real @ ( F2 @ X2 ) @ ( qbs_space_real @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_671_qbs__morphismE_I2_J,axiom,
    ! [F2: nat > real,X5: quasi_borel_nat,Y4: quasi_borel_real,X2: nat] :
      ( ( member_nat_real @ F2 @ ( qbs_mo2000642995705457910t_real @ X5 @ Y4 ) )
     => ( ( member_nat @ X2 @ ( qbs_space_nat @ X5 ) )
       => ( member_real @ ( F2 @ X2 ) @ ( qbs_space_real @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_672_qbs__morphismE_I2_J,axiom,
    ! [F2: a > b,X5: quasi_borel_a,Y4: quasi_borel_b,X2: a] :
      ( ( member_a_b @ F2 @ ( qbs_morphism_a_b @ X5 @ Y4 ) )
     => ( ( member_a @ X2 @ ( qbs_space_a @ X5 ) )
       => ( member_b @ ( F2 @ X2 ) @ ( qbs_space_b @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_673_qbs__morphismE_I2_J,axiom,
    ! [F2: c > d,X5: quasi_borel_c,Y4: quasi_borel_d,X2: c] :
      ( ( member_c_d @ F2 @ ( qbs_morphism_c_d @ X5 @ Y4 ) )
     => ( ( member_c @ X2 @ ( qbs_space_c @ X5 ) )
       => ( member_d @ ( F2 @ X2 ) @ ( qbs_space_d @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_674_qbs__morphismE_I2_J,axiom,
    ! [F2: set_nat > set_nat,X5: quasi_borel_set_nat,Y4: quasi_borel_set_nat,X2: set_nat] :
      ( ( member1686471427249568706et_nat @ F2 @ ( qbs_mo4089447833561950854et_nat @ X5 @ Y4 ) )
     => ( ( member_set_nat @ X2 @ ( qbs_space_set_nat @ X5 ) )
       => ( member_set_nat @ ( F2 @ X2 ) @ ( qbs_space_set_nat @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_675_qbs__morphismE_I2_J,axiom,
    ! [F2: set_nat > set_o,X5: quasi_borel_set_nat,Y4: quasi_borel_set_o,X2: set_nat] :
      ( ( member_set_nat_set_o @ F2 @ ( qbs_mo5249294451373078392_set_o @ X5 @ Y4 ) )
     => ( ( member_set_nat @ X2 @ ( qbs_space_set_nat @ X5 ) )
       => ( member_set_o @ ( F2 @ X2 ) @ ( qbs_space_set_o @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_676_qbs__morphismE_I2_J,axiom,
    ! [F2: set_nat > set_Ex3793607809372303086nnreal,X5: quasi_borel_set_nat,Y4: quasi_953260806197706462nnreal,X2: set_nat] :
      ( ( member2136063858382286634nnreal @ F2 @ ( qbs_mo3729789103443073518nnreal @ X5 @ Y4 ) )
     => ( ( member_set_nat @ X2 @ ( qbs_space_set_nat @ X5 ) )
       => ( member603777416030116741nnreal @ ( F2 @ X2 ) @ ( qbs_sp6328763913151411768nnreal @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_677_qbs__morphismE_I2_J,axiom,
    ! [F2: set_nat > set_real,X5: quasi_borel_set_nat,Y4: quasi_borel_set_real,X2: set_nat] :
      ( ( member6118920896213660830t_real @ F2 @ ( qbs_mo34581953175210466t_real @ X5 @ Y4 ) )
     => ( ( member_set_nat @ X2 @ ( qbs_space_set_nat @ X5 ) )
       => ( member_set_real @ ( F2 @ X2 ) @ ( qbs_space_set_real @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_678_qbs__morphismE_I2_J,axiom,
    ! [F2: set_o > set_nat,X5: quasi_borel_set_o,Y4: quasi_borel_set_nat,X2: set_o] :
      ( ( member_set_o_set_nat @ F2 @ ( qbs_mo4711702798760812422et_nat @ X5 @ Y4 ) )
     => ( ( member_set_o @ X2 @ ( qbs_space_set_o @ X5 ) )
       => ( member_set_nat @ ( F2 @ X2 ) @ ( qbs_space_set_nat @ Y4 ) ) ) ) ).

% qbs_morphismE(2)
thf(fact_679_sets_Oinsert__in__sets,axiom,
    ! [X2: real,M: sigma_measure_real,A: set_real] :
      ( ( member_set_real @ ( insert_real @ X2 @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real @ A @ ( sigma_sets_real @ M ) )
       => ( member_set_real @ ( insert_real @ X2 @ A ) @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_680_sets_Oinsert__in__sets,axiom,
    ! [X2: nat,M: sigma_measure_nat,A: set_nat] :
      ( ( member_set_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat @ A @ ( sigma_sets_nat @ M ) )
       => ( member_set_nat @ ( insert_nat @ X2 @ A ) @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_681_sets_Oinsert__in__sets,axiom,
    ! [X2: $o,M: sigma_measure_o,A: set_o] :
      ( ( member_set_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o @ A @ ( sigma_sets_o @ M ) )
       => ( member_set_o @ ( insert_o @ X2 @ A ) @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_682_sets_Oinsert__in__sets,axiom,
    ! [X2: extend8495563244428889912nnreal,M: sigma_7234349610311085201nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ A ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.insert_in_sets
thf(fact_683_borel__singleton,axiom,
    ! [A: set_real,X2: real] :
      ( ( member_set_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( member_set_real @ ( insert_real @ X2 @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% borel_singleton
thf(fact_684_borel__singleton,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal] :
      ( ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
     => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ A ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).

% borel_singleton
thf(fact_685_borel__singleton,axiom,
    ! [A: set_o,X2: $o] :
      ( ( member_set_o @ A @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) )
     => ( member_set_o @ ( insert_o @ X2 @ A ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% borel_singleton
thf(fact_686_borel__singleton,axiom,
    ! [A: set_nat,X2: nat] :
      ( ( member_set_nat @ A @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) )
     => ( member_set_nat @ ( insert_nat @ X2 @ A ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% borel_singleton
thf(fact_687_empty__quasi__borel__iff,axiom,
    ! [X5: quasi_borel_real] :
      ( ( ( qbs_space_real @ X5 )
        = bot_bot_set_real )
      = ( X5 = empty_1876425439295802446l_real ) ) ).

% empty_quasi_borel_iff
thf(fact_688_empty__quasi__borel__iff,axiom,
    ! [X5: quasi_borel_nat] :
      ( ( ( qbs_space_nat @ X5 )
        = bot_bot_set_nat )
      = ( X5 = empty_8278123436611590770el_nat ) ) ).

% empty_quasi_borel_iff
thf(fact_689_empty__quasi__borel__iff,axiom,
    ! [X5: quasi_borel_o] :
      ( ( ( qbs_space_o @ X5 )
        = bot_bot_set_o )
      = ( X5 = empty_quasi_borel_o ) ) ).

% empty_quasi_borel_iff
thf(fact_690_empty__quasi__borel__iff,axiom,
    ! [X5: quasi_9015997321629101608nnreal] :
      ( ( ( qbs_sp175953267596557954nnreal @ X5 )
        = bot_bo4854962954004695426nnreal )
      = ( X5 = empty_1788085430566700506nnreal ) ) ).

% empty_quasi_borel_iff
thf(fact_691_case__sum__map__sum,axiom,
    ! [L: real > real,R2: real > real,F2: real > real,G: real > real,X2: sum_sum_real_real] :
      ( ( sum_ca8732840427581260704l_real @ L @ R2 @ ( sum_ma9028575376852974268l_real @ F2 @ G @ X2 ) )
      = ( sum_ca8732840427581260704l_real @ ( comp_real_real_real @ L @ F2 ) @ ( comp_real_real_real @ R2 @ G ) @ X2 ) ) ).

% case_sum_map_sum
thf(fact_692_case__sum__map__sum,axiom,
    ! [L: real > $o,R2: real > $o,F2: $o > real,G: $o > real,X2: sum_sum_o_o] :
      ( ( sum_ca7497528202123613950o_real @ L @ R2 @ ( sum_ma2390433805885911944o_real @ F2 @ G @ X2 ) )
      = ( sum_case_sum_o_o_o @ ( comp_real_o_o @ L @ F2 ) @ ( comp_real_o_o @ R2 @ G ) @ X2 ) ) ).

% case_sum_map_sum
thf(fact_693_case__sum__map__sum,axiom,
    ! [L: real > nat,R2: real > nat,F2: nat > real,G: nat > real,X2: sum_sum_nat_nat] :
      ( ( sum_ca2049687194608230980t_real @ L @ R2 @ ( sum_ma3776711750342522116t_real @ F2 @ G @ X2 ) )
      = ( sum_ca6763686470577984908at_nat @ ( comp_real_nat_nat @ L @ F2 ) @ ( comp_real_nat_nat @ R2 @ G ) @ X2 ) ) ).

% case_sum_map_sum
thf(fact_694_case__sum__map__sum,axiom,
    ! [L: c > d,R2: c > d,F2: real > c,G: real > c,X2: sum_sum_real_real] :
      ( ( sum_case_sum_c_d_c @ L @ R2 @ ( sum_ma393449481388452872real_c @ F2 @ G @ X2 ) )
      = ( sum_ca2032921040086735207d_real @ ( comp_c_d_real @ L @ F2 ) @ ( comp_c_d_real @ R2 @ G ) @ X2 ) ) ).

% case_sum_map_sum
thf(fact_695_case__sum__map__sum,axiom,
    ! [L: a > b,R2: a > b,F2: real > a,G: real > a,X2: sum_sum_real_real] :
      ( ( sum_case_sum_a_b_a @ L @ R2 @ ( sum_ma6640897443653631624real_a @ F2 @ G @ X2 ) )
      = ( sum_ca63855846565249637b_real @ ( comp_a_b_real @ L @ F2 ) @ ( comp_a_b_real @ R2 @ G ) @ X2 ) ) ).

% case_sum_map_sum
thf(fact_696_o__case__sum,axiom,
    ! [H: real > real,F2: real > real,G: real > real] :
      ( ( comp_r5151246396438109300l_real @ H @ ( sum_ca8732840427581260704l_real @ F2 @ G ) )
      = ( sum_ca8732840427581260704l_real @ ( comp_real_real_real @ H @ F2 ) @ ( comp_real_real_real @ H @ G ) ) ) ).

% o_case_sum
thf(fact_697_o__case__sum,axiom,
    ! [H: real > $o,F2: $o > real,G: $o > real] :
      ( ( comp_r962287379166727918um_o_o @ H @ ( sum_ca5525272764133257196real_o @ F2 @ G ) )
      = ( sum_case_sum_o_o_o @ ( comp_real_o_o @ H @ F2 ) @ ( comp_real_o_o @ H @ G ) ) ) ).

% o_case_sum
thf(fact_698_o__case__sum,axiom,
    ! [H: real > nat,F2: nat > real,G: nat > real] :
      ( ( comp_r6376950801160420448at_nat @ H @ ( sum_ca8334624595930125032al_nat @ F2 @ G ) )
      = ( sum_ca6763686470577984908at_nat @ ( comp_real_nat_nat @ H @ F2 ) @ ( comp_real_nat_nat @ H @ G ) ) ) ).

% o_case_sum
thf(fact_699_o__case__sum,axiom,
    ! [H: c > d,F2: real > c,G: real > c] :
      ( ( comp_c8591738523173874331l_real @ H @ ( sum_ca5660074461753380326c_real @ F2 @ G ) )
      = ( sum_ca2032921040086735207d_real @ ( comp_c_d_real @ H @ F2 ) @ ( comp_c_d_real @ H @ G ) ) ) ).

% o_case_sum
thf(fact_700_o__case__sum,axiom,
    ! [H: a > b,F2: real > a,G: real > a] :
      ( ( comp_a5455185540716242459l_real @ H @ ( sum_ca3691009268231894756a_real @ F2 @ G ) )
      = ( sum_ca63855846565249637b_real @ ( comp_a_b_real @ H @ F2 ) @ ( comp_a_b_real @ H @ G ) ) ) ).

% o_case_sum
thf(fact_701_case__sum__preserves__morphisms,axiom,
    ! [F2: real > a,X5: quasi_borel_real,Z: quasi_borel_a,G: real > a,Y4: quasi_borel_real] :
      ( ( member_real_a @ F2 @ ( qbs_morphism_real_a @ X5 @ Z ) )
     => ( ( member_real_a @ G @ ( qbs_morphism_real_a @ Y4 @ Z ) )
       => ( member6630180758533022695real_a @ ( sum_ca3691009268231894756a_real @ F2 @ G ) @ ( qbs_mo4449801170876743949real_a @ ( binary3369073574056317543l_real @ X5 @ Y4 ) @ Z ) ) ) ) ).

% case_sum_preserves_morphisms
thf(fact_702_case__sum__preserves__morphisms,axiom,
    ! [F2: $o > real,X5: quasi_borel_o,Z: quasi_borel_real,G: $o > real,Y4: quasi_borel_o] :
      ( ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X5 @ Z ) )
     => ( ( member_o_real @ G @ ( qbs_morphism_o_real @ Y4 @ Z ) )
       => ( member3421099978642967687o_real @ ( sum_ca5525272764133257196real_o @ F2 @ G ) @ ( qbs_mo4523485202552518603o_real @ ( binary6836164603859296013bs_o_o @ X5 @ Y4 ) @ Z ) ) ) ) ).

% case_sum_preserves_morphisms
thf(fact_703_case__sum__preserves__morphisms,axiom,
    ! [F2: $o > real,X5: quasi_borel_o,Z: quasi_borel_real,G: nat > real,Y4: quasi_borel_nat] :
      ( ( member_o_real @ F2 @ ( qbs_morphism_o_real @ X5 @ Z ) )
     => ( ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z ) )
       => ( member8111550167385560159t_real @ ( sum_ca4335601045291449468al_nat @ F2 @ G ) @ ( qbs_mo7285195558795298381t_real @ ( binary2765902541835032475_o_nat @ X5 @ Y4 ) @ Z ) ) ) ) ).

% case_sum_preserves_morphisms
thf(fact_704_case__sum__preserves__morphisms,axiom,
    ! [F2: nat > real,X5: quasi_borel_nat,Z: quasi_borel_real,G: $o > real,Y4: quasi_borel_o] :
      ( ( member_nat_real @ F2 @ ( qbs_mo2000642995705457910t_real @ X5 @ Z ) )
     => ( ( member_o_real @ G @ ( qbs_morphism_o_real @ Y4 @ Z ) )
       => ( member5842278251609744133o_real @ ( sum_ca7527334803461514880real_o @ F2 @ G ) @ ( qbs_mo5015923643019482355o_real @ ( binary5868573902943217785_nat_o @ X5 @ Y4 ) @ Z ) ) ) ) ).

% case_sum_preserves_morphisms
thf(fact_705_case__sum__preserves__morphisms,axiom,
    ! [F2: nat > real,X5: quasi_borel_nat,Z: quasi_borel_real,G: nat > real,Y4: quasi_borel_nat] :
      ( ( member_nat_real @ F2 @ ( qbs_mo2000642995705457910t_real @ X5 @ Z ) )
     => ( ( member_nat_real @ G @ ( qbs_mo2000642995705457910t_real @ Y4 @ Z ) )
       => ( member1647107379014654305t_real @ ( sum_ca8334624595930125032al_nat @ F2 @ G ) @ ( qbs_mo3635929294422082341t_real @ ( binary5195869006974583087at_nat @ X5 @ Y4 ) @ Z ) ) ) ) ).

% case_sum_preserves_morphisms
thf(fact_706_case__sum__preserves__morphisms,axiom,
    ! [F2: a > b,X5: quasi_borel_a,Z: quasi_borel_b,G: a > b,Y4: quasi_borel_a] :
      ( ( member_a_b @ F2 @ ( qbs_morphism_a_b @ X5 @ Z ) )
     => ( ( member_a_b @ G @ ( qbs_morphism_a_b @ Y4 @ Z ) )
       => ( member_Sum_sum_a_a_b @ ( sum_case_sum_a_b_a @ F2 @ G ) @ ( qbs_mo1966825068709392160_a_a_b @ ( binary8555328655094383373bs_a_a @ X5 @ Y4 ) @ Z ) ) ) ) ).

% case_sum_preserves_morphisms
thf(fact_707_case__sum__preserves__morphisms,axiom,
    ! [F2: c > d,X5: quasi_borel_c,Z: quasi_borel_d,G: c > d,Y4: quasi_borel_c] :
      ( ( member_c_d @ F2 @ ( qbs_morphism_c_d @ X5 @ Z ) )
     => ( ( member_c_d @ G @ ( qbs_morphism_c_d @ Y4 @ Z ) )
       => ( member_Sum_sum_c_c_d @ ( sum_case_sum_c_d_c @ F2 @ G ) @ ( qbs_mo2664592261020904354_c_c_d @ ( binary2980417491149031309bs_c_c @ X5 @ Y4 ) @ Z ) ) ) ) ).

% case_sum_preserves_morphisms
thf(fact_708_function__factors__right,axiom,
    ! [G: real > real,F2: real > real] :
      ( ( ! [X3: real] :
          ? [Y5: real] :
            ( ( G @ Y5 )
            = ( F2 @ X3 ) ) )
      = ( ? [H3: real > real] :
            ( F2
            = ( comp_real_real_real @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_709_function__factors__right,axiom,
    ! [G: real > $o,F2: $o > $o] :
      ( ( ! [X3: $o] :
          ? [Y5: real] :
            ( ( G @ Y5 )
            = ( F2 @ X3 ) ) )
      = ( ? [H3: $o > real] :
            ( F2
            = ( comp_real_o_o @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_710_function__factors__right,axiom,
    ! [G: real > nat,F2: nat > nat] :
      ( ( ! [X3: nat] :
          ? [Y5: real] :
            ( ( G @ Y5 )
            = ( F2 @ X3 ) ) )
      = ( ? [H3: nat > real] :
            ( F2
            = ( comp_real_nat_nat @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_711_function__factors__right,axiom,
    ! [G: c > d,F2: real > d] :
      ( ( ! [X3: real] :
          ? [Y5: c] :
            ( ( G @ Y5 )
            = ( F2 @ X3 ) ) )
      = ( ? [H3: real > c] :
            ( F2
            = ( comp_c_d_real @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_712_function__factors__right,axiom,
    ! [G: a > b,F2: real > b] :
      ( ( ! [X3: real] :
          ? [Y5: a] :
            ( ( G @ Y5 )
            = ( F2 @ X3 ) ) )
      = ( ? [H3: real > a] :
            ( F2
            = ( comp_a_b_real @ G @ H3 ) ) ) ) ).

% function_factors_right
thf(fact_713_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_714_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_715_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_716_sets__bot,axiom,
    ( ( sigma_5465916536984168985nnreal @ bot_bo1740529460517930749nnreal )
    = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) ) ).

% sets_bot
thf(fact_717_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [X2: real] :
      ( top_top_set_real
     != ( insert_real @ X2 @ bot_bot_set_real ) ) ).

% perfect_space_class.UNIV_not_singleton
thf(fact_718_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( top_to7994903218803871134nnreal
     != ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) ).

% perfect_space_class.UNIV_not_singleton
thf(fact_719_is__singletonI,axiom,
    ! [X2: real] : ( is_singleton_real @ ( insert_real @ X2 @ bot_bot_set_real ) ) ).

% is_singletonI
thf(fact_720_is__singletonI,axiom,
    ! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_721_is__singletonI,axiom,
    ! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_722_is__singletonI,axiom,
    ! [X2: extend8495563244428889912nnreal] : ( is_sin3654761921782142788nnreal @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) ).

% is_singletonI
thf(fact_723_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_724_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_725_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_726_sets__eq__bot2,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal )
        = ( sigma_5465916536984168985nnreal @ M ) )
      = ( M = bot_bo1740529460517930749nnreal ) ) ).

% sets_eq_bot2
thf(fact_727_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_728_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_729_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_730_sets__eq__bot,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) )
      = ( M = bot_bo1740529460517930749nnreal ) ) ).

% sets_eq_bot
thf(fact_731_the__elem__eq,axiom,
    ! [X2: real] :
      ( ( the_elem_real @ ( insert_real @ X2 @ bot_bot_set_real ) )
      = X2 ) ).

% the_elem_eq
thf(fact_732_the__elem__eq,axiom,
    ! [X2: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
      = X2 ) ).

% the_elem_eq
thf(fact_733_the__elem__eq,axiom,
    ! [X2: $o] :
      ( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
      = X2 ) ).

% the_elem_eq
thf(fact_734_the__elem__eq,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( the_el3795950934141317635nnreal @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) )
      = X2 ) ).

% the_elem_eq
thf(fact_735_case__sum__o__map__sum__id,axiom,
    ! [G: real > real,F2: extend8495563244428889912nnreal > real,X2: sum_su3194684483830730051l_real] :
      ( ( comp_S3462165939084609547l_real @ ( sum_ca8732840427581260704l_real @ id_real @ G ) @ ( sum_ma436539099328933320l_real @ F2 @ id_real ) @ X2 )
      = ( sum_ca615535534601039148l_real @ ( comp_E6146708109475903745nnreal @ F2 @ id_Ext8331313139072774535nnreal ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_736_case__sum__o__map__sum__id,axiom,
    ! [G: real > real,F2: nat > real,X2: sum_sum_nat_real] :
      ( ( comp_S7774070043488426147t_real @ ( sum_ca8732840427581260704l_real @ id_real @ G ) @ ( sum_ma1321509161104881248l_real @ F2 @ id_real ) @ X2 )
      = ( sum_ca6225210458162260420l_real @ ( comp_nat_real_nat @ F2 @ id_nat ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_737_case__sum__o__map__sum__id,axiom,
    ! [G: real > real,F2: $o > real,X2: sum_sum_o_real] :
      ( ( comp_S1736399213409601069o_real @ ( sum_ca8732840427581260704l_real @ id_real @ G ) @ ( sum_ma7581390967160423062l_real @ F2 @ id_real ) @ X2 )
      = ( sum_ca576396664188569944l_real @ ( comp_o_real_o @ F2 @ id_o ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_738_case__sum__o__map__sum__id,axiom,
    ! [G: extend8495563244428889912nnreal > real,F2: extend8495563244428889912nnreal > real,X2: sum_su4415445757542774223nnreal] :
      ( ( comp_S8072141257669888163nnreal @ ( sum_ca5039698333674796588nnreal @ id_real @ G ) @ ( sum_ma7572881292572212320nnreal @ F2 @ id_Ext8331313139072774535nnreal ) @ X2 )
      = ( sum_ca1130728446324819384nnreal @ ( comp_E6146708109475903745nnreal @ F2 @ id_Ext8331313139072774535nnreal ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_739_case__sum__o__map__sum__id,axiom,
    ! [G: extend8495563244428889912nnreal > real,F2: nat > real,X2: sum_su3730406437774119527nnreal] :
      ( ( comp_S1133027055745533243nnreal @ ( sum_ca5039698333674796588nnreal @ id_real @ G ) @ ( sum_ma2610832426335247608nnreal @ F2 @ id_Ext8331313139072774535nnreal ) @ X2 )
      = ( sum_ca6708266607595310672nnreal @ ( comp_nat_real_nat @ F2 @ id_nat ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_740_case__sum__o__map__sum__id,axiom,
    ! [G: extend8495563244428889912nnreal > real,F2: $o > real,X2: sum_su7753341093581952129nnreal] :
      ( ( comp_S3118471914838181037nnreal @ ( sum_ca5039698333674796588nnreal @ id_real @ G ) @ ( sum_ma3146212978127159086nnreal @ F2 @ id_Ext8331313139072774535nnreal ) @ X2 )
      = ( sum_ca412452145296005092nnreal @ ( comp_o_real_o @ F2 @ id_o ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_741_case__sum__o__map__sum__id,axiom,
    ! [G: nat > real,F2: extend8495563244428889912nnreal > real,X2: sum_su1883948583941721703al_nat] :
      ( ( comp_S3004035650380784083al_nat @ ( sum_ca6616995767737129668al_nat @ id_real @ G ) @ ( sum_ma2425729402563468688at_nat @ F2 @ id_nat ) @ X2 )
      = ( sum_ca112682941461744208al_nat @ ( comp_E6146708109475903745nnreal @ F2 @ id_Ext8331313139072774535nnreal ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_742_case__sum__o__map__sum__id,axiom,
    ! [G: nat > real,F2: nat > real,X2: sum_sum_nat_nat] :
      ( ( comp_S3112721647402360427at_nat @ ( sum_ca6616995767737129668al_nat @ id_real @ G ) @ ( sum_ma2704325357117275688at_nat @ F2 @ id_nat ) @ X2 )
      = ( sum_ca8334624595930125032al_nat @ ( comp_nat_real_nat @ F2 @ id_nat ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_743_case__sum__o__map__sum__id,axiom,
    ! [G: nat > real,F2: $o > real,X2: sum_sum_o_nat] :
      ( ( comp_S1246679031322203821_o_nat @ ( sum_ca6616995767737129668al_nat @ id_real @ G ) @ ( sum_ma3983538065750851678at_nat @ F2 @ id_nat ) @ X2 )
      = ( sum_ca4335601045291449468al_nat @ ( comp_o_real_o @ F2 @ id_o ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_744_case__sum__o__map__sum__id,axiom,
    ! [G: $o > real,F2: extend8495563244428889912nnreal > real,X2: sum_su1359252823600007511real_o] :
      ( ( comp_S8575781285974020201real_o @ ( sum_ca9006592577164783524real_o @ id_real @ G ) @ ( sum_ma4122737713070071790al_o_o @ F2 @ id_o ) @ X2 )
      = ( sum_ca5495973872699983128real_o @ ( comp_E6146708109475903745nnreal @ F2 @ id_Ext8331313139072774535nnreal ) @ G @ X2 ) ) ).

% case_sum_o_map_sum_id
thf(fact_745_is__singletonE,axiom,
    ! [A: set_real] :
      ( ( is_singleton_real @ A )
     => ~ ! [X: real] :
            ( A
           != ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% is_singletonE
thf(fact_746_is__singletonE,axiom,
    ! [A: set_nat] :
      ( ( is_singleton_nat @ A )
     => ~ ! [X: nat] :
            ( A
           != ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_747_is__singletonE,axiom,
    ! [A: set_o] :
      ( ( is_singleton_o @ A )
     => ~ ! [X: $o] :
            ( A
           != ( insert_o @ X @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_748_is__singletonE,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( is_sin3654761921782142788nnreal @ A )
     => ~ ! [X: extend8495563244428889912nnreal] :
            ( A
           != ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) ) ) ).

% is_singletonE
thf(fact_749_id__apply,axiom,
    ( id_real
    = ( ^ [X3: real] : X3 ) ) ).

% id_apply
thf(fact_750_id__apply,axiom,
    ( id_Ext8331313139072774535nnreal
    = ( ^ [X3: extend8495563244428889912nnreal] : X3 ) ) ).

% id_apply
thf(fact_751_id__apply,axiom,
    ( id_nat
    = ( ^ [X3: nat] : X3 ) ) ).

% id_apply
thf(fact_752_id__apply,axiom,
    ( id_o
    = ( ^ [X3: $o] : X3 ) ) ).

% id_apply
thf(fact_753_fun_Omap__id,axiom,
    ! [T: real > real] :
      ( ( comp_real_real_real @ id_real @ T )
      = T ) ).

% fun.map_id
thf(fact_754_comp__id,axiom,
    ! [F2: real > real] :
      ( ( comp_real_real_real @ F2 @ id_real )
      = F2 ) ).

% comp_id
thf(fact_755_id__comp,axiom,
    ! [G: real > real] :
      ( ( comp_real_real_real @ id_real @ G )
      = G ) ).

% id_comp
thf(fact_756_fun_Omap__id0,axiom,
    ( ( comp_real_real_real @ id_real )
    = id_real_real ) ).

% fun.map_id0
thf(fact_757_eq__id__iff,axiom,
    ! [F2: real > real] :
      ( ( ! [X3: real] :
            ( ( F2 @ X3 )
            = X3 ) )
      = ( F2 = id_real ) ) ).

% eq_id_iff
thf(fact_758_eq__id__iff,axiom,
    ! [F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
      ( ( ! [X3: extend8495563244428889912nnreal] :
            ( ( F2 @ X3 )
            = X3 ) )
      = ( F2 = id_Ext8331313139072774535nnreal ) ) ).

% eq_id_iff
thf(fact_759_eq__id__iff,axiom,
    ! [F2: nat > nat] :
      ( ( ! [X3: nat] :
            ( ( F2 @ X3 )
            = X3 ) )
      = ( F2 = id_nat ) ) ).

% eq_id_iff
thf(fact_760_eq__id__iff,axiom,
    ! [F2: $o > $o] :
      ( ( ! [X3: $o] :
            ( ( F2 @ X3 )
            = X3 ) )
      = ( F2 = id_o ) ) ).

% eq_id_iff
thf(fact_761_id__def,axiom,
    ( id_real
    = ( ^ [X3: real] : X3 ) ) ).

% id_def
thf(fact_762_id__def,axiom,
    ( id_Ext8331313139072774535nnreal
    = ( ^ [X3: extend8495563244428889912nnreal] : X3 ) ) ).

% id_def
thf(fact_763_id__def,axiom,
    ( id_nat
    = ( ^ [X3: nat] : X3 ) ) ).

% id_def
thf(fact_764_id__def,axiom,
    ( id_o
    = ( ^ [X3: $o] : X3 ) ) ).

% id_def
thf(fact_765_sum_Omap__id0,axiom,
    ( ( sum_ma9028575376852974268l_real @ id_real @ id_real )
    = id_Sum_sum_real_real ) ).

% sum.map_id0
thf(fact_766_sum_Omap__id0,axiom,
    ( ( sum_ma4707617466468859220nnreal @ id_real @ id_Ext8331313139072774535nnreal )
    = id_Sum7836748629692818962nnreal ) ).

% sum.map_id0
thf(fact_767_sum_Omap__id0,axiom,
    ( ( sum_ma5138984332203957892at_nat @ id_real @ id_nat )
    = id_Sum_sum_real_nat ) ).

% sum.map_id0
thf(fact_768_sum_Omap__id0,axiom,
    ( ( sum_ma5398380231479159522al_o_o @ id_real @ id_o )
    = id_Sum_sum_real_o ) ).

% sum.map_id0
thf(fact_769_sum_Omap__id0,axiom,
    ( ( sum_ma5613765533332990804l_real @ id_Ext8331313139072774535nnreal @ id_real )
    = id_Sum7791200330348796306l_real ) ).

% sum.map_id0
thf(fact_770_sum_Omap__id0,axiom,
    ( ( sum_ma2548858965165278700nnreal @ id_Ext8331313139072774535nnreal @ id_Ext8331313139072774535nnreal )
    = id_Sum3615161427726631326nnreal ) ).

% sum.map_id0
thf(fact_771_sum_Omap__id0,axiom,
    ( ( sum_ma840650751729215260at_nat @ id_Ext8331313139072774535nnreal @ id_nat )
    = id_Sum4558731775607509558al_nat ) ).

% sum.map_id0
thf(fact_772_sum_Omap__id0,axiom,
    ( ( sum_ma5762510830048976762al_o_o @ id_Ext8331313139072774535nnreal @ id_o )
    = id_Sum6174715882650043336real_o ) ).

% sum.map_id0
thf(fact_773_sum_Omap__id0,axiom,
    ( ( sum_ma7952235013896551556l_real @ id_nat @ id_real )
    = id_Sum_sum_nat_real ) ).

% sum.map_id0
thf(fact_774_sum_Omap__id0,axiom,
    ( ( sum_ma5444172813184304924nnreal @ id_nat @ id_Ext8331313139072774535nnreal )
    = id_Sum6405189629439907382nnreal ) ).

% sum.map_id0
thf(fact_775_comp__eq__id__dest,axiom,
    ! [A2: real > real,B: real > real,C: real > real,V: real] :
      ( ( ( comp_real_real_real @ A2 @ B )
        = ( comp_real_real_real @ id_real @ C ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_776_comp__eq__id__dest,axiom,
    ! [A2: real > $o,B: $o > real,C: $o > $o,V: $o] :
      ( ( ( comp_real_o_o @ A2 @ B )
        = ( comp_o_o_o @ id_o @ C ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_777_comp__eq__id__dest,axiom,
    ! [A2: real > nat,B: nat > real,C: nat > nat,V: nat] :
      ( ( ( comp_real_nat_nat @ A2 @ B )
        = ( comp_nat_nat_nat @ id_nat @ C ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_778_comp__eq__id__dest,axiom,
    ! [A2: c > d,B: real > c,C: real > d,V: real] :
      ( ( ( comp_c_d_real @ A2 @ B )
        = ( comp_d_d_real @ id_d @ C ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_779_comp__eq__id__dest,axiom,
    ! [A2: a > b,B: real > a,C: real > b,V: real] :
      ( ( ( comp_a_b_real @ A2 @ B )
        = ( comp_b_b_real @ id_b @ C ) )
     => ( ( A2 @ ( B @ V ) )
        = ( C @ V ) ) ) ).

% comp_eq_id_dest
thf(fact_780_pointfree__idE,axiom,
    ! [F2: real > real,G: real > real,X2: real] :
      ( ( ( comp_real_real_real @ F2 @ G )
        = id_real )
     => ( ( F2 @ ( G @ X2 ) )
        = X2 ) ) ).

% pointfree_idE
thf(fact_781_pointfree__idE,axiom,
    ! [F2: real > $o,G: $o > real,X2: $o] :
      ( ( ( comp_real_o_o @ F2 @ G )
        = id_o )
     => ( ( F2 @ ( G @ X2 ) )
        = X2 ) ) ).

% pointfree_idE
thf(fact_782_pointfree__idE,axiom,
    ! [F2: real > nat,G: nat > real,X2: nat] :
      ( ( ( comp_real_nat_nat @ F2 @ G )
        = id_nat )
     => ( ( F2 @ ( G @ X2 ) )
        = X2 ) ) ).

% pointfree_idE
thf(fact_783_sum_Omap__id,axiom,
    ! [T: sum_sum_real_real] :
      ( ( sum_ma9028575376852974268l_real @ id_real @ id_real @ T )
      = T ) ).

% sum.map_id
thf(fact_784_sum_Omap__id,axiom,
    ! [T: sum_su3240232783174752707nnreal] :
      ( ( sum_ma4707617466468859220nnreal @ id_real @ id_Ext8331313139072774535nnreal @ T )
      = T ) ).

% sum.map_id
thf(fact_785_sum_Omap__id,axiom,
    ! [T: sum_sum_real_nat] :
      ( ( sum_ma5138984332203957892at_nat @ id_real @ id_nat @ T )
      = T ) ).

% sum.map_id
thf(fact_786_sum_Omap__id,axiom,
    ! [T: sum_sum_real_o] :
      ( ( sum_ma5398380231479159522al_o_o @ id_real @ id_o @ T )
      = T ) ).

% sum.map_id
thf(fact_787_sum_Omap__id,axiom,
    ! [T: sum_su3194684483830730051l_real] :
      ( ( sum_ma5613765533332990804l_real @ id_Ext8331313139072774535nnreal @ id_real @ T )
      = T ) ).

% sum.map_id
thf(fact_788_sum_Omap__id,axiom,
    ! [T: sum_su4415445757542774223nnreal] :
      ( ( sum_ma2548858965165278700nnreal @ id_Ext8331313139072774535nnreal @ id_Ext8331313139072774535nnreal @ T )
      = T ) ).

% sum.map_id
thf(fact_789_sum_Omap__id,axiom,
    ! [T: sum_su1883948583941721703al_nat] :
      ( ( sum_ma840650751729215260at_nat @ id_Ext8331313139072774535nnreal @ id_nat @ T )
      = T ) ).

% sum.map_id
thf(fact_790_sum_Omap__id,axiom,
    ! [T: sum_su1359252823600007511real_o] :
      ( ( sum_ma5762510830048976762al_o_o @ id_Ext8331313139072774535nnreal @ id_o @ T )
      = T ) ).

% sum.map_id
thf(fact_791_sum_Omap__id,axiom,
    ! [T: sum_sum_nat_real] :
      ( ( sum_ma7952235013896551556l_real @ id_nat @ id_real @ T )
      = T ) ).

% sum.map_id
thf(fact_792_sum_Omap__id,axiom,
    ! [T: sum_su3730406437774119527nnreal] :
      ( ( sum_ma5444172813184304924nnreal @ id_nat @ id_Ext8331313139072774535nnreal @ T )
      = T ) ).

% sum.map_id
thf(fact_793_qbs__morphism__ident,axiom,
    ! [X5: quasi_borel_real] : ( member_real_real @ id_real @ ( qbs_mo5229770564518008146l_real @ X5 @ X5 ) ) ).

% qbs_morphism_ident
thf(fact_794_qbs__morphism__ident,axiom,
    ! [X5: quasi_9015997321629101608nnreal] : ( member8329810500450651686nnreal @ id_Ext8331313139072774535nnreal @ ( qbs_mo660571752308592106nnreal @ X5 @ X5 ) ) ).

% qbs_morphism_ident
thf(fact_795_qbs__morphism__ident,axiom,
    ! [X5: quasi_borel_nat] : ( member_nat_nat @ id_nat @ ( qbs_morphism_nat_nat @ X5 @ X5 ) ) ).

% qbs_morphism_ident
thf(fact_796_qbs__morphism__ident,axiom,
    ! [X5: quasi_borel_o] : ( member_o_o @ id_o @ ( qbs_morphism_o_o @ X5 @ X5 ) ) ).

% qbs_morphism_ident
thf(fact_797_is__singleton__the__elem,axiom,
    ( is_singleton_real
    = ( ^ [A5: set_real] :
          ( A5
          = ( insert_real @ ( the_elem_real @ A5 ) @ bot_bot_set_real ) ) ) ) ).

% is_singleton_the_elem
thf(fact_798_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
          ( A5
          = ( insert_nat @ ( the_elem_nat @ A5 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_799_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
          ( A5
          = ( insert_o @ ( the_elem_o @ A5 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_800_is__singleton__the__elem,axiom,
    ( is_sin3654761921782142788nnreal
    = ( ^ [A5: set_Ex3793607809372303086nnreal] :
          ( A5
          = ( insert7407984058720857448nnreal @ ( the_el3795950934141317635nnreal @ A5 ) @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% is_singleton_the_elem
thf(fact_801_is__singletonI_H,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
     => ( ! [X: real,Y3: real] :
            ( ( member_real @ X @ A )
           => ( ( member_real @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_singleton_real @ A ) ) ) ).

% is_singletonI'
thf(fact_802_is__singletonI_H,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X: nat,Y3: nat] :
            ( ( member_nat @ X @ A )
           => ( ( member_nat @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_singleton_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_803_is__singletonI_H,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
     => ( ! [X: $o,Y3: $o] :
            ( ( member_o @ X @ A )
           => ( ( member_o @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_singleton_o @ A ) ) ) ).

% is_singletonI'
thf(fact_804_is__singletonI_H,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( A != bot_bo4854962954004695426nnreal )
     => ( ! [X: extend8495563244428889912nnreal,Y3: extend8495563244428889912nnreal] :
            ( ( member7908768830364227535nnreal @ X @ A )
           => ( ( member7908768830364227535nnreal @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_sin3654761921782142788nnreal @ A ) ) ) ).

% is_singletonI'
thf(fact_805_is__singletonI_H,axiom,
    ! [A: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => ( ! [X: set_nat,Y3: set_nat] :
            ( ( member_set_nat @ X @ A )
           => ( ( member_set_nat @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_singleton_set_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_806_is__singletonI_H,axiom,
    ! [A: set_set_o] :
      ( ( A != bot_bot_set_set_o )
     => ( ! [X: set_o,Y3: set_o] :
            ( ( member_set_o @ X @ A )
           => ( ( member_set_o @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_singleton_set_o @ A ) ) ) ).

% is_singletonI'
thf(fact_807_is__singletonI_H,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ( A != bot_bo2988155216863113784nnreal )
     => ( ! [X: set_Ex3793607809372303086nnreal,Y3: set_Ex3793607809372303086nnreal] :
            ( ( member603777416030116741nnreal @ X @ A )
           => ( ( member603777416030116741nnreal @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_sin9058363718368806650nnreal @ A ) ) ) ).

% is_singletonI'
thf(fact_808_is__singletonI_H,axiom,
    ! [A: set_set_real] :
      ( ( A != bot_bot_set_set_real )
     => ( ! [X: set_real,Y3: set_real] :
            ( ( member_set_real @ X @ A )
           => ( ( member_set_real @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_sin3548895728136638702t_real @ A ) ) ) ).

% is_singletonI'
thf(fact_809_is__singletonI_H,axiom,
    ! [A: set_real_a] :
      ( ( A != bot_bot_set_real_a )
     => ( ! [X: real > a,Y3: real > a] :
            ( ( member_real_a @ X @ A )
           => ( ( member_real_a @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_singleton_real_a @ A ) ) ) ).

% is_singletonI'
thf(fact_810_is__singletonI_H,axiom,
    ! [A: set_o_real] :
      ( ( A != bot_bot_set_o_real )
     => ( ! [X: $o > real,Y3: $o > real] :
            ( ( member_o_real @ X @ A )
           => ( ( member_o_real @ Y3 @ A )
             => ( X = Y3 ) ) )
       => ( is_singleton_o_real @ A ) ) ) ).

% is_singletonI'
thf(fact_811_is__singleton__def,axiom,
    ( is_singleton_real
    = ( ^ [A5: set_real] :
        ? [X3: real] :
          ( A5
          = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).

% is_singleton_def
thf(fact_812_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
        ? [X3: nat] :
          ( A5
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_813_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
        ? [X3: $o] :
          ( A5
          = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_814_is__singleton__def,axiom,
    ( is_sin3654761921782142788nnreal
    = ( ^ [A5: set_Ex3793607809372303086nnreal] :
        ? [X3: extend8495563244428889912nnreal] :
          ( A5
          = ( insert7407984058720857448nnreal @ X3 @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% is_singleton_def
thf(fact_815_isomorphism__expand,axiom,
    ! [F2: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal] :
      ( ( ( ( comp_E3822617923592311797l_real @ F2 @ G )
          = id_real )
        & ( ( comp_r6281409797179841921nnreal @ G @ F2 )
          = id_Ext8331313139072774535nnreal ) )
      = ( ! [X3: real] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: extend8495563244428889912nnreal] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_816_isomorphism__expand,axiom,
    ! [F2: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real] :
      ( ( ( ( comp_r6281409797179841921nnreal @ F2 @ G )
          = id_Ext8331313139072774535nnreal )
        & ( ( comp_E3822617923592311797l_real @ G @ F2 )
          = id_real ) )
      = ( ! [X3: extend8495563244428889912nnreal] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: real] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_817_isomorphism__expand,axiom,
    ! [F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
      ( ( ( ( comp_E7860224481218928525nnreal @ F2 @ G )
          = id_Ext8331313139072774535nnreal )
        & ( ( comp_E7860224481218928525nnreal @ G @ F2 )
          = id_Ext8331313139072774535nnreal ) )
      = ( ! [X3: extend8495563244428889912nnreal] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: extend8495563244428889912nnreal] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_818_isomorphism__expand,axiom,
    ! [F2: nat > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > nat] :
      ( ( ( ( comp_n4458900468863859749nnreal @ F2 @ G )
          = id_Ext8331313139072774535nnreal )
        & ( ( comp_E1259497758557687997at_nat @ G @ F2 )
          = id_nat ) )
      = ( ! [X3: extend8495563244428889912nnreal] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: nat] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_819_isomorphism__expand,axiom,
    ! [F2: $o > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > $o] :
      ( ( ( ( comp_o7356229531094944743nnreal @ F2 @ G )
          = id_Ext8331313139072774535nnreal )
        & ( ( comp_E3606505270377598363al_o_o @ G @ F2 )
          = id_o ) )
      = ( ! [X3: extend8495563244428889912nnreal] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: $o] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_820_isomorphism__expand,axiom,
    ! [F2: extend8495563244428889912nnreal > nat,G: nat > extend8495563244428889912nnreal] :
      ( ( ( ( comp_E1259497758557687997at_nat @ F2 @ G )
          = id_nat )
        & ( ( comp_n4458900468863859749nnreal @ G @ F2 )
          = id_Ext8331313139072774535nnreal ) )
      = ( ! [X3: nat] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: extend8495563244428889912nnreal] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_821_isomorphism__expand,axiom,
    ! [F2: nat > nat,G: nat > nat] :
      ( ( ( ( comp_nat_nat_nat @ F2 @ G )
          = id_nat )
        & ( ( comp_nat_nat_nat @ G @ F2 )
          = id_nat ) )
      = ( ! [X3: nat] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: nat] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_822_isomorphism__expand,axiom,
    ! [F2: $o > nat,G: nat > $o] :
      ( ( ( ( comp_o_nat_nat @ F2 @ G )
          = id_nat )
        & ( ( comp_nat_o_o @ G @ F2 )
          = id_o ) )
      = ( ! [X3: nat] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: $o] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_823_isomorphism__expand,axiom,
    ! [F2: extend8495563244428889912nnreal > $o,G: $o > extend8495563244428889912nnreal] :
      ( ( ( ( comp_E3606505270377598363al_o_o @ F2 @ G )
          = id_o )
        & ( ( comp_o7356229531094944743nnreal @ G @ F2 )
          = id_Ext8331313139072774535nnreal ) )
      = ( ! [X3: $o] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: extend8495563244428889912nnreal] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_824_isomorphism__expand,axiom,
    ! [F2: nat > $o,G: $o > nat] :
      ( ( ( ( comp_nat_o_o @ F2 @ G )
          = id_o )
        & ( ( comp_o_nat_nat @ G @ F2 )
          = id_nat ) )
      = ( ! [X3: $o] :
            ( ( F2 @ ( G @ X3 ) )
            = X3 )
        & ! [X3: nat] :
            ( ( G @ ( F2 @ X3 ) )
            = X3 ) ) ) ).

% isomorphism_expand
thf(fact_825_left__right__inverse__eq,axiom,
    ! [F2: extend8495563244428889912nnreal > real,G: real > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > real] :
      ( ( ( comp_E3822617923592311797l_real @ F2 @ G )
        = id_real )
     => ( ( ( comp_r6281409797179841921nnreal @ G @ H )
          = id_Ext8331313139072774535nnreal )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_826_left__right__inverse__eq,axiom,
    ! [F2: real > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > real,H: real > extend8495563244428889912nnreal] :
      ( ( ( comp_r6281409797179841921nnreal @ F2 @ G )
        = id_Ext8331313139072774535nnreal )
     => ( ( ( comp_E3822617923592311797l_real @ G @ H )
          = id_real )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_827_left__right__inverse__eq,axiom,
    ! [F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > extend8495563244428889912nnreal] :
      ( ( ( comp_E7860224481218928525nnreal @ F2 @ G )
        = id_Ext8331313139072774535nnreal )
     => ( ( ( comp_E7860224481218928525nnreal @ G @ H )
          = id_Ext8331313139072774535nnreal )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_828_left__right__inverse__eq,axiom,
    ! [F2: nat > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > nat,H: nat > extend8495563244428889912nnreal] :
      ( ( ( comp_n4458900468863859749nnreal @ F2 @ G )
        = id_Ext8331313139072774535nnreal )
     => ( ( ( comp_E1259497758557687997at_nat @ G @ H )
          = id_nat )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_829_left__right__inverse__eq,axiom,
    ! [F2: $o > extend8495563244428889912nnreal,G: extend8495563244428889912nnreal > $o,H: $o > extend8495563244428889912nnreal] :
      ( ( ( comp_o7356229531094944743nnreal @ F2 @ G )
        = id_Ext8331313139072774535nnreal )
     => ( ( ( comp_E3606505270377598363al_o_o @ G @ H )
          = id_o )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_830_left__right__inverse__eq,axiom,
    ! [F2: extend8495563244428889912nnreal > nat,G: nat > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > nat] :
      ( ( ( comp_E1259497758557687997at_nat @ F2 @ G )
        = id_nat )
     => ( ( ( comp_n4458900468863859749nnreal @ G @ H )
          = id_Ext8331313139072774535nnreal )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_831_left__right__inverse__eq,axiom,
    ! [F2: nat > nat,G: nat > nat,H: nat > nat] :
      ( ( ( comp_nat_nat_nat @ F2 @ G )
        = id_nat )
     => ( ( ( comp_nat_nat_nat @ G @ H )
          = id_nat )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_832_left__right__inverse__eq,axiom,
    ! [F2: $o > nat,G: nat > $o,H: $o > nat] :
      ( ( ( comp_o_nat_nat @ F2 @ G )
        = id_nat )
     => ( ( ( comp_nat_o_o @ G @ H )
          = id_o )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_833_left__right__inverse__eq,axiom,
    ! [F2: extend8495563244428889912nnreal > $o,G: $o > extend8495563244428889912nnreal,H: extend8495563244428889912nnreal > $o] :
      ( ( ( comp_E3606505270377598363al_o_o @ F2 @ G )
        = id_o )
     => ( ( ( comp_o7356229531094944743nnreal @ G @ H )
          = id_Ext8331313139072774535nnreal )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_834_left__right__inverse__eq,axiom,
    ! [F2: nat > $o,G: $o > nat,H: nat > $o] :
      ( ( ( comp_nat_o_o @ F2 @ G )
        = id_o )
     => ( ( ( comp_o_nat_nat @ G @ H )
          = id_nat )
       => ( F2 = H ) ) ) ).

% left_right_inverse_eq
thf(fact_835_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
    = ( ^ [A6: real,B6: real] :
        ? [Z4: real] :
          ( ( member_real @ Z4 @ top_top_set_real )
          & ( ( id_real @ Z4 )
            = A6 )
          & ( ( id_real @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_836_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
    = ( ^ [A6: $o,B6: $o] :
        ? [Z4: $o] :
          ( ( member_o @ Z4 @ top_top_set_o )
          & ( ( id_o @ Z4 )
            = A6 )
          & ( ( id_o @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_837_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
    = ( ^ [A6: nat,B6: nat] :
        ? [Z4: nat] :
          ( ( member_nat @ Z4 @ top_top_set_nat )
          & ( ( id_nat @ Z4 )
            = A6 )
          & ( ( id_nat @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_838_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: extend8495563244428889912nnreal,Z3: extend8495563244428889912nnreal] : ( Y6 = Z3 ) )
    = ( ^ [A6: extend8495563244428889912nnreal,B6: extend8495563244428889912nnreal] :
        ? [Z4: extend8495563244428889912nnreal] :
          ( ( member7908768830364227535nnreal @ Z4 @ top_to7994903218803871134nnreal )
          & ( ( id_Ext8331313139072774535nnreal @ Z4 )
            = A6 )
          & ( ( id_Ext8331313139072774535nnreal @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_839_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_nat,B6: set_nat] :
        ? [Z4: set_nat] :
          ( ( member_set_nat @ Z4 @ top_top_set_set_nat )
          & ( ( id_set_nat @ Z4 )
            = A6 )
          & ( ( id_set_nat @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_840_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: set_o,Z3: set_o] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_o,B6: set_o] :
        ? [Z4: set_o] :
          ( ( member_set_o @ Z4 @ top_top_set_set_o )
          & ( ( id_set_o @ Z4 )
            = A6 )
          & ( ( id_set_o @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_841_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: set_Ex3793607809372303086nnreal,Z3: set_Ex3793607809372303086nnreal] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_Ex3793607809372303086nnreal,B6: set_Ex3793607809372303086nnreal] :
        ? [Z4: set_Ex3793607809372303086nnreal] :
          ( ( member603777416030116741nnreal @ Z4 @ top_to3356475028079756884nnreal )
          & ( ( id_set2823833123132642621nnreal @ Z4 )
            = A6 )
          & ( ( id_set2823833123132642621nnreal @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_842_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: set_real,Z3: set_real] : ( Y6 = Z3 ) )
    = ( ^ [A6: set_real,B6: set_real] :
        ? [Z4: set_real] :
          ( ( member_set_real @ Z4 @ top_top_set_set_real )
          & ( ( id_set_real @ Z4 )
            = A6 )
          & ( ( id_set_real @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_843_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: real > a,Z3: real > a] : ( Y6 = Z3 ) )
    = ( ^ [A6: real > a,B6: real > a] :
        ? [Z4: real > a] :
          ( ( member_real_a @ Z4 @ top_top_set_real_a )
          & ( ( id_real_a @ Z4 )
            = A6 )
          & ( ( id_real_a @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_844_DEADID_Oin__rel,axiom,
    ( ( ^ [Y6: $o > real,Z3: $o > real] : ( Y6 = Z3 ) )
    = ( ^ [A6: $o > real,B6: $o > real] :
        ? [Z4: $o > real] :
          ( ( member_o_real @ Z4 @ top_top_set_o_real )
          & ( ( id_o_real @ Z4 )
            = A6 )
          & ( ( id_o_real @ Z4 )
            = B6 ) ) ) ) ).

% DEADID.in_rel
thf(fact_845_space__empty__iff,axiom,
    ! [N: sigma_measure_real] :
      ( ( ( sigma_space_real @ N )
        = bot_bot_set_real )
      = ( ( sigma_sets_real @ N )
        = ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ) ).

% space_empty_iff
thf(fact_846_space__empty__iff,axiom,
    ! [N: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ N )
        = bot_bot_set_nat )
      = ( ( sigma_sets_nat @ N )
        = ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ) ).

% space_empty_iff
thf(fact_847_space__empty__iff,axiom,
    ! [N: sigma_measure_o] :
      ( ( ( sigma_space_o @ N )
        = bot_bot_set_o )
      = ( ( sigma_sets_o @ N )
        = ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ) ).

% space_empty_iff
thf(fact_848_space__empty__iff,axiom,
    ! [N: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ N )
        = bot_bo4854962954004695426nnreal )
      = ( ( sigma_5465916536984168985nnreal @ N )
        = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) ) ) ).

% space_empty_iff
thf(fact_849_totally__bounded__empty,axiom,
    topolo4708875952704042879d_real @ bot_bot_set_real ).

% totally_bounded_empty
thf(fact_850_Pow__empty,axiom,
    ( ( pow_real @ bot_bot_set_real )
    = ( insert_set_real @ bot_bot_set_real @ bot_bot_set_set_real ) ) ).

% Pow_empty
thf(fact_851_Pow__empty,axiom,
    ( ( pow_nat @ bot_bot_set_nat )
    = ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).

% Pow_empty
thf(fact_852_Pow__empty,axiom,
    ( ( pow_o @ bot_bot_set_o )
    = ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).

% Pow_empty
thf(fact_853_Pow__empty,axiom,
    ( ( pow_Ex5372160365422184283nnreal @ bot_bo4854962954004695426nnreal )
    = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ bot_bo2988155216863113784nnreal ) ) ).

% Pow_empty
thf(fact_854_Pow__singleton__iff,axiom,
    ! [X5: set_real,Y4: set_real] :
      ( ( ( pow_real @ X5 )
        = ( insert_set_real @ Y4 @ bot_bot_set_set_real ) )
      = ( ( X5 = bot_bot_set_real )
        & ( Y4 = bot_bot_set_real ) ) ) ).

% Pow_singleton_iff
thf(fact_855_Pow__singleton__iff,axiom,
    ! [X5: set_nat,Y4: set_nat] :
      ( ( ( pow_nat @ X5 )
        = ( insert_set_nat @ Y4 @ bot_bot_set_set_nat ) )
      = ( ( X5 = bot_bot_set_nat )
        & ( Y4 = bot_bot_set_nat ) ) ) ).

% Pow_singleton_iff
thf(fact_856_Pow__singleton__iff,axiom,
    ! [X5: set_o,Y4: set_o] :
      ( ( ( pow_o @ X5 )
        = ( insert_set_o @ Y4 @ bot_bot_set_set_o ) )
      = ( ( X5 = bot_bot_set_o )
        & ( Y4 = bot_bot_set_o ) ) ) ).

% Pow_singleton_iff
thf(fact_857_Pow__singleton__iff,axiom,
    ! [X5: set_Ex3793607809372303086nnreal,Y4: set_Ex3793607809372303086nnreal] :
      ( ( ( pow_Ex5372160365422184283nnreal @ X5 )
        = ( insert1343806209672318238nnreal @ Y4 @ bot_bo2988155216863113784nnreal ) )
      = ( ( X5 = bot_bo4854962954004695426nnreal )
        & ( Y4 = bot_bo4854962954004695426nnreal ) ) ) ).

% Pow_singleton_iff
thf(fact_858_sigma__sets__single,axiom,
    ! [A: set_real] :
      ( ( sigma_7195353284648819924s_real @ A @ ( insert_set_real @ A @ bot_bot_set_set_real ) )
      = ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ A @ bot_bot_set_set_real ) ) ) ).

% sigma_sets_single
thf(fact_859_sigma__sets__single,axiom,
    ! [A: set_nat] :
      ( ( sigma_sigma_sets_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
      = ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ).

% sigma_sets_single
thf(fact_860_sigma__sets__single,axiom,
    ! [A: set_o] :
      ( ( sigma_sigma_sets_o @ A @ ( insert_set_o @ A @ bot_bot_set_set_o ) )
      = ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ A @ bot_bot_set_set_o ) ) ) ).

% sigma_sets_single
thf(fact_861_sigma__sets__single,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( sigma_7808855514367478112nnreal @ A @ ( insert1343806209672318238nnreal @ A @ bot_bo2988155216863113784nnreal ) )
      = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ A @ bot_bo2988155216863113784nnreal ) ) ) ).

% sigma_sets_single
thf(fact_862_sets_Otop,axiom,
    ! [M: sigma_measure_real] : ( member_set_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) ).

% sets.top
thf(fact_863_sets_Otop,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( member603777416030116741nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.top
thf(fact_864_sets_Otop,axiom,
    ! [M: sigma_measure_o] : ( member_set_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) ).

% sets.top
thf(fact_865_sets_Otop,axiom,
    ! [M: sigma_measure_nat] : ( member_set_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) ).

% sets.top
thf(fact_866_Pow__UNIV,axiom,
    ( ( pow_real @ top_top_set_real )
    = top_top_set_set_real ) ).

% Pow_UNIV
thf(fact_867_Pow__UNIV,axiom,
    ( ( pow_o @ top_top_set_o )
    = top_top_set_set_o ) ).

% Pow_UNIV
thf(fact_868_Pow__UNIV,axiom,
    ( ( pow_nat @ top_top_set_nat )
    = top_top_set_set_nat ) ).

% Pow_UNIV
thf(fact_869_Pow__UNIV,axiom,
    ( ( pow_Ex5372160365422184283nnreal @ top_to7994903218803871134nnreal )
    = top_to3356475028079756884nnreal ) ).

% Pow_UNIV
thf(fact_870_space__interval__measure,axiom,
    ! [F: real > real] :
      ( ( sigma_space_real @ ( lebesg8227263024992965735easure @ F ) )
      = top_top_set_real ) ).

% space_interval_measure
thf(fact_871_space__borel,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% space_borel
thf(fact_872_space__borel,axiom,
    ( ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal )
    = top_to7994903218803871134nnreal ) ).

% space_borel
thf(fact_873_space__borel,axiom,
    ( ( sigma_space_o @ borel_5500255247093592246orel_o )
    = top_top_set_o ) ).

% space_borel
thf(fact_874_space__borel,axiom,
    ( ( sigma_space_nat @ borel_8449730974584783410el_nat )
    = top_top_set_nat ) ).

% space_borel
thf(fact_875_space__convolution,axiom,
    ! [M: sigma_measure_real,N: sigma_measure_real] :
      ( ( sigma_space_real @ ( convolution_real @ M @ N ) )
      = ( sigma_space_real @ borel_5078946678739801102l_real ) ) ).

% space_convolution
thf(fact_876_space__bot,axiom,
    ( ( sigma_space_real @ bot_bo5982154664989874033e_real )
    = bot_bot_set_real ) ).

% space_bot
thf(fact_877_space__bot,axiom,
    ( ( sigma_space_nat @ bot_bo6718502177978453909re_nat )
    = bot_bot_set_nat ) ).

% space_bot
thf(fact_878_space__bot,axiom,
    ( ( sigma_space_o @ bot_bo5758314138661044393sure_o )
    = bot_bot_set_o ) ).

% space_bot
thf(fact_879_space__bot,axiom,
    ( ( sigma_3147302497200244656nnreal @ bot_bo1740529460517930749nnreal )
    = bot_bo4854962954004695426nnreal ) ).

% space_bot
thf(fact_880_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_measure_real] :
      ( ( sigma_7195353284648819924s_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) )
      = ( sigma_sets_real @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_881_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( sigma_7808855514367478112nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) )
      = ( sigma_5465916536984168985nnreal @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_882_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_measure_o] :
      ( ( sigma_sigma_sets_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) )
      = ( sigma_sets_o @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_883_sets_Osigma__sets__eq,axiom,
    ! [M: sigma_measure_nat] :
      ( ( sigma_sigma_sets_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) )
      = ( sigma_sets_nat @ M ) ) ).

% sets.sigma_sets_eq
thf(fact_884_sigma__sets_OBasic,axiom,
    ! [A2: set_nat,A: set_set_nat,Sp: set_nat] :
      ( ( member_set_nat @ A2 @ A )
     => ( member_set_nat @ A2 @ ( sigma_sigma_sets_nat @ Sp @ A ) ) ) ).

% sigma_sets.Basic
thf(fact_885_sigma__sets_OBasic,axiom,
    ! [A2: set_o,A: set_set_o,Sp: set_o] :
      ( ( member_set_o @ A2 @ A )
     => ( member_set_o @ A2 @ ( sigma_sigma_sets_o @ Sp @ A ) ) ) ).

% sigma_sets.Basic
thf(fact_886_sigma__sets_OBasic,axiom,
    ! [A2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,Sp: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ A2 @ A )
     => ( member603777416030116741nnreal @ A2 @ ( sigma_7808855514367478112nnreal @ Sp @ A ) ) ) ).

% sigma_sets.Basic
thf(fact_887_sigma__sets_OBasic,axiom,
    ! [A2: set_real,A: set_set_real,Sp: set_real] :
      ( ( member_set_real @ A2 @ A )
     => ( member_set_real @ A2 @ ( sigma_7195353284648819924s_real @ Sp @ A ) ) ) ).

% sigma_sets.Basic
thf(fact_888_sigma__sets__eqI,axiom,
    ! [A: set_set_nat,M: set_nat,B4: set_set_nat] :
      ( ! [A4: set_nat] :
          ( ( member_set_nat @ A4 @ A )
         => ( member_set_nat @ A4 @ ( sigma_sigma_sets_nat @ M @ B4 ) ) )
     => ( ! [B3: set_nat] :
            ( ( member_set_nat @ B3 @ B4 )
           => ( member_set_nat @ B3 @ ( sigma_sigma_sets_nat @ M @ A ) ) )
       => ( ( sigma_sigma_sets_nat @ M @ A )
          = ( sigma_sigma_sets_nat @ M @ B4 ) ) ) ) ).

% sigma_sets_eqI
thf(fact_889_sigma__sets__eqI,axiom,
    ! [A: set_set_o,M: set_o,B4: set_set_o] :
      ( ! [A4: set_o] :
          ( ( member_set_o @ A4 @ A )
         => ( member_set_o @ A4 @ ( sigma_sigma_sets_o @ M @ B4 ) ) )
     => ( ! [B3: set_o] :
            ( ( member_set_o @ B3 @ B4 )
           => ( member_set_o @ B3 @ ( sigma_sigma_sets_o @ M @ A ) ) )
       => ( ( sigma_sigma_sets_o @ M @ A )
          = ( sigma_sigma_sets_o @ M @ B4 ) ) ) ) ).

% sigma_sets_eqI
thf(fact_890_sigma__sets__eqI,axiom,
    ! [A: set_se4580700918925141924nnreal,M: set_Ex3793607809372303086nnreal,B4: set_se4580700918925141924nnreal] :
      ( ! [A4: set_Ex3793607809372303086nnreal] :
          ( ( member603777416030116741nnreal @ A4 @ A )
         => ( member603777416030116741nnreal @ A4 @ ( sigma_7808855514367478112nnreal @ M @ B4 ) ) )
     => ( ! [B3: set_Ex3793607809372303086nnreal] :
            ( ( member603777416030116741nnreal @ B3 @ B4 )
           => ( member603777416030116741nnreal @ B3 @ ( sigma_7808855514367478112nnreal @ M @ A ) ) )
       => ( ( sigma_7808855514367478112nnreal @ M @ A )
          = ( sigma_7808855514367478112nnreal @ M @ B4 ) ) ) ) ).

% sigma_sets_eqI
thf(fact_891_sigma__sets__eqI,axiom,
    ! [A: set_set_real,M: set_real,B4: set_set_real] :
      ( ! [A4: set_real] :
          ( ( member_set_real @ A4 @ A )
         => ( member_set_real @ A4 @ ( sigma_7195353284648819924s_real @ M @ B4 ) ) )
     => ( ! [B3: set_real] :
            ( ( member_set_real @ B3 @ B4 )
           => ( member_set_real @ B3 @ ( sigma_7195353284648819924s_real @ M @ A ) ) )
       => ( ( sigma_7195353284648819924s_real @ M @ A )
          = ( sigma_7195353284648819924s_real @ M @ B4 ) ) ) ) ).

% sigma_sets_eqI
thf(fact_892_sigma__sets__top,axiom,
    ! [Sp: set_nat,A: set_set_nat] : ( member_set_nat @ Sp @ ( sigma_sigma_sets_nat @ Sp @ A ) ) ).

% sigma_sets_top
thf(fact_893_sigma__sets__top,axiom,
    ! [Sp: set_o,A: set_set_o] : ( member_set_o @ Sp @ ( sigma_sigma_sets_o @ Sp @ A ) ) ).

% sigma_sets_top
thf(fact_894_sigma__sets__top,axiom,
    ! [Sp: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] : ( member603777416030116741nnreal @ Sp @ ( sigma_7808855514367478112nnreal @ Sp @ A ) ) ).

% sigma_sets_top
thf(fact_895_sigma__sets__top,axiom,
    ! [Sp: set_real,A: set_set_real] : ( member_set_real @ Sp @ ( sigma_7195353284648819924s_real @ Sp @ A ) ) ).

% sigma_sets_top
thf(fact_896_Pow__top,axiom,
    ! [A: set_nat] : ( member_set_nat @ A @ ( pow_nat @ A ) ) ).

% Pow_top
thf(fact_897_Pow__top,axiom,
    ! [A: set_o] : ( member_set_o @ A @ ( pow_o @ A ) ) ).

% Pow_top
thf(fact_898_Pow__top,axiom,
    ! [A: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ A @ ( pow_Ex5372160365422184283nnreal @ A ) ) ).

% Pow_top
thf(fact_899_Pow__top,axiom,
    ! [A: set_real] : ( member_set_real @ A @ ( pow_real @ A ) ) ).

% Pow_top
thf(fact_900_sigma__sets_OEmpty,axiom,
    ! [Sp: set_real,A: set_set_real] : ( member_set_real @ bot_bot_set_real @ ( sigma_7195353284648819924s_real @ Sp @ A ) ) ).

% sigma_sets.Empty
thf(fact_901_sigma__sets_OEmpty,axiom,
    ! [Sp: set_nat,A: set_set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( sigma_sigma_sets_nat @ Sp @ A ) ) ).

% sigma_sets.Empty
thf(fact_902_sigma__sets_OEmpty,axiom,
    ! [Sp: set_o,A: set_set_o] : ( member_set_o @ bot_bot_set_o @ ( sigma_sigma_sets_o @ Sp @ A ) ) ).

% sigma_sets.Empty
thf(fact_903_sigma__sets_OEmpty,axiom,
    ! [Sp: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( sigma_7808855514367478112nnreal @ Sp @ A ) ) ).

% sigma_sets.Empty
thf(fact_904_Pow__bottom,axiom,
    ! [B4: set_real] : ( member_set_real @ bot_bot_set_real @ ( pow_real @ B4 ) ) ).

% Pow_bottom
thf(fact_905_Pow__bottom,axiom,
    ! [B4: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B4 ) ) ).

% Pow_bottom
thf(fact_906_Pow__bottom,axiom,
    ! [B4: set_o] : ( member_set_o @ bot_bot_set_o @ ( pow_o @ B4 ) ) ).

% Pow_bottom
thf(fact_907_Pow__bottom,axiom,
    ! [B4: set_Ex3793607809372303086nnreal] : ( member603777416030116741nnreal @ bot_bo4854962954004695426nnreal @ ( pow_Ex5372160365422184283nnreal @ B4 ) ) ).

% Pow_bottom
thf(fact_908_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_real,M3: sigma_measure_real] :
      ( ( ( sigma_sets_real @ M )
        = ( sigma_sets_real @ M3 ) )
     => ( ( sigma_space_real @ M )
        = ( sigma_space_real @ M3 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_909_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_7234349610311085201nnreal,M3: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ M )
        = ( sigma_5465916536984168985nnreal @ M3 ) )
     => ( ( sigma_3147302497200244656nnreal @ M )
        = ( sigma_3147302497200244656nnreal @ M3 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_910_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_o,M3: sigma_measure_o] :
      ( ( ( sigma_sets_o @ M )
        = ( sigma_sets_o @ M3 ) )
     => ( ( sigma_space_o @ M )
        = ( sigma_space_o @ M3 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_911_sets__eq__imp__space__eq,axiom,
    ! [M: sigma_measure_nat,M3: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ M )
        = ( sigma_sets_nat @ M3 ) )
     => ( ( sigma_space_nat @ M )
        = ( sigma_space_nat @ M3 ) ) ) ).

% sets_eq_imp_space_eq
thf(fact_912_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_913_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_914_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_915_space__empty__eq__bot,axiom,
    ! [A2: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ A2 )
        = bot_bo4854962954004695426nnreal )
      = ( A2 = bot_bo1740529460517930749nnreal ) ) ).

% space_empty_eq_bot
thf(fact_916_sigma__sets__empty__eq,axiom,
    ! [A: set_real] :
      ( ( sigma_7195353284648819924s_real @ A @ bot_bot_set_set_real )
      = ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ A @ bot_bot_set_set_real ) ) ) ).

% sigma_sets_empty_eq
thf(fact_917_sigma__sets__empty__eq,axiom,
    ! [A: set_nat] :
      ( ( sigma_sigma_sets_nat @ A @ bot_bot_set_set_nat )
      = ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ).

% sigma_sets_empty_eq
thf(fact_918_sigma__sets__empty__eq,axiom,
    ! [A: set_o] :
      ( ( sigma_sigma_sets_o @ A @ bot_bot_set_set_o )
      = ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ A @ bot_bot_set_set_o ) ) ) ).

% sigma_sets_empty_eq
thf(fact_919_sigma__sets__empty__eq,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( sigma_7808855514367478112nnreal @ A @ bot_bo2988155216863113784nnreal )
      = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ A @ bot_bo2988155216863113784nnreal ) ) ) ).

% sigma_sets_empty_eq
thf(fact_920_type__copy__map__cong0,axiom,
    ! [M: $o > real,G: $o > $o,X2: $o,N: real > real,H: $o > real,F2: real > $o] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_o_o_o @ ( comp_real_o_o @ F2 @ M ) @ G @ X2 )
        = ( comp_real_o_o @ ( comp_real_o_real @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_921_type__copy__map__cong0,axiom,
    ! [M: nat > real,G: nat > nat,X2: nat,N: real > real,H: nat > real,F2: real > nat] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F2 @ M ) @ G @ X2 )
        = ( comp_real_nat_nat @ ( comp_real_nat_real @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_922_type__copy__map__cong0,axiom,
    ! [M: real > c,G: real > real,X2: real,N: c > c,H: real > c,F2: c > d] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_real_d_real @ ( comp_c_d_real @ F2 @ M ) @ G @ X2 )
        = ( comp_c_d_real @ ( comp_c_d_c @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_923_type__copy__map__cong0,axiom,
    ! [M: real > a,G: real > real,X2: real,N: a > a,H: real > a,F2: a > b] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_real_b_real @ ( comp_a_b_real @ F2 @ M ) @ G @ X2 )
        = ( comp_a_b_real @ ( comp_a_b_a @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_924_type__copy__map__cong0,axiom,
    ! [M: real > real,G: real > real,X2: real,N: real > real,H: real > real,F2: real > real] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_real_real_real @ ( comp_real_real_real @ F2 @ M ) @ G @ X2 )
        = ( comp_real_real_real @ ( comp_real_real_real @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_925_type__copy__map__cong0,axiom,
    ! [M: real > real,G: $o > real,X2: $o,N: $o > real,H: $o > $o,F2: real > $o] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_real_o_o @ ( comp_real_o_real @ F2 @ M ) @ G @ X2 )
        = ( comp_o_o_o @ ( comp_real_o_o @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_926_type__copy__map__cong0,axiom,
    ! [M: real > real,G: nat > real,X2: nat,N: nat > real,H: nat > nat,F2: real > nat] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_real_nat_nat @ ( comp_real_nat_real @ F2 @ M ) @ G @ X2 )
        = ( comp_nat_nat_nat @ ( comp_real_nat_nat @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_927_type__copy__map__cong0,axiom,
    ! [M: c > c,G: real > c,X2: real,N: real > c,H: real > real,F2: c > d] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_c_d_real @ ( comp_c_d_c @ F2 @ M ) @ G @ X2 )
        = ( comp_real_d_real @ ( comp_c_d_real @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_928_type__copy__map__cong0,axiom,
    ! [M: a > a,G: real > a,X2: real,N: real > a,H: real > real,F2: a > b] :
      ( ( ( M @ ( G @ X2 ) )
        = ( N @ ( H @ X2 ) ) )
     => ( ( comp_a_b_real @ ( comp_a_b_a @ F2 @ M ) @ G @ X2 )
        = ( comp_real_b_real @ ( comp_a_b_real @ F2 @ N ) @ H @ X2 ) ) ) ).

% type_copy_map_cong0
thf(fact_929_real_Osingleton__sets,axiom,
    ! [X2: real] :
      ( ( member_real @ X2 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( member_set_real @ ( insert_real @ X2 @ bot_bot_set_real ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% real.singleton_sets
thf(fact_930_real_Ospace__UNIV,axiom,
    ( ( sigma_space_real @ borel_5078946678739801102l_real )
    = top_top_set_real ) ).

% real.space_UNIV
thf(fact_931_space__sup__measure_H,axiom,
    ! [B4: sigma_measure_real,A: sigma_measure_real] :
      ( ( ( sigma_sets_real @ B4 )
        = ( sigma_sets_real @ A ) )
     => ( ( sigma_space_real @ ( measur2147279183506585690e_real @ A @ B4 ) )
        = ( sigma_space_real @ A ) ) ) ).

% space_sup_measure'
thf(fact_932_space__sup__measure_H,axiom,
    ! [B4: sigma_7234349610311085201nnreal,A: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ B4 )
        = ( sigma_5465916536984168985nnreal @ A ) )
     => ( ( sigma_3147302497200244656nnreal @ ( measur4473656680840910822nnreal @ A @ B4 ) )
        = ( sigma_3147302497200244656nnreal @ A ) ) ) ).

% space_sup_measure'
thf(fact_933_space__sup__measure_H,axiom,
    ! [B4: sigma_measure_o,A: sigma_measure_o] :
      ( ( ( sigma_sets_o @ B4 )
        = ( sigma_sets_o @ A ) )
     => ( ( sigma_space_o @ ( measur4529518739368704874sure_o @ A @ B4 ) )
        = ( sigma_space_o @ A ) ) ) ).

% space_sup_measure'
thf(fact_934_space__sup__measure_H,axiom,
    ! [B4: sigma_measure_nat,A: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ B4 )
        = ( sigma_sets_nat @ A ) )
     => ( ( sigma_space_nat @ ( measur876423496291765374re_nat @ A @ B4 ) )
        = ( sigma_space_nat @ A ) ) ) ).

% space_sup_measure'
thf(fact_935_space__Sup__measure_H,axiom,
    ! [M: set_Si6059263944882162789e_real,A: sigma_measure_real] :
      ( ! [M4: sigma_measure_real] :
          ( ( member4553183543495551918e_real @ M4 @ M )
         => ( ( sigma_sets_real @ M4 )
            = ( sigma_sets_real @ A ) ) )
     => ( ( M != bot_bo5686449298802467025e_real )
       => ( ( sigma_space_real @ ( measur8657758558638653562e_real @ M ) )
          = ( sigma_space_real @ A ) ) ) ) ).

% space_Sup_measure'
thf(fact_936_space__Sup__measure_H,axiom,
    ! [M: set_Si97717610131227249nnreal,A: sigma_7234349610311085201nnreal] :
      ( ! [M4: sigma_7234349610311085201nnreal] :
          ( ( member6261374078160781754nnreal @ M4 @ M )
         => ( ( sigma_5465916536984168985nnreal @ M4 )
            = ( sigma_5465916536984168985nnreal @ A ) ) )
     => ( ( M != bot_bo8227844048696536285nnreal )
       => ( ( sigma_3147302497200244656nnreal @ ( measur1651139276328235014nnreal @ M ) )
          = ( sigma_3147302497200244656nnreal @ A ) ) ) ) ).

% space_Sup_measure'
thf(fact_937_space__Sup__measure_H,axiom,
    ! [M: set_Sigma_measure_o,A: sigma_measure_o] :
      ( ! [M4: sigma_measure_o] :
          ( ( member1844656263901471916sure_o @ M4 @ M )
         => ( ( sigma_sets_o @ M4 )
            = ( sigma_sets_o @ A ) ) )
     => ( ( M != bot_bo7838039659004643295sure_o )
       => ( ( sigma_space_o @ ( measur1214336222341667658sure_o @ M ) )
          = ( sigma_space_o @ A ) ) ) ) ).

% space_Sup_measure'
thf(fact_938_space__Sup__measure_H,axiom,
    ! [M: set_Si3048223896905877257re_nat,A: sigma_measure_nat] :
      ( ! [M4: sigma_measure_nat] :
          ( ( member4416920341759242834re_nat @ M4 @ M )
         => ( ( sigma_sets_nat @ M4 )
            = ( sigma_sets_nat @ A ) ) )
     => ( ( M != bot_bo8872222457363190133re_nat )
       => ( ( sigma_space_nat @ ( measur3575099672463795358re_nat @ M ) )
          = ( sigma_space_nat @ A ) ) ) ) ).

% space_Sup_measure'
thf(fact_939_borel__sigma__sets__subset,axiom,
    ! [A: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ top_top_set_real @ A ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% borel_sigma_sets_subset
thf(fact_940_borel__sigma__sets__subset,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) )
     => ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ top_to7994903218803871134nnreal @ A ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).

% borel_sigma_sets_subset
thf(fact_941_borel__sigma__sets__subset,axiom,
    ! [A: set_set_o] :
      ( ( ord_le4374716579403074808_set_o @ A @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) )
     => ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ top_top_set_o @ A ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% borel_sigma_sets_subset
thf(fact_942_borel__sigma__sets__subset,axiom,
    ! [A: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) )
     => ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ top_top_set_nat @ A ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% borel_sigma_sets_subset
thf(fact_943_iso__tuple__update__accessor__eq__assist__idI,axiom,
    ! [V3: real,F2: real > real,V: real] :
      ( ( V3
        = ( F2 @ V ) )
     => ( iso_tu1764886537532566175l_real @ id_real_real @ id_real @ V @ F2 @ V3 @ V ) ) ).

% iso_tuple_update_accessor_eq_assist_idI
thf(fact_944_iso__tuple__update__accessor__eq__assist__idI,axiom,
    ! [V3: extend8495563244428889912nnreal,F2: extend8495563244428889912nnreal > extend8495563244428889912nnreal,V: extend8495563244428889912nnreal] :
      ( ( V3
        = ( F2 @ V ) )
     => ( iso_tu1500430003656523831nnreal @ id_Ext6301196394018042846nnreal @ id_Ext8331313139072774535nnreal @ V @ F2 @ V3 @ V ) ) ).

% iso_tuple_update_accessor_eq_assist_idI
thf(fact_945_iso__tuple__update__accessor__eq__assist__idI,axiom,
    ! [V3: nat,F2: nat > nat,V: nat] :
      ( ( V3
        = ( F2 @ V ) )
     => ( iso_tu3079242893353865575at_nat @ id_nat_nat @ id_nat @ V @ F2 @ V3 @ V ) ) ).

% iso_tuple_update_accessor_eq_assist_idI
thf(fact_946_iso__tuple__update__accessor__eq__assist__idI,axiom,
    ! [V3: $o,F2: $o > $o,V: $o] :
      ( ( V3
        = ( F2 @ V ) )
     => ( iso_tu5524162117909064517st_o_o @ id_o_o @ id_o @ V @ F2 @ V3 @ V ) ) ).

% iso_tuple_update_accessor_eq_assist_idI
thf(fact_947_sets__eq__countable,axiom,
    ! [A: set_real,M: sigma_measure_real] :
      ( ( counta7319604579010473777e_real @ A )
     => ( ( ( sigma_space_real @ M )
          = A )
       => ( ! [X: real] :
              ( ( member_real @ X @ A )
             => ( member_set_real @ ( insert_real @ X @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
         => ( ( sigma_sets_real @ M )
            = ( pow_real @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_948_sets__eq__countable,axiom,
    ! [A: set_nat,M: sigma_measure_nat] :
      ( ( counta1168086296615599829le_nat @ A )
     => ( ( ( sigma_space_nat @ M )
          = A )
       => ( ! [X: nat] :
              ( ( member_nat @ X @ A )
             => ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) ) )
         => ( ( sigma_sets_nat @ M )
            = ( pow_nat @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_949_sets__eq__countable,axiom,
    ! [A: set_o,M: sigma_measure_o] :
      ( ( counta5976203206615340371able_o @ A )
     => ( ( ( sigma_space_o @ M )
          = A )
       => ( ! [X: $o] :
              ( ( member_o @ X @ A )
             => ( member_set_o @ ( insert_o @ X @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) ) )
         => ( ( sigma_sets_o @ M )
            = ( pow_o @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_950_sets__eq__countable,axiom,
    ! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( counta8439243037236335165nnreal @ A )
     => ( ( ( sigma_3147302497200244656nnreal @ M )
          = A )
       => ( ! [X: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ X @ A )
             => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
         => ( ( sigma_5465916536984168985nnreal @ M )
            = ( pow_Ex5372160365422184283nnreal @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_951_sets__eq__countable,axiom,
    ! [A: set_set_nat,M: sigma_3334325623652945375et_nat] :
      ( ( counta3299167949292459659et_nat @ A )
     => ( ( ( sigma_space_set_nat @ M )
          = A )
       => ( ! [X: set_nat] :
              ( ( member_set_nat @ X @ A )
             => ( member_set_set_nat @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) @ ( sigma_sets_set_nat @ M ) ) )
         => ( ( sigma_sets_set_nat @ M )
            = ( pow_set_nat @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_952_sets__eq__countable,axiom,
    ! [A: set_set_o,M: sigma_measure_set_o] :
      ( ( counta9002483607034949683_set_o @ A )
     => ( ( ( sigma_space_set_o @ M )
          = A )
       => ( ! [X: set_o] :
              ( ( member_set_o @ X @ A )
             => ( member_set_set_o @ ( insert_set_o @ X @ bot_bot_set_set_o ) @ ( sigma_sets_set_o @ M ) ) )
         => ( ( sigma_sets_set_o @ M )
            = ( pow_set_o @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_953_sets__eq__countable,axiom,
    ! [A: set_se4580700918925141924nnreal,M: sigma_523634232904505671nnreal] :
      ( ( counta2425349316461633011nnreal @ A )
     => ( ( ( sigma_2539764534872131430nnreal @ M )
          = A )
       => ( ! [X: set_Ex3793607809372303086nnreal] :
              ( ( member603777416030116741nnreal @ X @ A )
             => ( member6568240578637133883nnreal @ ( insert1343806209672318238nnreal @ X @ bot_bo2988155216863113784nnreal ) @ ( sigma_5308793447563920847nnreal @ M ) ) )
         => ( ( sigma_5308793447563920847nnreal @ M )
            = ( pow_se7371645914972035857nnreal @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_954_sets__eq__countable,axiom,
    ! [A: set_set_real,M: sigma_3733394171116455995t_real] :
      ( ( counta8054315614235329383t_real @ A )
     => ( ( ( sigma_space_set_real @ M )
          = A )
       => ( ! [X: set_real] :
              ( ( member_set_real @ X @ A )
             => ( member_set_set_real @ ( insert_set_real @ X @ bot_bot_set_set_real ) @ ( sigma_sets_set_real @ M ) ) )
         => ( ( sigma_sets_set_real @ M )
            = ( pow_set_real @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_955_sets__eq__countable,axiom,
    ! [A: set_real_a,M: sigma_measure_real_a] :
      ( ( counta6639396083684174020real_a @ A )
     => ( ( ( sigma_space_real_a @ M )
          = A )
       => ( ! [X: real > a] :
              ( ( member_real_a @ X @ A )
             => ( member_set_real_a @ ( insert_real_a @ X @ bot_bot_set_real_a ) @ ( sigma_sets_real_a @ M ) ) )
         => ( ( sigma_sets_real_a @ M )
            = ( pow_real_a @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_956_sets__eq__countable,axiom,
    ! [A: set_o_real,M: sigma_measure_o_real] :
      ( ( counta8783200249485735024o_real @ A )
     => ( ( ( sigma_space_o_real @ M )
          = A )
       => ( ! [X: $o > real] :
              ( ( member_o_real @ X @ A )
             => ( member_set_o_real @ ( insert_o_real @ X @ bot_bot_set_o_real ) @ ( sigma_sets_o_real @ M ) ) )
         => ( ( sigma_sets_o_real @ M )
            = ( pow_o_real @ A ) ) ) ) ) ).

% sets_eq_countable
thf(fact_957_sets__sup__measure_H,axiom,
    ! [B4: sigma_measure_real,A: sigma_measure_real] :
      ( ( ( sigma_sets_real @ B4 )
        = ( sigma_sets_real @ A ) )
     => ( ( sigma_sets_real @ ( measur2147279183506585690e_real @ A @ B4 ) )
        = ( sigma_sets_real @ A ) ) ) ).

% sets_sup_measure'
thf(fact_958_sets__sup__measure_H,axiom,
    ! [B4: sigma_measure_nat,A: sigma_measure_nat] :
      ( ( ( sigma_sets_nat @ B4 )
        = ( sigma_sets_nat @ A ) )
     => ( ( sigma_sets_nat @ ( measur876423496291765374re_nat @ A @ B4 ) )
        = ( sigma_sets_nat @ A ) ) ) ).

% sets_sup_measure'
thf(fact_959_sets__sup__measure_H,axiom,
    ! [B4: sigma_measure_o,A: sigma_measure_o] :
      ( ( ( sigma_sets_o @ B4 )
        = ( sigma_sets_o @ A ) )
     => ( ( sigma_sets_o @ ( measur4529518739368704874sure_o @ A @ B4 ) )
        = ( sigma_sets_o @ A ) ) ) ).

% sets_sup_measure'
thf(fact_960_sets__sup__measure_H,axiom,
    ! [B4: sigma_7234349610311085201nnreal,A: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_5465916536984168985nnreal @ B4 )
        = ( sigma_5465916536984168985nnreal @ A ) )
     => ( ( sigma_5465916536984168985nnreal @ ( measur4473656680840910822nnreal @ A @ B4 ) )
        = ( sigma_5465916536984168985nnreal @ A ) ) ) ).

% sets_sup_measure'
thf(fact_961_dual__order_Orefl,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).

% dual_order.refl
thf(fact_962_order__refl,axiom,
    ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).

% order_refl
thf(fact_963_subsetI,axiom,
    ! [A: set_real_a,B4: set_real_a] :
      ( ! [X: real > a] :
          ( ( member_real_a @ X @ A )
         => ( member_real_a @ X @ B4 ) )
     => ( ord_le5743406823621094409real_a @ A @ B4 ) ) ).

% subsetI
thf(fact_964_subsetI,axiom,
    ! [A: set_o_real,B4: set_o_real] :
      ( ! [X: $o > real] :
          ( ( member_o_real @ X @ A )
         => ( member_o_real @ X @ B4 ) )
     => ( ord_le3251842697534426805o_real @ A @ B4 ) ) ).

% subsetI
thf(fact_965_subsetI,axiom,
    ! [A: set_nat_real,B4: set_nat_real] :
      ( ! [X: nat > real] :
          ( ( member_nat_real @ X @ A )
         => ( member_nat_real @ X @ B4 ) )
     => ( ord_le2908806416726583473t_real @ A @ B4 ) ) ).

% subsetI
thf(fact_966_subsetI,axiom,
    ! [A: set_c_d,B4: set_c_d] :
      ( ! [X: c > d] :
          ( ( member_c_d @ X @ A )
         => ( member_c_d @ X @ B4 ) )
     => ( ord_less_eq_set_c_d @ A @ B4 ) ) ).

% subsetI
thf(fact_967_subsetI,axiom,
    ! [A: set_a_b,B4: set_a_b] :
      ( ! [X: a > b] :
          ( ( member_a_b @ X @ A )
         => ( member_a_b @ X @ B4 ) )
     => ( ord_less_eq_set_a_b @ A @ B4 ) ) ).

% subsetI
thf(fact_968_subsetI,axiom,
    ! [A: set_set_nat,B4: set_set_nat] :
      ( ! [X: set_nat] :
          ( ( member_set_nat @ X @ A )
         => ( member_set_nat @ X @ B4 ) )
     => ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ).

% subsetI
thf(fact_969_subsetI,axiom,
    ! [A: set_set_o,B4: set_set_o] :
      ( ! [X: set_o] :
          ( ( member_set_o @ X @ A )
         => ( member_set_o @ X @ B4 ) )
     => ( ord_le4374716579403074808_set_o @ A @ B4 ) ) ).

% subsetI
thf(fact_970_subsetI,axiom,
    ! [A: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal] :
      ( ! [X: set_Ex3793607809372303086nnreal] :
          ( ( member603777416030116741nnreal @ X @ A )
         => ( member603777416030116741nnreal @ X @ B4 ) )
     => ( ord_le3366939622266546180nnreal @ A @ B4 ) ) ).

% subsetI
thf(fact_971_subsetI,axiom,
    ! [A: set_set_real,B4: set_set_real] :
      ( ! [X: set_real] :
          ( ( member_set_real @ X @ A )
         => ( member_set_real @ X @ B4 ) )
     => ( ord_le3558479182127378552t_real @ A @ B4 ) ) ).

% subsetI
thf(fact_972_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_973_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_974_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_975_subset__empty,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ bot_bo4854962954004695426nnreal )
      = ( A = bot_bo4854962954004695426nnreal ) ) ).

% subset_empty
thf(fact_976_empty__subsetI,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% empty_subsetI
thf(fact_977_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_978_empty__subsetI,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% empty_subsetI
thf(fact_979_empty__subsetI,axiom,
    ! [A: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ bot_bo4854962954004695426nnreal @ A ) ).

% empty_subsetI
thf(fact_980_insert__subset,axiom,
    ! [X2: real,A: set_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X2 @ A ) @ B4 )
      = ( ( member_real @ X2 @ B4 )
        & ( ord_less_eq_set_real @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_981_insert__subset,axiom,
    ! [X2: nat,A: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A ) @ B4 )
      = ( ( member_nat @ X2 @ B4 )
        & ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_982_insert__subset,axiom,
    ! [X2: $o,A: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X2 @ A ) @ B4 )
      = ( ( member_o @ X2 @ B4 )
        & ( ord_less_eq_set_o @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_983_insert__subset,axiom,
    ! [X2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ ( insert7407984058720857448nnreal @ X2 @ A ) @ B4 )
      = ( ( member7908768830364227535nnreal @ X2 @ B4 )
        & ( ord_le6787938422905777998nnreal @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_984_insert__subset,axiom,
    ! [X2: set_nat,A: set_set_nat,B4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ A ) @ B4 )
      = ( ( member_set_nat @ X2 @ B4 )
        & ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_985_insert__subset,axiom,
    ! [X2: set_o,A: set_set_o,B4: set_set_o] :
      ( ( ord_le4374716579403074808_set_o @ ( insert_set_o @ X2 @ A ) @ B4 )
      = ( ( member_set_o @ X2 @ B4 )
        & ( ord_le4374716579403074808_set_o @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_986_insert__subset,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal] :
      ( ( ord_le3366939622266546180nnreal @ ( insert1343806209672318238nnreal @ X2 @ A ) @ B4 )
      = ( ( member603777416030116741nnreal @ X2 @ B4 )
        & ( ord_le3366939622266546180nnreal @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_987_insert__subset,axiom,
    ! [X2: set_real,A: set_set_real,B4: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ ( insert_set_real @ X2 @ A ) @ B4 )
      = ( ( member_set_real @ X2 @ B4 )
        & ( ord_le3558479182127378552t_real @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_988_insert__subset,axiom,
    ! [X2: real > a,A: set_real_a,B4: set_real_a] :
      ( ( ord_le5743406823621094409real_a @ ( insert_real_a @ X2 @ A ) @ B4 )
      = ( ( member_real_a @ X2 @ B4 )
        & ( ord_le5743406823621094409real_a @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_989_insert__subset,axiom,
    ! [X2: $o > real,A: set_o_real,B4: set_o_real] :
      ( ( ord_le3251842697534426805o_real @ ( insert_o_real @ X2 @ A ) @ B4 )
      = ( ( member_o_real @ X2 @ B4 )
        & ( ord_le3251842697534426805o_real @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_990_Pow__iff,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
      = ( ord_less_eq_set_nat @ A @ B4 ) ) ).

% Pow_iff
thf(fact_991_Pow__iff,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( member_set_o @ A @ ( pow_o @ B4 ) )
      = ( ord_less_eq_set_o @ A @ B4 ) ) ).

% Pow_iff
thf(fact_992_Pow__iff,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ A @ ( pow_Ex5372160365422184283nnreal @ B4 ) )
      = ( ord_le6787938422905777998nnreal @ A @ B4 ) ) ).

% Pow_iff
thf(fact_993_Pow__iff,axiom,
    ! [A: set_real,B4: set_real] :
      ( ( member_set_real @ A @ ( pow_real @ B4 ) )
      = ( ord_less_eq_set_real @ A @ B4 ) ) ).

% Pow_iff
thf(fact_994_PowI,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B4 )
     => ( member_set_nat @ A @ ( pow_nat @ B4 ) ) ) ).

% PowI
thf(fact_995_PowI,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A @ B4 )
     => ( member_set_o @ A @ ( pow_o @ B4 ) ) ) ).

% PowI
thf(fact_996_PowI,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B4 )
     => ( member603777416030116741nnreal @ A @ ( pow_Ex5372160365422184283nnreal @ B4 ) ) ) ).

% PowI
thf(fact_997_PowI,axiom,
    ! [A: set_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ A @ B4 )
     => ( member_set_real @ A @ ( pow_real @ B4 ) ) ) ).

% PowI
thf(fact_998_singleton__insert__inj__eq_H,axiom,
    ! [A2: real,A: set_real,B: real] :
      ( ( ( insert_real @ A2 @ A )
        = ( insert_real @ B @ bot_bot_set_real ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_real @ A @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_999_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B: nat] :
      ( ( ( insert_nat @ A2 @ A )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1000_singleton__insert__inj__eq_H,axiom,
    ! [A2: $o,A: set_o,B: $o] :
      ( ( ( insert_o @ A2 @ A )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1001_singleton__insert__inj__eq_H,axiom,
    ! [A2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( ( insert7407984058720857448nnreal @ A2 @ A )
        = ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) )
      = ( ( A2 = B )
        & ( ord_le6787938422905777998nnreal @ A @ ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1002_singleton__insert__inj__eq,axiom,
    ! [B: real,A2: real,A: set_real] :
      ( ( ( insert_real @ B @ bot_bot_set_real )
        = ( insert_real @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_real @ A @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1003_singleton__insert__inj__eq,axiom,
    ! [B: nat,A2: nat,A: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1004_singleton__insert__inj__eq,axiom,
    ! [B: $o,A2: $o,A: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_o @ A @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1005_singleton__insert__inj__eq,axiom,
    ! [B: extend8495563244428889912nnreal,A2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal] :
      ( ( ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal )
        = ( insert7407984058720857448nnreal @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_le6787938422905777998nnreal @ A @ ( insert7407984058720857448nnreal @ B @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1006_order__antisym__conv,axiom,
    ! [Y: nat,X2: nat] :
      ( ( ord_less_eq_nat @ Y @ X2 )
     => ( ( ord_less_eq_nat @ X2 @ Y )
        = ( X2 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1007_linorder__le__cases,axiom,
    ! [X2: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X2 @ Y )
     => ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_le_cases
thf(fact_1008_ord__le__eq__subst,axiom,
    ! [A2: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ( F2 @ B )
          = C )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1009_ord__eq__le__subst,axiom,
    ! [A2: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( A2
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1010_linorder__linear,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
      | ( ord_less_eq_nat @ Y @ X2 ) ) ).

% linorder_linear
thf(fact_1011_order__eq__refl,axiom,
    ! [X2: nat,Y: nat] :
      ( ( X2 = Y )
     => ( ord_less_eq_nat @ X2 @ Y ) ) ).

% order_eq_refl
thf(fact_1012_order__subst2,axiom,
    ! [A2: nat,B: nat,F2: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A2 ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1013_order__subst1,axiom,
    ! [A2: nat,F2: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A2 @ ( F2 @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1014_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
          & ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1015_antisym,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ B @ A2 )
       => ( A2 = B ) ) ) ).

% antisym
thf(fact_1016_subset__iff,axiom,
    ( ord_le5743406823621094409real_a
    = ( ^ [A5: set_real_a,B7: set_real_a] :
        ! [T2: real > a] :
          ( ( member_real_a @ T2 @ A5 )
         => ( member_real_a @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1017_subset__iff,axiom,
    ( ord_le3251842697534426805o_real
    = ( ^ [A5: set_o_real,B7: set_o_real] :
        ! [T2: $o > real] :
          ( ( member_o_real @ T2 @ A5 )
         => ( member_o_real @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1018_subset__iff,axiom,
    ( ord_le2908806416726583473t_real
    = ( ^ [A5: set_nat_real,B7: set_nat_real] :
        ! [T2: nat > real] :
          ( ( member_nat_real @ T2 @ A5 )
         => ( member_nat_real @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1019_subset__iff,axiom,
    ( ord_less_eq_set_c_d
    = ( ^ [A5: set_c_d,B7: set_c_d] :
        ! [T2: c > d] :
          ( ( member_c_d @ T2 @ A5 )
         => ( member_c_d @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1020_subset__iff,axiom,
    ( ord_less_eq_set_a_b
    = ( ^ [A5: set_a_b,B7: set_a_b] :
        ! [T2: a > b] :
          ( ( member_a_b @ T2 @ A5 )
         => ( member_a_b @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1021_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B7: set_set_nat] :
        ! [T2: set_nat] :
          ( ( member_set_nat @ T2 @ A5 )
         => ( member_set_nat @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1022_subset__iff,axiom,
    ( ord_le4374716579403074808_set_o
    = ( ^ [A5: set_set_o,B7: set_set_o] :
        ! [T2: set_o] :
          ( ( member_set_o @ T2 @ A5 )
         => ( member_set_o @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1023_subset__iff,axiom,
    ( ord_le3366939622266546180nnreal
    = ( ^ [A5: set_se4580700918925141924nnreal,B7: set_se4580700918925141924nnreal] :
        ! [T2: set_Ex3793607809372303086nnreal] :
          ( ( member603777416030116741nnreal @ T2 @ A5 )
         => ( member603777416030116741nnreal @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1024_subset__iff,axiom,
    ( ord_le3558479182127378552t_real
    = ( ^ [A5: set_set_real,B7: set_set_real] :
        ! [T2: set_real] :
          ( ( member_set_real @ T2 @ A5 )
         => ( member_set_real @ T2 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1025_subset__eq,axiom,
    ( ord_le5743406823621094409real_a
    = ( ^ [A5: set_real_a,B7: set_real_a] :
        ! [X3: real > a] :
          ( ( member_real_a @ X3 @ A5 )
         => ( member_real_a @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1026_subset__eq,axiom,
    ( ord_le3251842697534426805o_real
    = ( ^ [A5: set_o_real,B7: set_o_real] :
        ! [X3: $o > real] :
          ( ( member_o_real @ X3 @ A5 )
         => ( member_o_real @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1027_subset__eq,axiom,
    ( ord_le2908806416726583473t_real
    = ( ^ [A5: set_nat_real,B7: set_nat_real] :
        ! [X3: nat > real] :
          ( ( member_nat_real @ X3 @ A5 )
         => ( member_nat_real @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1028_subset__eq,axiom,
    ( ord_less_eq_set_c_d
    = ( ^ [A5: set_c_d,B7: set_c_d] :
        ! [X3: c > d] :
          ( ( member_c_d @ X3 @ A5 )
         => ( member_c_d @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1029_subset__eq,axiom,
    ( ord_less_eq_set_a_b
    = ( ^ [A5: set_a_b,B7: set_a_b] :
        ! [X3: a > b] :
          ( ( member_a_b @ X3 @ A5 )
         => ( member_a_b @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1030_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B7: set_set_nat] :
        ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A5 )
         => ( member_set_nat @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1031_subset__eq,axiom,
    ( ord_le4374716579403074808_set_o
    = ( ^ [A5: set_set_o,B7: set_set_o] :
        ! [X3: set_o] :
          ( ( member_set_o @ X3 @ A5 )
         => ( member_set_o @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1032_subset__eq,axiom,
    ( ord_le3366939622266546180nnreal
    = ( ^ [A5: set_se4580700918925141924nnreal,B7: set_se4580700918925141924nnreal] :
        ! [X3: set_Ex3793607809372303086nnreal] :
          ( ( member603777416030116741nnreal @ X3 @ A5 )
         => ( member603777416030116741nnreal @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1033_subset__eq,axiom,
    ( ord_le3558479182127378552t_real
    = ( ^ [A5: set_set_real,B7: set_set_real] :
        ! [X3: set_real] :
          ( ( member_set_real @ X3 @ A5 )
         => ( member_set_real @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1034_subsetD,axiom,
    ! [A: set_real_a,B4: set_real_a,C: real > a] :
      ( ( ord_le5743406823621094409real_a @ A @ B4 )
     => ( ( member_real_a @ C @ A )
       => ( member_real_a @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1035_subsetD,axiom,
    ! [A: set_o_real,B4: set_o_real,C: $o > real] :
      ( ( ord_le3251842697534426805o_real @ A @ B4 )
     => ( ( member_o_real @ C @ A )
       => ( member_o_real @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1036_subsetD,axiom,
    ! [A: set_nat_real,B4: set_nat_real,C: nat > real] :
      ( ( ord_le2908806416726583473t_real @ A @ B4 )
     => ( ( member_nat_real @ C @ A )
       => ( member_nat_real @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1037_subsetD,axiom,
    ! [A: set_c_d,B4: set_c_d,C: c > d] :
      ( ( ord_less_eq_set_c_d @ A @ B4 )
     => ( ( member_c_d @ C @ A )
       => ( member_c_d @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1038_subsetD,axiom,
    ! [A: set_a_b,B4: set_a_b,C: a > b] :
      ( ( ord_less_eq_set_a_b @ A @ B4 )
     => ( ( member_a_b @ C @ A )
       => ( member_a_b @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1039_subsetD,axiom,
    ! [A: set_set_nat,B4: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B4 )
     => ( ( member_set_nat @ C @ A )
       => ( member_set_nat @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1040_subsetD,axiom,
    ! [A: set_set_o,B4: set_set_o,C: set_o] :
      ( ( ord_le4374716579403074808_set_o @ A @ B4 )
     => ( ( member_set_o @ C @ A )
       => ( member_set_o @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1041_subsetD,axiom,
    ! [A: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal,C: set_Ex3793607809372303086nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A @ B4 )
     => ( ( member603777416030116741nnreal @ C @ A )
       => ( member603777416030116741nnreal @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1042_subsetD,axiom,
    ! [A: set_set_real,B4: set_set_real,C: set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ B4 )
     => ( ( member_set_real @ C @ A )
       => ( member_set_real @ C @ B4 ) ) ) ).

% subsetD
thf(fact_1043_in__mono,axiom,
    ! [A: set_real_a,B4: set_real_a,X2: real > a] :
      ( ( ord_le5743406823621094409real_a @ A @ B4 )
     => ( ( member_real_a @ X2 @ A )
       => ( member_real_a @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_1044_in__mono,axiom,
    ! [A: set_o_real,B4: set_o_real,X2: $o > real] :
      ( ( ord_le3251842697534426805o_real @ A @ B4 )
     => ( ( member_o_real @ X2 @ A )
       => ( member_o_real @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_1045_in__mono,axiom,
    ! [A: set_nat_real,B4: set_nat_real,X2: nat > real] :
      ( ( ord_le2908806416726583473t_real @ A @ B4 )
     => ( ( member_nat_real @ X2 @ A )
       => ( member_nat_real @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_1046_in__mono,axiom,
    ! [A: set_c_d,B4: set_c_d,X2: c > d] :
      ( ( ord_less_eq_set_c_d @ A @ B4 )
     => ( ( member_c_d @ X2 @ A )
       => ( member_c_d @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_1047_in__mono,axiom,
    ! [A: set_a_b,B4: set_a_b,X2: a > b] :
      ( ( ord_less_eq_set_a_b @ A @ B4 )
     => ( ( member_a_b @ X2 @ A )
       => ( member_a_b @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_1048_in__mono,axiom,
    ! [A: set_set_nat,B4: set_set_nat,X2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B4 )
     => ( ( member_set_nat @ X2 @ A )
       => ( member_set_nat @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_1049_in__mono,axiom,
    ! [A: set_set_o,B4: set_set_o,X2: set_o] :
      ( ( ord_le4374716579403074808_set_o @ A @ B4 )
     => ( ( member_set_o @ X2 @ A )
       => ( member_set_o @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_1050_in__mono,axiom,
    ! [A: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal,X2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A @ B4 )
     => ( ( member603777416030116741nnreal @ X2 @ A )
       => ( member603777416030116741nnreal @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_1051_in__mono,axiom,
    ! [A: set_set_real,B4: set_set_real,X2: set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ B4 )
     => ( ( member_set_real @ X2 @ A )
       => ( member_set_real @ X2 @ B4 ) ) ) ).

% in_mono
thf(fact_1052_dual__order_Otrans,axiom,
    ! [B: nat,A2: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A2 ) ) ) ).

% dual_order.trans
thf(fact_1053_dual__order_Oantisym,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( A2 = B ) ) ) ).

% dual_order.antisym
thf(fact_1054_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1055_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A2: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A4 @ B3 )
         => ( P @ A4 @ B3 ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( P @ B3 @ A4 )
           => ( P @ A4 @ B3 ) )
       => ( P @ A2 @ B ) ) ) ).

% linorder_wlog
thf(fact_1056_order__trans,axiom,
    ! [X2: nat,Y: nat,Z5: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z5 )
       => ( ord_less_eq_nat @ X2 @ Z5 ) ) ) ).

% order_trans
thf(fact_1057_order_Otrans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% order.trans
thf(fact_1058_order__antisym,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X2 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X2 )
       => ( X2 = Y ) ) ) ).

% order_antisym
thf(fact_1059_ord__le__eq__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_1060_ord__eq__le__trans,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( A2 = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A2 @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_1061_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
    = ( ^ [X3: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1062_le__cases3,axiom,
    ! [X2: nat,Y: nat,Z5: nat] :
      ( ( ( ord_less_eq_nat @ X2 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z5 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X2 )
         => ~ ( ord_less_eq_nat @ X2 @ Z5 ) )
       => ( ( ( ord_less_eq_nat @ X2 @ Z5 )
           => ~ ( ord_less_eq_nat @ Z5 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z5 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X2 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z5 )
               => ~ ( ord_less_eq_nat @ Z5 @ X2 ) )
             => ~ ( ( ord_less_eq_nat @ Z5 @ X2 )
                 => ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1063_nle__le,axiom,
    ! [A2: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
      = ( ( ord_less_eq_nat @ B @ A2 )
        & ( B != A2 ) ) ) ).

% nle_le
thf(fact_1064_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_real,B4: sigma_measure_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ A ) @ ( sigma_sets_real @ B4 ) )
     => ( ord_less_eq_set_real @ ( sigma_space_real @ A ) @ ( sigma_space_real @ B4 ) ) ) ).

% sets_le_imp_space_le
thf(fact_1065_sets__le__imp__space__le,axiom,
    ! [A: sigma_7234349610311085201nnreal,B4: sigma_7234349610311085201nnreal] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ A ) @ ( sigma_5465916536984168985nnreal @ B4 ) )
     => ( ord_le6787938422905777998nnreal @ ( sigma_3147302497200244656nnreal @ A ) @ ( sigma_3147302497200244656nnreal @ B4 ) ) ) ).

% sets_le_imp_space_le
thf(fact_1066_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_o,B4: sigma_measure_o] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ A ) @ ( sigma_sets_o @ B4 ) )
     => ( ord_less_eq_set_o @ ( sigma_space_o @ A ) @ ( sigma_space_o @ B4 ) ) ) ).

% sets_le_imp_space_le
thf(fact_1067_sets__le__imp__space__le,axiom,
    ! [A: sigma_measure_nat,B4: sigma_measure_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ A ) @ ( sigma_sets_nat @ B4 ) )
     => ( ord_less_eq_set_nat @ ( sigma_space_nat @ A ) @ ( sigma_space_nat @ B4 ) ) ) ).

% sets_le_imp_space_le
thf(fact_1068_sigma__sets__into__sp,axiom,
    ! [A: set_set_nat,Sp: set_nat,X2: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ ( pow_nat @ Sp ) )
     => ( ( member_set_nat @ X2 @ ( sigma_sigma_sets_nat @ Sp @ A ) )
       => ( ord_less_eq_set_nat @ X2 @ Sp ) ) ) ).

% sigma_sets_into_sp
thf(fact_1069_sigma__sets__into__sp,axiom,
    ! [A: set_set_o,Sp: set_o,X2: set_o] :
      ( ( ord_le4374716579403074808_set_o @ A @ ( pow_o @ Sp ) )
     => ( ( member_set_o @ X2 @ ( sigma_sigma_sets_o @ Sp @ A ) )
       => ( ord_less_eq_set_o @ X2 @ Sp ) ) ) ).

% sigma_sets_into_sp
thf(fact_1070_sigma__sets__into__sp,axiom,
    ! [A: set_se4580700918925141924nnreal,Sp: set_Ex3793607809372303086nnreal,X2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A @ ( pow_Ex5372160365422184283nnreal @ Sp ) )
     => ( ( member603777416030116741nnreal @ X2 @ ( sigma_7808855514367478112nnreal @ Sp @ A ) )
       => ( ord_le6787938422905777998nnreal @ X2 @ Sp ) ) ) ).

% sigma_sets_into_sp
thf(fact_1071_sigma__sets__into__sp,axiom,
    ! [A: set_set_real,Sp: set_real,X2: set_real] :
      ( ( ord_le3558479182127378552t_real @ A @ ( pow_real @ Sp ) )
     => ( ( member_set_real @ X2 @ ( sigma_7195353284648819924s_real @ Sp @ A ) )
       => ( ord_less_eq_set_real @ X2 @ Sp ) ) ) ).

% sigma_sets_into_sp
thf(fact_1072_top__greatest,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ A2 @ top_top_set_real ) ).

% top_greatest
thf(fact_1073_top__greatest,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).

% top_greatest
thf(fact_1074_top__greatest,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).

% top_greatest
thf(fact_1075_top__greatest,axiom,
    ! [A2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A2 @ top_to7994903218803871134nnreal ) ).

% top_greatest
thf(fact_1076_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_1077_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_1078_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_1079_top_Oextremum__unique,axiom,
    ! [A2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ top_to7994903218803871134nnreal @ A2 )
      = ( A2 = top_to7994903218803871134nnreal ) ) ).

% top.extremum_unique
thf(fact_1080_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_1081_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_1082_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_1083_top_Oextremum__uniqueI,axiom,
    ! [A2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ top_to7994903218803871134nnreal @ A2 )
     => ( A2 = top_to7994903218803871134nnreal ) ) ).

% top.extremum_uniqueI
thf(fact_1084_bot_Oextremum,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% bot.extremum
thf(fact_1085_bot_Oextremum,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% bot.extremum
thf(fact_1086_bot_Oextremum,axiom,
    ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).

% bot.extremum
thf(fact_1087_bot_Oextremum,axiom,
    ! [A2: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ bot_bo4854962954004695426nnreal @ A2 ) ).

% bot.extremum
thf(fact_1088_bot_Oextremum,axiom,
    ! [A2: extend8495563244428889912nnreal] : ( ord_le3935885782089961368nnreal @ bot_bo841427958541957580nnreal @ A2 ) ).

% bot.extremum
thf(fact_1089_bot_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).

% bot.extremum
thf(fact_1090_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_1091_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_1092_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_1093_bot_Oextremum__unique,axiom,
    ! [A2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A2 @ bot_bo4854962954004695426nnreal )
      = ( A2 = bot_bo4854962954004695426nnreal ) ) ).

% bot.extremum_unique
thf(fact_1094_bot_Oextremum__unique,axiom,
    ! [A2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ bot_bo841427958541957580nnreal )
      = ( A2 = bot_bo841427958541957580nnreal ) ) ).

% bot.extremum_unique
thf(fact_1095_bot_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
      = ( A2 = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1096_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_1097_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_1098_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_1099_bot_Oextremum__uniqueI,axiom,
    ! [A2: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A2 @ bot_bo4854962954004695426nnreal )
     => ( A2 = bot_bo4854962954004695426nnreal ) ) ).

% bot.extremum_uniqueI
thf(fact_1100_bot_Oextremum__uniqueI,axiom,
    ! [A2: extend8495563244428889912nnreal] :
      ( ( ord_le3935885782089961368nnreal @ A2 @ bot_bo841427958541957580nnreal )
     => ( A2 = bot_bo841427958541957580nnreal ) ) ).

% bot.extremum_uniqueI
thf(fact_1101_bot_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
     => ( A2 = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1102_subset__UNIV,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ A @ top_top_set_real ) ).

% subset_UNIV
thf(fact_1103_subset__UNIV,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).

% subset_UNIV
thf(fact_1104_subset__UNIV,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).

% subset_UNIV
thf(fact_1105_subset__UNIV,axiom,
    ! [A: set_Ex3793607809372303086nnreal] : ( ord_le6787938422905777998nnreal @ A @ top_to7994903218803871134nnreal ) ).

% subset_UNIV
thf(fact_1106_subset__insertI2,axiom,
    ! [A: set_real,B4: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A @ B4 )
     => ( ord_less_eq_set_real @ A @ ( insert_real @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_1107_subset__insertI2,axiom,
    ! [A: set_nat,B4: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A @ B4 )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_1108_subset__insertI2,axiom,
    ! [A: set_o,B4: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A @ B4 )
     => ( ord_less_eq_set_o @ A @ ( insert_o @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_1109_subset__insertI2,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal,B: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ B4 )
     => ( ord_le6787938422905777998nnreal @ A @ ( insert7407984058720857448nnreal @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_1110_subset__insertI,axiom,
    ! [B4: set_real,A2: real] : ( ord_less_eq_set_real @ B4 @ ( insert_real @ A2 @ B4 ) ) ).

% subset_insertI
thf(fact_1111_subset__insertI,axiom,
    ! [B4: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A2 @ B4 ) ) ).

% subset_insertI
thf(fact_1112_subset__insertI,axiom,
    ! [B4: set_o,A2: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A2 @ B4 ) ) ).

% subset_insertI
thf(fact_1113_subset__insertI,axiom,
    ! [B4: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] : ( ord_le6787938422905777998nnreal @ B4 @ ( insert7407984058720857448nnreal @ A2 @ B4 ) ) ).

% subset_insertI
thf(fact_1114_subset__insert,axiom,
    ! [X2: real,A: set_real,B4: set_real] :
      ( ~ ( member_real @ X2 @ A )
     => ( ( ord_less_eq_set_real @ A @ ( insert_real @ X2 @ B4 ) )
        = ( ord_less_eq_set_real @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1115_subset__insert,axiom,
    ! [X2: nat,A: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X2 @ A )
     => ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X2 @ B4 ) )
        = ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1116_subset__insert,axiom,
    ! [X2: $o,A: set_o,B4: set_o] :
      ( ~ ( member_o @ X2 @ A )
     => ( ( ord_less_eq_set_o @ A @ ( insert_o @ X2 @ B4 ) )
        = ( ord_less_eq_set_o @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1117_subset__insert,axiom,
    ! [X2: extend8495563244428889912nnreal,A: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ~ ( member7908768830364227535nnreal @ X2 @ A )
     => ( ( ord_le6787938422905777998nnreal @ A @ ( insert7407984058720857448nnreal @ X2 @ B4 ) )
        = ( ord_le6787938422905777998nnreal @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1118_subset__insert,axiom,
    ! [X2: set_nat,A: set_set_nat,B4: set_set_nat] :
      ( ~ ( member_set_nat @ X2 @ A )
     => ( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X2 @ B4 ) )
        = ( ord_le6893508408891458716et_nat @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1119_subset__insert,axiom,
    ! [X2: set_o,A: set_set_o,B4: set_set_o] :
      ( ~ ( member_set_o @ X2 @ A )
     => ( ( ord_le4374716579403074808_set_o @ A @ ( insert_set_o @ X2 @ B4 ) )
        = ( ord_le4374716579403074808_set_o @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1120_subset__insert,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal] :
      ( ~ ( member603777416030116741nnreal @ X2 @ A )
     => ( ( ord_le3366939622266546180nnreal @ A @ ( insert1343806209672318238nnreal @ X2 @ B4 ) )
        = ( ord_le3366939622266546180nnreal @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1121_subset__insert,axiom,
    ! [X2: set_real,A: set_set_real,B4: set_set_real] :
      ( ~ ( member_set_real @ X2 @ A )
     => ( ( ord_le3558479182127378552t_real @ A @ ( insert_set_real @ X2 @ B4 ) )
        = ( ord_le3558479182127378552t_real @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1122_subset__insert,axiom,
    ! [X2: real > a,A: set_real_a,B4: set_real_a] :
      ( ~ ( member_real_a @ X2 @ A )
     => ( ( ord_le5743406823621094409real_a @ A @ ( insert_real_a @ X2 @ B4 ) )
        = ( ord_le5743406823621094409real_a @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1123_subset__insert,axiom,
    ! [X2: $o > real,A: set_o_real,B4: set_o_real] :
      ( ~ ( member_o_real @ X2 @ A )
     => ( ( ord_le3251842697534426805o_real @ A @ ( insert_o_real @ X2 @ B4 ) )
        = ( ord_le3251842697534426805o_real @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_1124_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_1125_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_1126_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_1127_Set_Oinsert__mono,axiom,
    ! [C3: set_Ex3793607809372303086nnreal,D2: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ C3 @ D2 )
     => ( ord_le6787938422905777998nnreal @ ( insert7407984058720857448nnreal @ A2 @ C3 ) @ ( insert7407984058720857448nnreal @ A2 @ D2 ) ) ) ).

% Set.insert_mono
thf(fact_1128_PowD,axiom,
    ! [A: set_nat,B4: set_nat] :
      ( ( member_set_nat @ A @ ( pow_nat @ B4 ) )
     => ( ord_less_eq_set_nat @ A @ B4 ) ) ).

% PowD
thf(fact_1129_PowD,axiom,
    ! [A: set_o,B4: set_o] :
      ( ( member_set_o @ A @ ( pow_o @ B4 ) )
     => ( ord_less_eq_set_o @ A @ B4 ) ) ).

% PowD
thf(fact_1130_PowD,axiom,
    ! [A: set_Ex3793607809372303086nnreal,B4: set_Ex3793607809372303086nnreal] :
      ( ( member603777416030116741nnreal @ A @ ( pow_Ex5372160365422184283nnreal @ B4 ) )
     => ( ord_le6787938422905777998nnreal @ A @ B4 ) ) ).

% PowD
thf(fact_1131_PowD,axiom,
    ! [A: set_real,B4: set_real] :
      ( ( member_set_real @ A @ ( pow_real @ B4 ) )
     => ( ord_less_eq_set_real @ A @ B4 ) ) ).

% PowD
thf(fact_1132_subset__singleton__iff,axiom,
    ! [X5: set_real,A2: real] :
      ( ( ord_less_eq_set_real @ X5 @ ( insert_real @ A2 @ bot_bot_set_real ) )
      = ( ( X5 = bot_bot_set_real )
        | ( X5
          = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_1133_subset__singleton__iff,axiom,
    ! [X5: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( ( X5 = bot_bot_set_nat )
        | ( X5
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_1134_subset__singleton__iff,axiom,
    ! [X5: set_o,A2: $o] :
      ( ( ord_less_eq_set_o @ X5 @ ( insert_o @ A2 @ bot_bot_set_o ) )
      = ( ( X5 = bot_bot_set_o )
        | ( X5
          = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_1135_subset__singleton__iff,axiom,
    ! [X5: set_Ex3793607809372303086nnreal,A2: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ X5 @ ( insert7407984058720857448nnreal @ A2 @ bot_bo4854962954004695426nnreal ) )
      = ( ( X5 = bot_bo4854962954004695426nnreal )
        | ( X5
          = ( insert7407984058720857448nnreal @ A2 @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% subset_singleton_iff
thf(fact_1136_subset__singletonD,axiom,
    ! [A: set_real,X2: real] :
      ( ( ord_less_eq_set_real @ A @ ( insert_real @ X2 @ bot_bot_set_real ) )
     => ( ( A = bot_bot_set_real )
        | ( A
          = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_1137_subset__singletonD,axiom,
    ! [A: set_nat,X2: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
     => ( ( A = bot_bot_set_nat )
        | ( A
          = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_1138_subset__singletonD,axiom,
    ! [A: set_o,X2: $o] :
      ( ( ord_less_eq_set_o @ A @ ( insert_o @ X2 @ bot_bot_set_o ) )
     => ( ( A = bot_bot_set_o )
        | ( A
          = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_1139_subset__singletonD,axiom,
    ! [A: set_Ex3793607809372303086nnreal,X2: extend8495563244428889912nnreal] :
      ( ( ord_le6787938422905777998nnreal @ A @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) )
     => ( ( A = bot_bo4854962954004695426nnreal )
        | ( A
          = ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) ) ) ) ).

% subset_singletonD
thf(fact_1140_sets_Osets__into__space,axiom,
    ! [X2: set_real,M: sigma_measure_real] :
      ( ( member_set_real @ X2 @ ( sigma_sets_real @ M ) )
     => ( ord_less_eq_set_real @ X2 @ ( sigma_space_real @ M ) ) ) ).

% sets.sets_into_space
thf(fact_1141_sets_Osets__into__space,axiom,
    ! [X2: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( member603777416030116741nnreal @ X2 @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ord_le6787938422905777998nnreal @ X2 @ ( sigma_3147302497200244656nnreal @ M ) ) ) ).

% sets.sets_into_space
thf(fact_1142_sets_Osets__into__space,axiom,
    ! [X2: set_o,M: sigma_measure_o] :
      ( ( member_set_o @ X2 @ ( sigma_sets_o @ M ) )
     => ( ord_less_eq_set_o @ X2 @ ( sigma_space_o @ M ) ) ) ).

% sets.sets_into_space
thf(fact_1143_sets_Osets__into__space,axiom,
    ! [X2: set_nat,M: sigma_measure_nat] :
      ( ( member_set_nat @ X2 @ ( sigma_sets_nat @ M ) )
     => ( ord_less_eq_set_nat @ X2 @ ( sigma_space_nat @ M ) ) ) ).

% sets.sets_into_space
thf(fact_1144_sets_Osigma__sets__subset_H,axiom,
    ! [A2: set_set_real,M: sigma_measure_real,Omega: set_real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ ( sigma_sets_real @ M ) )
     => ( ( member_set_real @ Omega @ ( sigma_sets_real @ M ) )
       => ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ Omega @ A2 ) @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_1145_sets_Osigma__sets__subset_H,axiom,
    ! [A2: set_set_nat,M: sigma_measure_nat,Omega: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( sigma_sets_nat @ M ) )
     => ( ( member_set_nat @ Omega @ ( sigma_sets_nat @ M ) )
       => ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ Omega @ A2 ) @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_1146_sets_Osigma__sets__subset_H,axiom,
    ! [A2: set_set_o,M: sigma_measure_o,Omega: set_o] :
      ( ( ord_le4374716579403074808_set_o @ A2 @ ( sigma_sets_o @ M ) )
     => ( ( member_set_o @ Omega @ ( sigma_sets_o @ M ) )
       => ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ Omega @ A2 ) @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_1147_sets_Osigma__sets__subset_H,axiom,
    ! [A2: set_se4580700918925141924nnreal,M: sigma_7234349610311085201nnreal,Omega: set_Ex3793607809372303086nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A2 @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ( member603777416030116741nnreal @ Omega @ ( sigma_5465916536984168985nnreal @ M ) )
       => ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ Omega @ A2 ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.sigma_sets_subset'
thf(fact_1148_sigma__sets__mono_H_H,axiom,
    ! [A: set_nat,C3: set_nat,D2: set_set_nat,B4: set_set_nat] :
      ( ( member_set_nat @ A @ ( sigma_sigma_sets_nat @ C3 @ D2 ) )
     => ( ( ord_le6893508408891458716et_nat @ B4 @ D2 )
       => ( ( ord_le6893508408891458716et_nat @ D2 @ ( pow_nat @ C3 ) )
         => ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ A @ B4 ) @ ( sigma_sigma_sets_nat @ C3 @ D2 ) ) ) ) ) ).

% sigma_sets_mono''
thf(fact_1149_sigma__sets__mono_H_H,axiom,
    ! [A: set_o,C3: set_o,D2: set_set_o,B4: set_set_o] :
      ( ( member_set_o @ A @ ( sigma_sigma_sets_o @ C3 @ D2 ) )
     => ( ( ord_le4374716579403074808_set_o @ B4 @ D2 )
       => ( ( ord_le4374716579403074808_set_o @ D2 @ ( pow_o @ C3 ) )
         => ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ A @ B4 ) @ ( sigma_sigma_sets_o @ C3 @ D2 ) ) ) ) ) ).

% sigma_sets_mono''
thf(fact_1150_sigma__sets__mono_H_H,axiom,
    ! [A: set_Ex3793607809372303086nnreal,C3: set_Ex3793607809372303086nnreal,D2: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal] :
      ( ( member603777416030116741nnreal @ A @ ( sigma_7808855514367478112nnreal @ C3 @ D2 ) )
     => ( ( ord_le3366939622266546180nnreal @ B4 @ D2 )
       => ( ( ord_le3366939622266546180nnreal @ D2 @ ( pow_Ex5372160365422184283nnreal @ C3 ) )
         => ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ A @ B4 ) @ ( sigma_7808855514367478112nnreal @ C3 @ D2 ) ) ) ) ) ).

% sigma_sets_mono''
thf(fact_1151_sigma__sets__mono_H_H,axiom,
    ! [A: set_real,C3: set_real,D2: set_set_real,B4: set_set_real] :
      ( ( member_set_real @ A @ ( sigma_7195353284648819924s_real @ C3 @ D2 ) )
     => ( ( ord_le3558479182127378552t_real @ B4 @ D2 )
       => ( ( ord_le3558479182127378552t_real @ D2 @ ( pow_real @ C3 ) )
         => ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ A @ B4 ) @ ( sigma_7195353284648819924s_real @ C3 @ D2 ) ) ) ) ) ).

% sigma_sets_mono''
thf(fact_1152_sets__Sup__measure_H,axiom,
    ! [M: set_Si6059263944882162789e_real,A: sigma_measure_real] :
      ( ! [M4: sigma_measure_real] :
          ( ( member4553183543495551918e_real @ M4 @ M )
         => ( ( sigma_sets_real @ M4 )
            = ( sigma_sets_real @ A ) ) )
     => ( ( M != bot_bo5686449298802467025e_real )
       => ( ( sigma_sets_real @ ( measur8657758558638653562e_real @ M ) )
          = ( sigma_sets_real @ A ) ) ) ) ).

% sets_Sup_measure'
thf(fact_1153_sets__Sup__measure_H,axiom,
    ! [M: set_Si3048223896905877257re_nat,A: sigma_measure_nat] :
      ( ! [M4: sigma_measure_nat] :
          ( ( member4416920341759242834re_nat @ M4 @ M )
         => ( ( sigma_sets_nat @ M4 )
            = ( sigma_sets_nat @ A ) ) )
     => ( ( M != bot_bo8872222457363190133re_nat )
       => ( ( sigma_sets_nat @ ( measur3575099672463795358re_nat @ M ) )
          = ( sigma_sets_nat @ A ) ) ) ) ).

% sets_Sup_measure'
thf(fact_1154_sets__Sup__measure_H,axiom,
    ! [M: set_Sigma_measure_o,A: sigma_measure_o] :
      ( ! [M4: sigma_measure_o] :
          ( ( member1844656263901471916sure_o @ M4 @ M )
         => ( ( sigma_sets_o @ M4 )
            = ( sigma_sets_o @ A ) ) )
     => ( ( M != bot_bo7838039659004643295sure_o )
       => ( ( sigma_sets_o @ ( measur1214336222341667658sure_o @ M ) )
          = ( sigma_sets_o @ A ) ) ) ) ).

% sets_Sup_measure'
thf(fact_1155_sets__Sup__measure_H,axiom,
    ! [M: set_Si97717610131227249nnreal,A: sigma_7234349610311085201nnreal] :
      ( ! [M4: sigma_7234349610311085201nnreal] :
          ( ( member6261374078160781754nnreal @ M4 @ M )
         => ( ( sigma_5465916536984168985nnreal @ M4 )
            = ( sigma_5465916536984168985nnreal @ A ) ) )
     => ( ( M != bot_bo8227844048696536285nnreal )
       => ( ( sigma_5465916536984168985nnreal @ ( measur1651139276328235014nnreal @ M ) )
          = ( sigma_5465916536984168985nnreal @ A ) ) ) ) ).

% sets_Sup_measure'
thf(fact_1156_sets_Ocountable,axiom,
    ! [A: set_real,M: sigma_measure_real] :
      ( ! [A4: real] :
          ( ( member_real @ A4 @ A )
         => ( member_set_real @ ( insert_real @ A4 @ bot_bot_set_real ) @ ( sigma_sets_real @ M ) ) )
     => ( ( counta7319604579010473777e_real @ A )
       => ( member_set_real @ A @ ( sigma_sets_real @ M ) ) ) ) ).

% sets.countable
thf(fact_1157_sets_Ocountable,axiom,
    ! [A: set_nat,M: sigma_measure_nat] :
      ( ! [A4: nat] :
          ( ( member_nat @ A4 @ A )
         => ( member_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) @ ( sigma_sets_nat @ M ) ) )
     => ( ( counta1168086296615599829le_nat @ A )
       => ( member_set_nat @ A @ ( sigma_sets_nat @ M ) ) ) ) ).

% sets.countable
thf(fact_1158_sets_Ocountable,axiom,
    ! [A: set_o,M: sigma_measure_o] :
      ( ! [A4: $o] :
          ( ( member_o @ A4 @ A )
         => ( member_set_o @ ( insert_o @ A4 @ bot_bot_set_o ) @ ( sigma_sets_o @ M ) ) )
     => ( ( counta5976203206615340371able_o @ A )
       => ( member_set_o @ A @ ( sigma_sets_o @ M ) ) ) ) ).

% sets.countable
thf(fact_1159_sets_Ocountable,axiom,
    ! [A: set_Ex3793607809372303086nnreal,M: sigma_7234349610311085201nnreal] :
      ( ! [A4: extend8495563244428889912nnreal] :
          ( ( member7908768830364227535nnreal @ A4 @ A )
         => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ A4 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ M ) ) )
     => ( ( counta8439243037236335165nnreal @ A )
       => ( member603777416030116741nnreal @ A @ ( sigma_5465916536984168985nnreal @ M ) ) ) ) ).

% sets.countable
thf(fact_1160_sets_Ocountable,axiom,
    ! [A: set_set_nat,M: sigma_3334325623652945375et_nat] :
      ( ! [A4: set_nat] :
          ( ( member_set_nat @ A4 @ A )
         => ( member_set_set_nat @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) @ ( sigma_sets_set_nat @ M ) ) )
     => ( ( counta3299167949292459659et_nat @ A )
       => ( member_set_set_nat @ A @ ( sigma_sets_set_nat @ M ) ) ) ) ).

% sets.countable
thf(fact_1161_sets_Ocountable,axiom,
    ! [A: set_set_o,M: sigma_measure_set_o] :
      ( ! [A4: set_o] :
          ( ( member_set_o @ A4 @ A )
         => ( member_set_set_o @ ( insert_set_o @ A4 @ bot_bot_set_set_o ) @ ( sigma_sets_set_o @ M ) ) )
     => ( ( counta9002483607034949683_set_o @ A )
       => ( member_set_set_o @ A @ ( sigma_sets_set_o @ M ) ) ) ) ).

% sets.countable
thf(fact_1162_sets_Ocountable,axiom,
    ! [A: set_se4580700918925141924nnreal,M: sigma_523634232904505671nnreal] :
      ( ! [A4: set_Ex3793607809372303086nnreal] :
          ( ( member603777416030116741nnreal @ A4 @ A )
         => ( member6568240578637133883nnreal @ ( insert1343806209672318238nnreal @ A4 @ bot_bo2988155216863113784nnreal ) @ ( sigma_5308793447563920847nnreal @ M ) ) )
     => ( ( counta2425349316461633011nnreal @ A )
       => ( member6568240578637133883nnreal @ A @ ( sigma_5308793447563920847nnreal @ M ) ) ) ) ).

% sets.countable
thf(fact_1163_sets_Ocountable,axiom,
    ! [A: set_set_real,M: sigma_3733394171116455995t_real] :
      ( ! [A4: set_real] :
          ( ( member_set_real @ A4 @ A )
         => ( member_set_set_real @ ( insert_set_real @ A4 @ bot_bot_set_set_real ) @ ( sigma_sets_set_real @ M ) ) )
     => ( ( counta8054315614235329383t_real @ A )
       => ( member_set_set_real @ A @ ( sigma_sets_set_real @ M ) ) ) ) ).

% sets.countable
thf(fact_1164_sets_Ocountable,axiom,
    ! [A: set_real_a,M: sigma_measure_real_a] :
      ( ! [A4: real > a] :
          ( ( member_real_a @ A4 @ A )
         => ( member_set_real_a @ ( insert_real_a @ A4 @ bot_bot_set_real_a ) @ ( sigma_sets_real_a @ M ) ) )
     => ( ( counta6639396083684174020real_a @ A )
       => ( member_set_real_a @ A @ ( sigma_sets_real_a @ M ) ) ) ) ).

% sets.countable
thf(fact_1165_sets_Ocountable,axiom,
    ! [A: set_o_real,M: sigma_measure_o_real] :
      ( ! [A4: $o > real] :
          ( ( member_o_real @ A4 @ A )
         => ( member_set_o_real @ ( insert_o_real @ A4 @ bot_bot_set_o_real ) @ ( sigma_sets_o_real @ M ) ) )
     => ( ( counta8783200249485735024o_real @ A )
       => ( member_set_o_real @ A @ ( sigma_sets_o_real @ M ) ) ) ) ).

% sets.countable
thf(fact_1166_sets_Ospace__closed,axiom,
    ! [M: sigma_measure_real] : ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ ( pow_real @ ( sigma_space_real @ M ) ) ) ).

% sets.space_closed
thf(fact_1167_sets_Ospace__closed,axiom,
    ! [M: sigma_7234349610311085201nnreal] : ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ ( pow_Ex5372160365422184283nnreal @ ( sigma_3147302497200244656nnreal @ M ) ) ) ).

% sets.space_closed
thf(fact_1168_sets_Ospace__closed,axiom,
    ! [M: sigma_measure_o] : ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ M ) @ ( pow_o @ ( sigma_space_o @ M ) ) ) ).

% sets.space_closed
thf(fact_1169_sets_Ospace__closed,axiom,
    ! [M: sigma_measure_nat] : ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ M ) @ ( pow_nat @ ( sigma_space_nat @ M ) ) ) ).

% sets.space_closed
thf(fact_1170_sets_Osigma__sets__subset,axiom,
    ! [A2: set_set_real,M: sigma_measure_real] :
      ( ( ord_le3558479182127378552t_real @ A2 @ ( sigma_sets_real @ M ) )
     => ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ ( sigma_space_real @ M ) @ A2 ) @ ( sigma_sets_real @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_1171_sets_Osigma__sets__subset,axiom,
    ! [A2: set_se4580700918925141924nnreal,M: sigma_7234349610311085201nnreal] :
      ( ( ord_le3366939622266546180nnreal @ A2 @ ( sigma_5465916536984168985nnreal @ M ) )
     => ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ A2 ) @ ( sigma_5465916536984168985nnreal @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_1172_sets_Osigma__sets__subset,axiom,
    ! [A2: set_set_o,M: sigma_measure_o] :
      ( ( ord_le4374716579403074808_set_o @ A2 @ ( sigma_sets_o @ M ) )
     => ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ ( sigma_space_o @ M ) @ A2 ) @ ( sigma_sets_o @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_1173_sets_Osigma__sets__subset,axiom,
    ! [A2: set_set_nat,M: sigma_measure_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( sigma_sets_nat @ M ) )
     => ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ ( sigma_space_nat @ M ) @ A2 ) @ ( sigma_sets_nat @ M ) ) ) ).

% sets.sigma_sets_subset
thf(fact_1174_sigma__sets__le__sets__iff,axiom,
    ! [X2: sigma_measure_real,A7: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ ( sigma_space_real @ X2 ) @ A7 ) @ ( sigma_sets_real @ X2 ) )
      = ( ord_le3558479182127378552t_real @ A7 @ ( sigma_sets_real @ X2 ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_1175_sigma__sets__le__sets__iff,axiom,
    ! [X2: sigma_7234349610311085201nnreal,A7: set_se4580700918925141924nnreal] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ ( sigma_3147302497200244656nnreal @ X2 ) @ A7 ) @ ( sigma_5465916536984168985nnreal @ X2 ) )
      = ( ord_le3366939622266546180nnreal @ A7 @ ( sigma_5465916536984168985nnreal @ X2 ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_1176_sigma__sets__le__sets__iff,axiom,
    ! [X2: sigma_measure_o,A7: set_set_o] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ ( sigma_space_o @ X2 ) @ A7 ) @ ( sigma_sets_o @ X2 ) )
      = ( ord_le4374716579403074808_set_o @ A7 @ ( sigma_sets_o @ X2 ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_1177_sigma__sets__le__sets__iff,axiom,
    ! [X2: sigma_measure_nat,A7: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ ( sigma_space_nat @ X2 ) @ A7 ) @ ( sigma_sets_nat @ X2 ) )
      = ( ord_le6893508408891458716et_nat @ A7 @ ( sigma_sets_nat @ X2 ) ) ) ).

% sigma_sets_le_sets_iff
thf(fact_1178_countable__empty,axiom,
    counta7319604579010473777e_real @ bot_bot_set_real ).

% countable_empty
thf(fact_1179_countable__empty,axiom,
    counta1168086296615599829le_nat @ bot_bot_set_nat ).

% countable_empty
thf(fact_1180_countable__empty,axiom,
    counta5976203206615340371able_o @ bot_bot_set_o ).

% countable_empty
thf(fact_1181_countable__empty,axiom,
    counta8439243037236335165nnreal @ bot_bo4854962954004695426nnreal ).

% countable_empty
thf(fact_1182_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_measure_real,C3: set_set_real] :
      ( ( ord_le3558479182127378552t_real @ ( sigma_sets_real @ M ) @ C3 )
     => ( ( sigma_227922725797042522i_real @ ( sigma_space_real @ M ) @ C3 )
       => ( ord_le3558479182127378552t_real @ ( sigma_7195353284648819924s_real @ ( sigma_space_real @ M ) @ ( sigma_sets_real @ M ) ) @ C3 ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_1183_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_7234349610311085201nnreal,C3: set_se4580700918925141924nnreal] :
      ( ( ord_le3366939622266546180nnreal @ ( sigma_5465916536984168985nnreal @ M ) @ C3 )
     => ( ( sigma_114563780369365222nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ C3 )
       => ( ord_le3366939622266546180nnreal @ ( sigma_7808855514367478112nnreal @ ( sigma_3147302497200244656nnreal @ M ) @ ( sigma_5465916536984168985nnreal @ M ) ) @ C3 ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_1184_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_measure_o,C3: set_set_o] :
      ( ( ord_le4374716579403074808_set_o @ ( sigma_sets_o @ M ) @ C3 )
     => ( ( sigma_closed_cdi_o @ ( sigma_space_o @ M ) @ C3 )
       => ( ord_le4374716579403074808_set_o @ ( sigma_sigma_sets_o @ ( sigma_space_o @ M ) @ ( sigma_sets_o @ M ) ) @ C3 ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_1185_sets_Osigma__property__disjoint__lemma,axiom,
    ! [M: sigma_measure_nat,C3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( sigma_sets_nat @ M ) @ C3 )
     => ( ( sigma_closed_cdi_nat @ ( sigma_space_nat @ M ) @ C3 )
       => ( ord_le6893508408891458716et_nat @ ( sigma_sigma_sets_nat @ ( sigma_space_nat @ M ) @ ( sigma_sets_nat @ M ) ) @ C3 ) ) ) ).

% sets.sigma_property_disjoint_lemma
thf(fact_1186_subset__emptyI,axiom,
    ! [A: set_real] :
      ( ! [X: real] :
          ~ ( member_real @ X @ A )
     => ( ord_less_eq_set_real @ A @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_1187_subset__emptyI,axiom,
    ! [A: set_nat] :
      ( ! [X: nat] :
          ~ ( member_nat @ X @ A )
     => ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_1188_subset__emptyI,axiom,
    ! [A: set_o] :
      ( ! [X: $o] :
          ~ ( member_o @ X @ A )
     => ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_1189_subset__emptyI,axiom,
    ! [A: set_Ex3793607809372303086nnreal] :
      ( ! [X: extend8495563244428889912nnreal] :
          ~ ( member7908768830364227535nnreal @ X @ A )
     => ( ord_le6787938422905777998nnreal @ A @ bot_bo4854962954004695426nnreal ) ) ).

% subset_emptyI
thf(fact_1190_subset__emptyI,axiom,
    ! [A: set_set_nat] :
      ( ! [X: set_nat] :
          ~ ( member_set_nat @ X @ A )
     => ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_1191_subset__emptyI,axiom,
    ! [A: set_set_o] :
      ( ! [X: set_o] :
          ~ ( member_set_o @ X @ A )
     => ( ord_le4374716579403074808_set_o @ A @ bot_bot_set_set_o ) ) ).

% subset_emptyI
thf(fact_1192_subset__emptyI,axiom,
    ! [A: set_se4580700918925141924nnreal] :
      ( ! [X: set_Ex3793607809372303086nnreal] :
          ~ ( member603777416030116741nnreal @ X @ A )
     => ( ord_le3366939622266546180nnreal @ A @ bot_bo2988155216863113784nnreal ) ) ).

% subset_emptyI
thf(fact_1193_subset__emptyI,axiom,
    ! [A: set_set_real] :
      ( ! [X: set_real] :
          ~ ( member_set_real @ X @ A )
     => ( ord_le3558479182127378552t_real @ A @ bot_bot_set_set_real ) ) ).

% subset_emptyI
thf(fact_1194_subset__emptyI,axiom,
    ! [A: set_real_a] :
      ( ! [X: real > a] :
          ~ ( member_real_a @ X @ A )
     => ( ord_le5743406823621094409real_a @ A @ bot_bot_set_real_a ) ) ).

% subset_emptyI
thf(fact_1195_subset__emptyI,axiom,
    ! [A: set_o_real] :
      ( ! [X: $o > real] :
          ~ ( member_o_real @ X @ A )
     => ( ord_le3251842697534426805o_real @ A @ bot_bot_set_o_real ) ) ).

% subset_emptyI
thf(fact_1196_sets__measure__of__conv,axiom,
    ! [A: set_set_real,Omega2: set_real,Mu: set_real > extend8495563244428889912nnreal] :
      ( ( ( ord_le3558479182127378552t_real @ A @ ( pow_real @ Omega2 ) )
       => ( ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega2 @ A @ Mu ) )
          = ( sigma_7195353284648819924s_real @ Omega2 @ A ) ) )
      & ( ~ ( ord_le3558479182127378552t_real @ A @ ( pow_real @ Omega2 ) )
       => ( ( sigma_sets_real @ ( sigma_2693083824694760531f_real @ Omega2 @ A @ Mu ) )
          = ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ Omega2 @ bot_bot_set_set_real ) ) ) ) ) ).

% sets_measure_of_conv
thf(fact_1197_sets__measure__of__conv,axiom,
    ! [A: set_set_nat,Omega2: set_nat,Mu: set_nat > extend8495563244428889912nnreal] :
      ( ( ( ord_le6893508408891458716et_nat @ A @ ( pow_nat @ Omega2 ) )
       => ( ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega2 @ A @ Mu ) )
          = ( sigma_sigma_sets_nat @ Omega2 @ A ) ) )
      & ( ~ ( ord_le6893508408891458716et_nat @ A @ ( pow_nat @ Omega2 ) )
       => ( ( sigma_sets_nat @ ( sigma_measure_of_nat @ Omega2 @ A @ Mu ) )
          = ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ Omega2 @ bot_bot_set_set_nat ) ) ) ) ) ).

% sets_measure_of_conv
thf(fact_1198_sets__measure__of__conv,axiom,
    ! [A: set_set_o,Omega2: set_o,Mu: set_o > extend8495563244428889912nnreal] :
      ( ( ( ord_le4374716579403074808_set_o @ A @ ( pow_o @ Omega2 ) )
       => ( ( sigma_sets_o @ ( sigma_measure_of_o @ Omega2 @ A @ Mu ) )
          = ( sigma_sigma_sets_o @ Omega2 @ A ) ) )
      & ( ~ ( ord_le4374716579403074808_set_o @ A @ ( pow_o @ Omega2 ) )
       => ( ( sigma_sets_o @ ( sigma_measure_of_o @ Omega2 @ A @ Mu ) )
          = ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ Omega2 @ bot_bot_set_set_o ) ) ) ) ) ).

% sets_measure_of_conv
thf(fact_1199_sets__measure__of__conv,axiom,
    ! [A: set_se4580700918925141924nnreal,Omega2: set_Ex3793607809372303086nnreal,Mu: set_Ex3793607809372303086nnreal > extend8495563244428889912nnreal] :
      ( ( ( ord_le3366939622266546180nnreal @ A @ ( pow_Ex5372160365422184283nnreal @ Omega2 ) )
       => ( ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega2 @ A @ Mu ) )
          = ( sigma_7808855514367478112nnreal @ Omega2 @ A ) ) )
      & ( ~ ( ord_le3366939622266546180nnreal @ A @ ( pow_Ex5372160365422184283nnreal @ Omega2 ) )
       => ( ( sigma_5465916536984168985nnreal @ ( sigma_8167827323036178527nnreal @ Omega2 @ A @ Mu ) )
          = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ Omega2 @ bot_bo2988155216863113784nnreal ) ) ) ) ) ).

% sets_measure_of_conv
thf(fact_1200_sigma__sets__singleton,axiom,
    ! [X5: set_real,S3: set_real] :
      ( ( ord_less_eq_set_real @ X5 @ S3 )
     => ( ( sigma_7195353284648819924s_real @ S3 @ ( insert_set_real @ X5 @ bot_bot_set_set_real ) )
        = ( insert_set_real @ bot_bot_set_real @ ( insert_set_real @ X5 @ ( insert_set_real @ ( minus_minus_set_real @ S3 @ X5 ) @ ( insert_set_real @ S3 @ bot_bot_set_set_real ) ) ) ) ) ) ).

% sigma_sets_singleton
thf(fact_1201_sigma__sets__singleton,axiom,
    ! [X5: set_nat,S3: set_nat] :
      ( ( ord_less_eq_set_nat @ X5 @ S3 )
     => ( ( sigma_sigma_sets_nat @ S3 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) )
        = ( insert_set_nat @ bot_bot_set_nat @ ( insert_set_nat @ X5 @ ( insert_set_nat @ ( minus_minus_set_nat @ S3 @ X5 ) @ ( insert_set_nat @ S3 @ bot_bot_set_set_nat ) ) ) ) ) ) ).

% sigma_sets_singleton
thf(fact_1202_sigma__sets__singleton,axiom,
    ! [X5: set_o,S3: set_o] :
      ( ( ord_less_eq_set_o @ X5 @ S3 )
     => ( ( sigma_sigma_sets_o @ S3 @ ( insert_set_o @ X5 @ bot_bot_set_set_o ) )
        = ( insert_set_o @ bot_bot_set_o @ ( insert_set_o @ X5 @ ( insert_set_o @ ( minus_minus_set_o @ S3 @ X5 ) @ ( insert_set_o @ S3 @ bot_bot_set_set_o ) ) ) ) ) ) ).

% sigma_sets_singleton
thf(fact_1203_sigma__sets__singleton,axiom,
    ! [X5: set_Ex3793607809372303086nnreal,S3: set_Ex3793607809372303086nnreal] :
      ( ( ord_le6787938422905777998nnreal @ X5 @ S3 )
     => ( ( sigma_7808855514367478112nnreal @ S3 @ ( insert1343806209672318238nnreal @ X5 @ bot_bo2988155216863113784nnreal ) )
        = ( insert1343806209672318238nnreal @ bot_bo4854962954004695426nnreal @ ( insert1343806209672318238nnreal @ X5 @ ( insert1343806209672318238nnreal @ ( minus_104578273773384135nnreal @ S3 @ X5 ) @ ( insert1343806209672318238nnreal @ S3 @ bot_bo2988155216863113784nnreal ) ) ) ) ) ) ).

% sigma_sets_singleton
thf(fact_1204_standard__borel__space__UNIV__axioms__def,axiom,
    ( standa1498722272452280784s_real
    = ( ^ [M2: sigma_measure_real] :
          ( ( sigma_space_real @ M2 )
          = top_top_set_real ) ) ) ).

% standard_borel_space_UNIV_axioms_def
thf(fact_1205_standard__borel__space__UNIV__axioms__def,axiom,
    ( standa4575222554423029108ioms_o
    = ( ^ [M2: sigma_measure_o] :
          ( ( sigma_space_o @ M2 )
          = top_top_set_o ) ) ) ).

% standard_borel_space_UNIV_axioms_def
thf(fact_1206_standard__borel__space__UNIV__axioms__def,axiom,
    ( standa4898135366436483316ms_nat
    = ( ^ [M2: sigma_measure_nat] :
          ( ( sigma_space_nat @ M2 )
          = top_top_set_nat ) ) ) ).

% standard_borel_space_UNIV_axioms_def
thf(fact_1207_standard__borel__space__UNIV__axioms__def,axiom,
    ( standa602082540683807836nnreal
    = ( ^ [M2: sigma_7234349610311085201nnreal] :
          ( ( sigma_3147302497200244656nnreal @ M2 )
          = top_to7994903218803871134nnreal ) ) ) ).

% standard_borel_space_UNIV_axioms_def
thf(fact_1208_standard__borel__space__UNIV__axioms_Ointro,axiom,
    ! [M: sigma_measure_real] :
      ( ( ( sigma_space_real @ M )
        = top_top_set_real )
     => ( standa1498722272452280784s_real @ M ) ) ).

% standard_borel_space_UNIV_axioms.intro
thf(fact_1209_standard__borel__space__UNIV__axioms_Ointro,axiom,
    ! [M: sigma_measure_o] :
      ( ( ( sigma_space_o @ M )
        = top_top_set_o )
     => ( standa4575222554423029108ioms_o @ M ) ) ).

% standard_borel_space_UNIV_axioms.intro
thf(fact_1210_standard__borel__space__UNIV__axioms_Ointro,axiom,
    ! [M: sigma_measure_nat] :
      ( ( ( sigma_space_nat @ M )
        = top_top_set_nat )
     => ( standa4898135366436483316ms_nat @ M ) ) ).

% standard_borel_space_UNIV_axioms.intro
thf(fact_1211_standard__borel__space__UNIV__axioms_Ointro,axiom,
    ! [M: sigma_7234349610311085201nnreal] :
      ( ( ( sigma_3147302497200244656nnreal @ M )
        = top_to7994903218803871134nnreal )
     => ( standa602082540683807836nnreal @ M ) ) ).

% standard_borel_space_UNIV_axioms.intro
thf(fact_1212_DiffI,axiom,
    ! [C: real > a,A: set_real_a,B4: set_real_a] :
      ( ( member_real_a @ C @ A )
     => ( ~ ( member_real_a @ C @ B4 )
       => ( member_real_a @ C @ ( minus_6532636778494125008real_a @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_1213_DiffI,axiom,
    ! [C: $o > real,A: set_o_real,B4: set_o_real] :
      ( ( member_o_real @ C @ A )
     => ( ~ ( member_o_real @ C @ B4 )
       => ( member_o_real @ C @ ( minus_2870878895999678972o_real @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_1214_DiffI,axiom,
    ! [C: nat > real,A: set_nat_real,B4: set_nat_real] :
      ( ( member_nat_real @ C @ A )
     => ( ~ ( member_nat_real @ C @ B4 )
       => ( member_nat_real @ C @ ( minus_3492551254948764970t_real @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_1215_DiffI,axiom,
    ! [C: c > d,A: set_c_d,B4: set_c_d] :
      ( ( member_c_d @ C @ A )
     => ( ~ ( member_c_d @ C @ B4 )
       => ( member_c_d @ C @ ( minus_minus_set_c_d @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_1216_DiffI,axiom,
    ! [C: a > b,A: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ A )
     => ( ~ ( member_a_b @ C @ B4 )
       => ( member_a_b @ C @ ( minus_minus_set_a_b @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_1217_DiffI,axiom,
    ! [C: set_nat,A: set_set_nat,B4: set_set_nat] :
      ( ( member_set_nat @ C @ A )
     => ( ~ ( member_set_nat @ C @ B4 )
       => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_1218_DiffI,axiom,
    ! [C: set_o,A: set_set_o,B4: set_set_o] :
      ( ( member_set_o @ C @ A )
     => ( ~ ( member_set_o @ C @ B4 )
       => ( member_set_o @ C @ ( minus_4899875422681990719_set_o @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_1219_DiffI,axiom,
    ! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal] :
      ( ( member603777416030116741nnreal @ C @ A )
     => ( ~ ( member603777416030116741nnreal @ C @ B4 )
       => ( member603777416030116741nnreal @ C @ ( minus_5908140721592501885nnreal @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_1220_DiffI,axiom,
    ! [C: set_real,A: set_set_real,B4: set_set_real] :
      ( ( member_set_real @ C @ A )
     => ( ~ ( member_set_real @ C @ B4 )
       => ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_1221_Diff__iff,axiom,
    ! [C: c > d,A: set_c_d,B4: set_c_d] :
      ( ( member_c_d @ C @ ( minus_minus_set_c_d @ A @ B4 ) )
      = ( ( member_c_d @ C @ A )
        & ~ ( member_c_d @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1222_Diff__iff,axiom,
    ! [C: a > b,A: set_a_b,B4: set_a_b] :
      ( ( member_a_b @ C @ ( minus_minus_set_a_b @ A @ B4 ) )
      = ( ( member_a_b @ C @ A )
        & ~ ( member_a_b @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1223_Diff__iff,axiom,
    ! [C: set_nat,A: set_set_nat,B4: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B4 ) )
      = ( ( member_set_nat @ C @ A )
        & ~ ( member_set_nat @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1224_Diff__iff,axiom,
    ! [C: set_o,A: set_set_o,B4: set_set_o] :
      ( ( member_set_o @ C @ ( minus_4899875422681990719_set_o @ A @ B4 ) )
      = ( ( member_set_o @ C @ A )
        & ~ ( member_set_o @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1225_Diff__iff,axiom,
    ! [C: set_Ex3793607809372303086nnreal,A: set_se4580700918925141924nnreal,B4: set_se4580700918925141924nnreal] :
      ( ( member603777416030116741nnreal @ C @ ( minus_5908140721592501885nnreal @ A @ B4 ) )
      = ( ( member603777416030116741nnreal @ C @ A )
        & ~ ( member603777416030116741nnreal @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1226_Diff__iff,axiom,
    ! [C: set_real,A: set_set_real,B4: set_set_real] :
      ( ( member_set_real @ C @ ( minus_5467046032205032049t_real @ A @ B4 ) )
      = ( ( member_set_real @ C @ A )
        & ~ ( member_set_real @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1227_ennreal_Oexist__fg,axiom,
    ? [X: extend8495563244428889912nnreal > real] :
      ( ( member2874014351250825754l_real @ X @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > extend8495563244428889912nnreal] :
          ( ( member2919562650594848410nnreal @ Xa @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) )
          & ! [Xb: extend8495563244428889912nnreal] :
              ( ( member7908768830364227535nnreal @ Xb @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
             => ( ( comp_r6281409797179841921nnreal @ Xa @ X @ Xb )
                = Xb ) ) ) ) ).

% ennreal.exist_fg
thf(fact_1228_bool_Oexist__fg,axiom,
    ? [X: $o > real] :
      ( ( member_o_real @ X @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > $o] :
          ( ( member_real_o @ Xa @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ borel_5500255247093592246orel_o ) )
          & ! [Xb: $o] :
              ( ( member_o @ Xb @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
             => ( ( comp_real_o_o @ Xa @ X @ Xb )
                = Xb ) ) ) ) ).

% bool.exist_fg
thf(fact_1229_nat_Oexist__fg,axiom,
    ? [X: nat > real] :
      ( ( member_nat_real @ X @ ( sigma_1747752005702207822t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > nat] :
          ( ( member_real_nat @ Xa @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
          & ! [Xb: nat] :
              ( ( member_nat @ Xb @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
             => ( ( comp_real_nat_nat @ Xa @ X @ Xb )
                = Xb ) ) ) ) ).

% nat.exist_fg
thf(fact_1230_real_Oexist__fg,axiom,
    ? [X: real > real] :
      ( ( member_real_real @ X @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
      & ? [Xa: real > real] :
          ( ( member_real_real @ Xa @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) )
          & ! [Xb: real] :
              ( ( member_real @ Xb @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
             => ( ( comp_real_real_real @ Xa @ X @ Xb )
                = Xb ) ) ) ) ).

% real.exist_fg
thf(fact_1231_nat_Osingleton__sets,axiom,
    ! [X2: nat] :
      ( ( member_nat @ X2 @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
     => ( member_set_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) @ ( sigma_sets_nat @ borel_8449730974584783410el_nat ) ) ) ).

% nat.singleton_sets
thf(fact_1232_bool_Osingleton__sets,axiom,
    ! [X2: $o] :
      ( ( member_o @ X2 @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
     => ( member_set_o @ ( insert_o @ X2 @ bot_bot_set_o ) @ ( sigma_sets_o @ borel_5500255247093592246orel_o ) ) ) ).

% bool.singleton_sets
thf(fact_1233_ennreal_Osingleton__sets,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ X2 @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
     => ( member603777416030116741nnreal @ ( insert7407984058720857448nnreal @ X2 @ bot_bo4854962954004695426nnreal ) @ ( sigma_5465916536984168985nnreal @ borel_6524799422816628122nnreal ) ) ) ).

% ennreal.singleton_sets
thf(fact_1234_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_1235_real_Ostandard__borel__space__UNIV__axioms,axiom,
    standa1306199911732814765V_real @ borel_5078946678739801102l_real ).

% real.standard_borel_space_UNIV_axioms
thf(fact_1236_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_1237_r01__to__r01__r01__fst__measurable,axiom,
    member_real_real @ r01_to_r01_r01_fst @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% r01_to_r01_r01_fst_measurable
thf(fact_1238_r01__to__r01__r01__snd__measurable,axiom,
    member_real_real @ r01_to_r01_r01_snd @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% r01_to_r01_r01_snd_measurable
thf(fact_1239_real_Ostandard__borel__axioms,axiom,
    standard_borel_real @ borel_5078946678739801102l_real ).

% real.standard_borel_axioms
thf(fact_1240_real_Ostandard__borel__sets,axiom,
    ! [Y4: sigma_measure_real] :
      ( ( ( sigma_sets_real @ borel_5078946678739801102l_real )
        = ( sigma_sets_real @ Y4 ) )
     => ( standard_borel_real @ Y4 ) ) ).

% real.standard_borel_sets
thf(fact_1241_uncountable__UNIV__real,axiom,
    ~ ( counta7319604579010473777e_real @ top_top_set_real ) ).

% uncountable_UNIV_real
thf(fact_1242_measurable__separate,axiom,
    ! [P: real > nat,I2: nat] :
      ( ( member_real_nat @ P @ ( sigma_6315060578831106510al_nat @ borel_5078946678739801102l_real @ borel_8449730974584783410el_nat ) )
     => ( member_set_real @ ( vimage_real_nat @ P @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ) ) ).

% measurable_separate
thf(fact_1243_separate__measurable,axiom,
    ! [P: real > nat] :
      ( ! [I3: nat] : ( member_set_real @ ( vimage_real_nat @ P @ ( insert_nat @ I3 @ 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_1244_real_Og__meas,axiom,
    member_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% real.g_meas
thf(fact_1245_ennreal_Og__meas,axiom,
    member2919562650594848410nnreal @ ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal ) @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).

% ennreal.g_meas
thf(fact_1246_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_1247_bool_Og__meas,axiom,
    member_real_o @ ( standard_g_o @ borel_5500255247093592246orel_o ) @ ( sigma_3939073009482781210real_o @ borel_5078946678739801102l_real @ borel_5500255247093592246orel_o ) ).

% bool.g_meas
thf(fact_1248_bool_Ogf__comp__id_I1_J,axiom,
    ! [X2: $o] :
      ( ( member_o @ X2 @ ( sigma_space_o @ borel_5500255247093592246orel_o ) )
     => ( ( comp_real_o_o @ ( standard_g_o @ borel_5500255247093592246orel_o ) @ ( standard_f_o @ borel_5500255247093592246orel_o ) @ X2 )
        = X2 ) ) ).

% bool.gf_comp_id(1)
thf(fact_1249_nat_Ogf__comp__id_I1_J,axiom,
    ! [X2: nat] :
      ( ( member_nat @ X2 @ ( sigma_space_nat @ borel_8449730974584783410el_nat ) )
     => ( ( comp_real_nat_nat @ ( standard_g_nat @ borel_8449730974584783410el_nat ) @ ( standard_f_nat @ borel_8449730974584783410el_nat ) @ X2 )
        = X2 ) ) ).

% nat.gf_comp_id(1)
thf(fact_1250_real_Ogf__comp__id_H_I2_J,axiom,
    ! [X2: real] :
      ( ( standard_g_real @ borel_5078946678739801102l_real @ ( standard_f_real @ borel_5078946678739801102l_real @ X2 ) )
      = X2 ) ).

% real.gf_comp_id'(2)
thf(fact_1251_real_Ogf__comp__id_I2_J,axiom,
    ! [X2: real] :
      ( ( member_real @ X2 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( ( standard_g_real @ borel_5078946678739801102l_real @ ( standard_f_real @ borel_5078946678739801102l_real @ X2 ) )
        = X2 ) ) ).

% real.gf_comp_id(2)
thf(fact_1252_real_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_real_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( standard_f_real @ borel_5078946678739801102l_real ) )
    = id_real ) ).

% real.gf_comp_id'(1)
thf(fact_1253_ennreal_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_r6281409797179841921nnreal @ ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal ) @ ( standa4501783974915749827nnreal @ borel_6524799422816628122nnreal ) )
    = id_Ext8331313139072774535nnreal ) ).

% ennreal.gf_comp_id'(1)
thf(fact_1254_real_Ogf__comp__id_I1_J,axiom,
    ! [X2: real] :
      ( ( member_real @ X2 @ ( sigma_space_real @ borel_5078946678739801102l_real ) )
     => ( ( comp_real_real_real @ ( standard_g_real @ borel_5078946678739801102l_real ) @ ( standard_f_real @ borel_5078946678739801102l_real ) @ X2 )
        = X2 ) ) ).

% real.gf_comp_id(1)
thf(fact_1255_ennreal_Ogf__comp__id_I1_J,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( member7908768830364227535nnreal @ X2 @ ( sigma_3147302497200244656nnreal @ borel_6524799422816628122nnreal ) )
     => ( ( comp_r6281409797179841921nnreal @ ( standa1398259892199664580nnreal @ borel_6524799422816628122nnreal ) @ ( standa4501783974915749827nnreal @ borel_6524799422816628122nnreal ) @ X2 )
        = X2 ) ) ).

% ennreal.gf_comp_id(1)
thf(fact_1256_nat_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_real_nat_nat @ ( standard_g_nat @ borel_8449730974584783410el_nat ) @ ( standard_f_nat @ borel_8449730974584783410el_nat ) )
    = id_nat ) ).

% nat.gf_comp_id'(1)
thf(fact_1257_bool_Ogf__comp__id_H_I1_J,axiom,
    ( ( comp_real_o_o @ ( standard_g_o @ borel_5500255247093592246orel_o ) @ ( standard_f_o @ borel_5500255247093592246orel_o ) )
    = id_o ) ).

% bool.gf_comp_id'(1)
thf(fact_1258_real_Of__meas,axiom,
    member_real_real @ ( standard_f_real @ borel_5078946678739801102l_real ) @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ).

% real.f_meas
thf(fact_1259_ennreal_Of__meas,axiom,
    member2874014351250825754l_real @ ( standa4501783974915749827nnreal @ borel_6524799422816628122nnreal ) @ ( sigma_7049758200512112822l_real @ borel_6524799422816628122nnreal @ borel_5078946678739801102l_real ) ).

% ennreal.f_meas
thf(fact_1260_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_1261_bool_Of__meas,axiom,
    member_o_real @ ( standard_f_o @ borel_5500255247093592246orel_o ) @ ( sigma_2430008634441611636o_real @ borel_5500255247093592246orel_o @ borel_5078946678739801102l_real ) ).

% bool.f_meas
thf(fact_1262_open__minus__countable,axiom,
    ! [A: set_real,S3: set_real] :
      ( ( counta7319604579010473777e_real @ A )
     => ( ( S3 != bot_bot_set_real )
       => ( ( topolo4860482606490270245n_real @ S3 )
         => ? [X: real] :
              ( ( member_real @ X @ S3 )
              & ~ ( member_real @ X @ A ) ) ) ) ) ).

% open_minus_countable
thf(fact_1263_nat__real_Opair__standard__borel__space__UNIV__axioms,axiom,
    pair_s5107880421860391064t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ).

% nat_real.pair_standard_borel_space_UNIV_axioms
thf(fact_1264_nat__real_Opair__standard__borel__axioms,axiom,
    pair_s8264832550775477520t_real @ borel_8449730974584783410el_nat @ borel_5078946678739801102l_real ).

% nat_real.pair_standard_borel_axioms
thf(fact_1265_ennreal__add__bot,axiom,
    ! [X2: extend8495563244428889912nnreal] :
      ( ( plus_p1859984266308609217nnreal @ bot_bo841427958541957580nnreal @ X2 )
      = X2 ) ).

% ennreal_add_bot
thf(fact_1266_bot__ennreal,axiom,
    bot_bo841427958541957580nnreal = zero_z7100319975126383169nnreal ).

% bot_ennreal
thf(fact_1267_f01__borel__measurable,axiom,
    ! [F2: real > real] :
      ( ( member_set_real @ ( vimage_real_real @ F2 @ ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
     => ( ( member_set_real @ ( vimage_real_real @ F2 @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) @ ( sigma_sets_real @ borel_5078946678739801102l_real ) )
       => ( ! [R3: real] : ( member_real @ ( F2 @ R3 ) @ ( insert_real @ zero_zero_real @ ( insert_real @ one_one_real @ bot_bot_set_real ) ) )
         => ( member_real_real @ F2 @ ( sigma_5267869275261027754l_real @ borel_5078946678739801102l_real @ borel_5078946678739801102l_real ) ) ) ) ) ).

% f01_borel_measurable
thf(fact_1268_r01__to__r01__r01__fst_Hin01,axiom,
    ! [R2: real,N2: nat] : ( member_nat @ ( r01_to_r01_r01_fst2 @ R2 @ N2 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% r01_to_r01_r01_fst'in01
thf(fact_1269_r01__to__r01__r01__snd_Hin01,axiom,
    ! [R2: real,N2: nat] : ( member_nat @ ( r01_to_r01_r01_snd2 @ R2 @ N2 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% r01_to_r01_r01_snd'in01
thf(fact_1270_r01__r01__to__r01_Hin01,axiom,
    ! [Rs: produc2422161461964618553l_real,N2: nat] : ( member_nat @ ( r01_r01_to_r01 @ Rs @ N2 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% r01_r01_to_r01'in01
thf(fact_1271_biexp01__well__formedE,axiom,
    ! [A2: nat > nat] :
      ( ( biexp01_well_formed @ A2 )
     => ( ! [N3: nat] : ( member_nat @ ( A2 @ N3 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
        & ! [N3: nat] :
          ? [M4: nat] :
            ( ( ord_less_eq_nat @ N3 @ M4 )
            & ( ( A2 @ M4 )
              = zero_zero_nat ) ) ) ) ).

% biexp01_well_formedE
thf(fact_1272_biexp01__well__formed__def,axiom,
    ( biexp01_well_formed
    = ( ^ [A6: nat > nat] :
          ( ! [N4: nat] : ( member_nat @ ( A6 @ N4 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
          & ! [N4: nat] :
            ? [M5: nat] :
              ( ( ord_less_eq_nat @ N4 @ M5 )
              & ( ( A6 @ M5 )
                = zero_zero_nat ) ) ) ) ) ).

% biexp01_well_formed_def
thf(fact_1273_biexp01__well__formedI,axiom,
    ! [A2: nat > nat] :
      ( ! [N5: nat] : ( member_nat @ ( A2 @ N5 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) )
     => ( ! [N5: nat] :
          ? [M6: nat] :
            ( ( ord_less_eq_nat @ N5 @ M6 )
            & ( ( A2 @ M6 )
              = zero_zero_nat ) )
       => ( biexp01_well_formed @ A2 ) ) ) ).

% biexp01_well_formedI
thf(fact_1274_real01__binary__expansion_H__0or1,axiom,
    ! [R2: real,N2: nat] : ( member_nat @ ( r01_binary_expansion @ R2 @ N2 ) @ ( insert_nat @ zero_zero_nat @ ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ) ).

% real01_binary_expansion'_0or1
thf(fact_1275_measurable__ennreal,axiom,
    member2919562650594848410nnreal @ extend7643940197134561352nnreal @ ( sigma_9017504469962657078nnreal @ borel_5078946678739801102l_real @ borel_6524799422816628122nnreal ) ).

% measurable_ennreal
thf(fact_1276_set__borel__integrable__singleton,axiom,
    ! [X2: real,F2: real > real] : ( set_se5970144800844511125l_real @ lebesgue_lborel_real @ ( insert_real @ X2 @ bot_bot_set_real ) @ F2 ) ).

% set_borel_integrable_singleton

% Helper facts (5)
thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_T,axiom,
    ! [X2: sum_sum_a_c,Y: sum_sum_a_c] :
      ( ( if_Sum_sum_a_c @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mtf__c_J_T,axiom,
    ! [X2: sum_sum_a_c,Y: sum_sum_a_c] :
      ( ( if_Sum_sum_a_c @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_1_If_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_T,axiom,
    ! [X2: sum_sum_b_d,Y: sum_sum_b_d] :
      ( ( if_Sum_sum_b_d @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__b_Mtf__d_J_T,axiom,
    ! [X2: sum_sum_b_d,Y: sum_sum_b_d] :
      ( ( if_Sum_sum_b_d @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    member_set_real @ s @ ( sigma_sets_real @ borel_5078946678739801102l_real ) ).

%------------------------------------------------------------------------------